Teoría Total simplificada, Revista Chilena de Ingeniería, Vol. 16, Nº1 [PDF]

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(/doc/8947101/physISSN 0718-3291 Versión impresa 520b--ISSN 0718-3305 Versión en línea INGENIARE electromagneticRevista Chilena de Ingeniería theory) INGENIARE Revista Chilena de Ingeniería ISSN 0718-3291 Printed version ISSN 0718-3305 On line version Volume 16, N° 1 January - March 2008 I (/doc/8057882/electromagneticN fields) D E Í N D I C E X EDITORIAL EDITORIAL MODERN ELECTROMAGNETIC ENGINEERING Carlos Villarroel González 4 CHIRAL ELECTRODYNAMIC: CONNECTION FOR THE UNIFICATION OF ELECTROMAGNETISM AND GRAVITATION (/doc/2910966/3.4- (/doc/8758516/) H. Torres-Silva solve-systems-ofNEW INTERPRETATION OF THE ATOMIC SPECTRA OF THE HYDROGEN ATOM: A MIXED MECHANISM linear-equationsOF CLASSICAL LC CIRCUITS AND QUANTUM WAVE-PARTICLE DUALITY H. Torres-Silva in-three-variable) MAXWELL’S THEORY WITH CHIRAL CURRENTS H. Torres-Silva CHIRAL FIELD IDEAS FOR A THEORY OF MATTER H. Torres-Silva THE CLOSE RELATION BETWEEN THE MAXWELL SYSTEM AND THE DIRAC EQUATION WHEN THE ELECTRIC FIELD IS PARALLEL TO THE MAGNETIC FIELD H. Torres-Silva DIRAC MATRICES IN CHIRAL REPRESENTATION AND THE CONNECTION WITH THE ELECTRIC FIELD PARALLEL TO THE MAGNETIC FIELD H. Torres-Silva 6 ELECTRODINÁMICA QUIRAL: ESLABÓN PARA LA UNIFICACIÓN DEL ELECTROMAGNETISMO Y LA GRAVITACIÓN H. Torres-Silva NUEVA INTERPRETACIÓN DEL ESPECTRO ATÓMICO DEL ÁTOMO DE HIDRÓGENO: UN MECANISMO MIXTO DE CIRCUITOS LC Y LA DUALIDAD ONDA CUÁNTICA-PARTÍCULA 2 6 24 H. Torres-Silva 24 31 TEORÍA DE MAXWELL CON CORRIENTES QUIRALES H. Torres-Silva 31 36 IDEAS DE CAMPO QUIRAL PARA UNA TEORÍA DE LA MATERIA H. Torres-Silva 36 43 LA ESTRECHA RELACIÓN ENTRE EL SISTEMA DE MAXWELL Y LA ECUACIÓN DE DIRAC, CUANDO EL CAMPO ELÉCTRICO ES PARALELO AL CAMPO MAGNÉTICO H. Torres-Silva 43 48 MATRICES DE DIRAC EN REPRESENTACIÓN QUIRAL Y LA CONEXIÓN CON EL CAMPO ELÉCTRICO PARALELO AL CAMPO MAGNÉTICO H. Torres-Silva 48 ECUACIONES DE MAXWELL PARA UNA FUNCIONAL DE LAGRANGE GENERALIZADA H. Torres-Silva 53 CALIBRE QUIRAL PARA AUMENTAR EL COEFICIENTE DE RENDIMIENTO DE MOTORES MAGNÉTICOS H. Torres-Silva 60 ELECTRODINÁMICA DE PODOLSKY BAJO UN ENFOQUE QUIRAL H. Torres-Silva 65 ESPÍN Y RELATIVIDAD: UN MODELO SEMICLÁSICO PARA EL ESPÍN DEL ELECTRÓN H. Torres-Silva 72 TEORÍA EXTENDIDA DE ONDAS DE EINSTEIN EN LA PRESENCIA DE TENSIONES EN EL ESPACIO-TIEMPO H. Torres-Silva 78 MAXWELL EQUATIONS FOR A GENERALISED LAGRANGIAN FUNCTIONAL H. Torres-Silva 53 ASYMMETRICAL CHIRAL GAUGING TO INCREASE THE COEFFICIENT OF PERFORMANCE OF MAGNETIC MOTORS H. Torres-Silva 60 PODOLSKY'S ELECTRODYNAMICS UNDER A CHIRAL APPROACH H. Torres-Silva 65 SPIN AND RELATIVITY: A SEMICLASSICAL MODEL FOR ELECTRON SPIN H. Torres-Silva 72 EXTENDED EINSTEIN`S THEORY OF WAVES IN THE PRESENCE OF SPACE-TIME TENSIONS H. Torres-Silva 78 EINSTEIN EQUATIONS FOR TETRAD FIELDS H. Torres-Silva 85 A METRIC FOR A CHIRAL POTENTIAL FIELD H. Torres-Silva 91 CHIRAL UNIVERSES AND QUANTUM EFFECTS PRODUCED BY ELECTROMAGNETIC FIELDS H. Torres-Silva 99 CHIRAL WAVES IN A METAMATERIAL MEDIUM H. Torres-Silva LA INGENIERÍA ELECTROMAGNÉTICA MODERNA Carlos Villarroel González ARTÍCULOS ARTICLES A NEW RELATIVISTIC FIELD THEORY OF THE ELECTRON H. Torres-Silva VO L U M E N 1 6 - N º 1 E N E RO - M A R Z O 2 0 0 8 ECUACIONES DE EINSTEIN PARA CAMPOS TETRADOS H. Torres-Silva 85 UNA MÉTRICA PARA UN CAMPO POTENCIAL QUIRAL H. Torres-Silva 91 UNIVERSOS QUIRALES Y EFECTOS CUÁNTICOS PRODUCIDOS POR CAMPOS ELECTROMAGNÉTICOS H. Torres-Silva 99 UNA NUEVA TEORÍA RELATIVÍSTICA DE CAMPO PARA EL ELECTRÓN 111 H. Torres-Silva 111 119 ONDAS QUIRALES EN UN MEDIO METAMATERIAL H. Torres-Silva 119 DE TARAPACÁ ARICA-CHILE portada impresion.indd 6 24/3/08 11:14:21 DIRIGIR CORRESPONDENCIA A: Comité Editor Ingeniare. Revista chilena de ingeniería Universidad de Tarapacá Casilla 6-D, Arica - Chile e-mail: [email protected] INFORMACIÓN GENERAL Ingeniare. Revista chilena de ingeniería, edita tres números al año (cuatrimestral), publica estudios originales e inéditos de académicos y profesionales pertenecientes a entidades públicas y privadas, chilenas o extranjeras, que deseen difundir sus experiencias sobre ciencias de la ingeniería, tecnología y disciplinas afines. La responsabilidad sobre el contenido de los artículos es exclusivamente de los autores. Procedimientos y precios para la adquisición de la versión impresa: Suscripciones: Chile: $15.000 por año. Extranjero: US$ 25 por año. Incluyendo franqueo por correo ordinario. Todas las suscripciones y cambios de dirección se deben enviar a la dirección señalada. La versión online de la revista es preparada con metodología SciELO. Todos los materiales publicados en ese sitio están disponibles en forma gratuita. COMMUNICATIONS AND SUBSCRIPTIONS SHOULD BE ADDRESSED TO: Editor Committee Ingeniare. Revista chilena de ingeniería Universidad de Tarapacá Casilla 6-D, Arica - Chile e-mail: [email protected] GENERAL INFORMATION Ingeniare. Revista chilena de ingeniería is published periodically, is printed in three issues per volume annually, publishing original articles by professional and academic authors belonging to public or private organizations, from Chile and the rest of the world, with the purpose of disseminating their experiences in engineering science and technology. Subscription to the printed version of journal can be purchased as follows: Chile: Annual subscription: $15.000. Elsewhere: Annual subscription: US$ 25. Including ordinary mail post SciELO online version of the journal is based upon the SciELO methodology. All online material published by this journal’s site is available free of charge. TAPA 2 3 16-1.indd 2 24/3/08 11:10:54 TAPA 2 3 16-1.indd 3 24/3/08 11:10:54 INGENIARE. REVISTA CHILENA DE INGENIERÍA1 VOLUMEN 16 Nº 1, ENERO – MARZO 2008 VOLUME 16 Nº 1, JANUARY – MARCH 2008 2 COMITÉ EDITOR ASESOR ADVISORY EDITOR COMMITTEE EDITOR Carlos Villarroel González Universidad de Tarapacá Enrique Fuentes Heinrich (President) Raúl Borjas Montero Jaime Gómez Douzet Ingrid Guillén Figueroa Ernesto Ponce López Héctor Valdés González COEDITOR Adelheid Mahla Álvarez Universidad de Santiago de Chile PRODUCCIÓN EDITORIAL EDITORIAL PRODUCTION Carolina Cautín Barría COMITÉ EDITOR EDITOR COMMITTEE Yurilev Chalco Cano Universidad de Tarapacá, Chile Eva María Navarro Instituto Mexicano del Petróleo, México Luis Cifuentes Seves Universidad de Chile, Chile Liliana Pedraja Rejas Universidad de Tarapacá, Chile Gerardo Espinosa Pérez Universidad Autónoma de México, México Manuel Recuero López Universidad Politécnica de Madrid, España Sylviane Gentil Institut National Polytechnique de Grenoble, France Miguel Ríos Ojeda Pontificia Universidad Católica de Chile, Chile Hugo Hernández Figueroa Universidade de Campinas, Brasil Marko Rojas Medar Universidade de Campinas, Brasil Cynthia Junqueira General-Command of Aerospace Technology, Institute of Aeronautics and Space, Brazil Heriberto Román Flores Universidad de Tarapacá, Chile Andre Koch Torres Assis Universidade de Campinas, Brasil Mario Letelier Sotomayor Universidad de Santiago de Chile, Chile Orestes Llanes Santiago Instituto Superior Politécnico José Antonio Echeverría, Cuba Sebastián Lorca Pizarro Universidad de Tarapacá, Chile 1 Fideromo Saavedra Guzmán Universidad de Santiago de Chile, Chile Osvaldo Saavedra Mendez Universidade Federal do Maranhão, Brasil Mario Salgado Brocal Universidad Técnica Federico Santa María, Chile Salah S. A. Obayya Brunel University, United Kingdom Linda Madsen European Journal of Engineering Education, Denmark Héctor Torres Silva Universidad de Tarapacá, Chile Adelheid Mahla Álvarez Universidad de Santiago de Chile, Chile Miguel Villablanca Martínez Universidad de Santiago de Chile, Chile João Marcos Romano Universidade de Campinas, Brasil Andrés Weintraub Pohorille Universidad de Chile, Chile Antonio Martins Soares Universidade de Brasilia, Brasil Juan Zolezzi Cid Universidad de Santiago de Chile, Chile Nelson Moraga Benavides Universidad de Santiago de Chile, Chile Ernesto Zumelzu D. Universidad Austral de Chile, Chile Indizada en Risk Abstract, Safety Science & Risk Abstract, Environmental Sciences & Pollution Management Abstract, Applied Science & Technology Index, Latindex, RedALyC, The Serials Directory - Ebsco. Esta obra está bajo una licencia de Creative Commons Chile 2.0. Electrónicamente se encuentra en Scientific Library Online (www.scielo.cl) e incluida en el Sistema Regional de Información en Línea para Revistas Científicas de América Latina, el Caribe, España y Portugal (http://www.latindex.unam.mx); Revistas especializadas Al Día - Universidad de Chile (http://www.al-dia.cl); ProQuest Information and Learning ProQuest (www.proquest.com); EBSCO Information Services (http://www.ebsco.com) Ingeniare. Revista chilena de ingeniería, vol. 16 15 Nº 1, 3, 2008, 2007 pp. 2-3 EDITORIAL LA INGENIERÍA ELECTROMAGNÉTICA MODERNA La ingeniería electromagnética es una rama de la física aplicada, con tal velocidad de desarrollo que en un futuro inmediato los ingenieros electromagnéticos serán indispensables en una nueva e importante área emergente: la ahora denominada Estructura de Onda de la Materia (Wave Structure of Matter, WSM). La razón principal es la capacidad de penetración de la tecnología electromagnética, donde ingenieros especializados serán necesarios para el diseño de sistemas relacionados con la tecnología WSM. Por ejemplo, en muy altas frecuencias, en sistemas tales como, redes inalámbricas de comunicaciones, chips de computadores, redes ópticas, antenas, y en frecuencias muy bajas, en extracción de energía, en dispositivos de almacenamiento de energía y en sistemas relacionados con un Enfoque Métrico de la Ingeniería (Metric Engineering Approach, MEA), en propulsión con campos electromagnéticos. La dificultad y la complejidad de las leyes que gobiernan el diseño de sistemas relacionados con la ingeniería electromagnética indican que la teoría y el análisis del electromagnetismo es una ciencia en continua evolución y es un área activa de investigación que ha atraído el interés de matemáticos, científicos de la computación y de los ingenieros. Sin embargo, un buen entendimiento del análisis electromagnético moderno requiere de un profundo conocimiento de la física, habilidad para el análisis matemático y del conocimiento de los algoritmos numéricos utilizados en computación. Aunque algunas universidades enfatizan en el análisis computacional del electromagnetismo, tenemos que ser conscientes de que un estudiante de ingeniería electromagnética debe entender los conceptos de física involucrados y desarrollar intuición y entendimiento de los problemas a resolver. Estas habilidades son importantes tanto para el análisis como para el diseño. Por lo tanto, es importante formar a los estudiantes de postgrado en los métodos modernos del análisis electromagnético, y en las nuevas teorías tales como: metamateriales, electrodinámica quiral y electrogravedad. Por ejemplo, el análisis electromagnético quiral debe incluir, entre otros, los conceptos de ondas polarizadas circularmente, ondas superficiales, ondas que se arrastran (creeping waves), ondas laterales, modos guiados, modos evanescentes, modos radiantes y los modos filtrados (leaky modes). Todo esto en la física de altas frecuencias donde la dualidad onda/partícula emerge como un nuevo enfoque físico de las interacciones electromagnéticas de la WSM. Recientemente se han producido avances en la WSM, por ejemplo, en microcircuitos industriales y en electrodinámica, donde existen corrientes de lazos cerrados de ondas de electrones, siendo el electrón no una partícula puntual sino una estructura de onda. Aquí la mayoría de las aplicaciones, como ser nanotubos quirales y sustratos de metamateriales para uso en microcircuitos, requiere de la comprensión del comportamiento de la materia en “dimensiones muy pequeñas”, donde la aproximación de la partícula falla y la WSM se hace necesaria para entender qué ocurre cuando interactúan diferentes sustratos, a nivel químico, eléctrico o biológico. A nivel de microestructuras, empresas como Intel están empezando a utilizar biología y genética en las técnicas de fabricación de dispositivos orgánicos, usando partes biológicas para sintetizar filamentos quirales de DNA (Deoxyribonucleic acid), donde las ondas que se propagan son equivalentes a las WSM. Por otra parte a nivel macroscópico, para entender adecuadamente la naturaleza de la interacción entre un campo electromagnético de muy alta frecuencia con la materia, debemos considerar la electrodinámica quiral relacionada con la relatividad. Un ejemplo relevante es el diseño de nuevos sistemas GPS (Global Positioning System) con distinta polarización circular, que son más exactos, con la finalidad de mejorar los sistemas actuales. Un estudiante de electromagnetismo debe estar consciente de la metamorfosis, que ocurre en la física, cuando trabajamos en distintas longitudes de onda o en distintas frecuencias. Cuando la longitud de onda es muy larga, nos encontramos en el dominio de la electro estática y de la magneto estática; aquí se aplica la teoría de circuitos, un ejemplo es el área de los dispositivos de almacenamiento de energía, desde baterías comunes a sofisticados dispositivos híbridos 2 Villarroel: La ingeniería electromagnética moderna utilizados para almacenamiento de energía. Concretamente, en los automóviles modernos, dichos elementos están hechos de mezclas químicas cuyas energías vinculantes son diferentes. Si se conoce la forma en que los elementos de la mezcla se unen, se podrían diseñar baterías para fines específicos, con cálculos basados en la WSM. En el futuro, la WSM requerirá de nuevas técnicas de aplicación, cálculo y diseño de la ingeniería electromagnética. Por otro lado, la mayoría de las aleaciones más valiosas que se utilizan ampliamente en las aplicaciones industriales, como ser el acero, el bronce y el duraluminio, son mezclas simples de elementos básicos, esto es posible gracias a que las uniones de las aleaciones son del tipo Estructuras de Onda. En relación con todo esto tenemos la MEA, enfoque que será muy importante en las próximas décadas. Esta metodología, para tratar los cambios métricos, ha surgido a través de años de estudio de las teorías electro gravitacionales. Este enfoque es isomórfico con la representación general de la relatividad del vacío, tratando el vacío como un medio polarizable con cambios métricos internos, en términos de la permisividad y la permeabilidad consideradas constantes en el vacío. Este enfoque es básico para obtener energía a partir del vacío (motores magnéticos). Aquí, las ecuaciones de Maxwell en el espacio curvo se modelan como un medio polarizable de índice de refracción variable en el espacio plano, donde la curvatura de un rayo de luz y la reducción de la velocidad de la luz en un potencial gravitacional se representan por un aumento efectivo del índice de refracción. Con este método es posible estudiar los Sistemas de Propulsión de Campos Electromagnéticos, donde la propagación de fotones posee momentum producido por los campos magnéticos y eléctricos ortogonales entre sí (vector de Poynting). Estos desafíos tecnológicos nos hacen ver que es importante atraer para este campo a personas más calificadas y creativas, reclutando los mejores estudiantes y estimulando su creatividad. En esta perspectiva de la enseñanza de la ingeniería electromagnética, la gente joven siempre puede generar buenas ideas, forjar nuevas fronteras, crear nuevas áreas de trabajo y cultivar el pensamiento independiente, estimulados por el profesor en el desafío de pensar. Puesto que el análisis electromagnético ha sido usado como una importante herramienta de predicción en muchas ramas de la ingeniería eléctrica, seguirá siendo aún más importante en las nuevas tecnologías. La larga y rica historia del electromagnetismo nos ofrece un desafío sobre cómo debemos educar a nuestros estudiantes de postgrado en esta área. El total del conocimiento requerido no se puede entregar dentro del corto período de su enseñanza universitaria. Por lo tanto, es fundamental educarlos en los conocimientos básicos, ya que aprender todo lo pertinente a la tecnología electromagnética requiere de toda una vida de aprendizaje. Asimismo, es importante educar a dichos estudiantes como pensadores, en vez de adquirir los conocimientos en forma mecánica, siendo esta forma de enseñar un importante aporte para nuestra sociedad. Es así como, en este número, presentamos el aporte del doctor Héctor Torres-Silva en esta área del electromagnetismo moderno, mediante la electrodinámica vinculada a la mecánica cuántica y a la gravitación. Este trabajo incluye aspectos fundamentales de la WSM, al unificar al electromagnetismo y a la gravitación a través de la electrodinámica quiral, mostrando en este estudio, en forma rigurosa, que la mecánica cuántica de Dirac es una consecuencia lógica de dicha unificación. Carlos Villarroel González Editor Ingeniare. Revista chilena de ingeniería Universidad de Tarapacá Arica, Chile 3 Ingeniare. Revista chilena de ingeniería, vol. 15 16 Nº 3, 1, 2008, 2007 pp. 4-5 EDITORIAL MODERN ELECTROMAGNETIC ENGINEERING Electromagnetic engineering is a branch of applied physics which is developing so fast that in the near future electromagnetic engineers will be indispensable in an important emerging area, that which is now known as Wave Structure Matter (WSM). The main reason for this is the penetration capacity of electromagnetic technology where specialized engineers will be needed for the design of systems related to WSM technology. Examples of such systems with very high frequencies are wireless communication networks, computer chips, optical networks, antennae and at low frequencies, energy storage devices and systems related to a Metric Engineering Approach (MEA) with electromagnetic fields. The difficulty and complexity of the laws that govern the design of systems related with electromagnetic engineering indicate that the theory and analysis of electromagnetism is a continually-evolving science and an area of active research that has attracted the interest of mathematicians, computer scientists and engineers. However, a good understanding of modern electromagnetic analysis requires a deep knowledge of physics, a capacity for mathematical analysis and knowledge of the numerical algorithms used in computing. Even though some universities emphasize the computational analysis of electromagnetism, we must be aware that a student of electromagnetic engineering should understand the concepts of physics involved and develop intuition and understanding of the problems to be solved. These abilities are as important in design as they are in analysis. Therefore, it is important to train postgraduate students in modern methods of electromagnetic analysis and new theories such as: metamaterials, electrodynamism chiral electrogravity. For example, chiral electromagnetic analysis ought to include, amongst other things, the concepts of circular-polarized waves, superficial waves, creeping waves, lateral waves, guided modes, evanescent modes, radiant modes and leaky modes. All of this in high-frequency physics where the wave-particle duality is emerging as a new physical focus of the electromagnetic interactions of WSM. Recently, advances have been made in WSM, for example in industrial micro-circuits and electrodynamics where there are currents in closed loop of real electron waves, since the electron is not a point particle but rather a wave structure. Here the majority of applications, such as chiral nanotubes and metamaterial substrates for use in microcircuits, require the understanding of material in “very small dimensions” where an approximation of the particle fails and WSM makes it necessary to understand what happens when different substrates interact at the chemical, biological and physical level. A the microstructural level, companies such as Intel are beginning to use biology and genetics in the production techniques for organic devices, using living elements to synthesize chiral filaments of DNA (deoxyribonucleic acid) where the propagating waves are equivalent to WSM. Moreover, at the microscopic level, in order to adequately understand the nature of the interaction between a very high frequency electromagnetic field and the material, we should consider the chiral electrodynamic related to the relativity. A pertinent example is that of the design of new GPS (Global Positioning Systems) with distinct circular polarization that are more accurate with the aim of improving the current systems. A student of electromagnetism should be aware of the metamorphosis that occurs in physics when we work in different wavelengths or frequencies. When wavelength is very long, we find ourselves in the electrostatic and magnostatic domain, where circuit theory applies. An example from this area is that of energy storage devices that range from ordinary batteries to sophisticated hybrid devices used for storing energy. To give a concrete example, in modern automobiles, these elements are made from chemicals with different bond energies. If we know the way in which the elements of the mixture bond, batteries can be designed for specific purposes with calculations based on WSM. In the future, WSM will require new techniques for application, calculation and design from electromagnetic engineering. Furthermore, the majority of the most costly alloys that are widely used in industrial applications, such 4 Villarroel: La ingeniería electromagnética moderna as steel, bronze and hard aluminium are simple mixtures of basic elements. They serve their purposes thanks to the bonds between the alloys that have wave structure. Connected to all of this is MEA, an area that will be extremely significant during the next few decades. This methodology for treating metric changes has arisen through years of study of electrogravitational theories. This focus is isomorphic with the general representation of vacuums, treating them as polarizable media with internal metric changes in terms of the permittivity and permeability considered to be constant in the vacuum. This focus is the basis for obtaining energy from vacuum (magnetic motors). Here, Maxwell’s equations in curved space are modelled as a polarizable medium of the variable refraction index in flat space where the curvature of a ray of light and the reduction in the speed of life in a gravitational potential are represented by an increase in the refractive index. With this method it is possible to study Electromagnetic Field Propulsion Systems, where the propagation of photons has a momentum produced by the crossed electric and magnetic fields (Poynting’s vector). These technical challenges enable us to see that for this field it is important to attract qualified and creative people, to recruit the best students and stimulate their creativity. From the perspective of electromagnetic engineering teaching, young people can generate good ideas, stretch boundaries, create new areas for study and cultivate independent thought, stimulated by teachers that challenge them to think. Given that electromagnetic analysis has been used as an important tool in prediction in many areas of electric engineering, it will continue to be one of the most important tools in new technologies. The long and rich history of electromagnetism makes the question of how to train our postgraduate students in this area challenging. All the knowledge required cannot be conveyed during the short period of university education. Thus, it is fundamental to provide the most essential knowledge; learning about everything related to electromagnetic technology would take a whole life-time of learning. Moreover, it is important to train these students to be thinkers, rather than mechanically acquiring knowledge, and thus contribute significantly to our society. It is for these reasons, that in this issue, we present Dr. Héctor Silva-Torres’s contribution to the area of modern electromagnetism through electrodynamics linked to quantum mechanics and gravity. This work includes fundamental aspects of WSM, to unify the electromagnetism and gravity through electrodynamics chiral showing rigorously in this study, that the Dirac’s quantum mechanics is a logical consequence of this unification. Carlos Villarroel González Editor Ingeniare. Revista chilena de ingeniería Universidad de Tarapacá Arica, Chile 5 Ingeniare. Revista chilena de ingeniería, vol. 16 vol. 16 Nº 1, Nº 1, 2008, 2008 pp. 6-23 ELECTRODINÁMICA QUIRAL: ESLABÓN PARA LA UNIFICACIÓN DEL ELECTROMAGNETISMO Y LA GRAVITACIÓN CHIRAL ELECTRODYNAMIC: CONNECTION FOR THE UNIFICATION OF ELECTROMAGNETISM AND GRAVITATION H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 12 de diciembre de 2007 Received: September 5, 2007Accepted: December 12, 2007 RESUMEN Una alternativa a la teoría cuántica de la gravedad, aún no descubierta, es la Teoría Total Simplificada (TTS) aquí propuesta, que postula unificar la gravedad con el electromagnetismo (EM) teniendo como corolario fundamental la ecuación cuántica de Dirac. Con ello, aquí se propone todo un programa de unificación en el cual el electromagnetismo quiral juega el rol central. La TTS se deriva de las ecuaciones originales de Einstein-Hilbert G µv = kTµv, donde el tensor de Einstein no se modifica. El tensor EM en cambio es quiral y la masa de las partículas es de naturaleza electromagnética. Para el caso del electrón se tiene como consecuencia que, por primera vez, se obtiene la ecuación de Dirac a partir de ondas EM con el campo eléctrico paralelo espacialmente al campo magnético. Como modelo del universo se propone una interfaz o membrana de separación donde ocurren solamente eventos cuánticos. Hay dos regiones enantioméricas de un universo cerrado, o un universo derecho y un universo izquierdo, relacionados por un elemento de simetría PCT (paridad, carga, tiempo) a lo largo de la interfaz. Las ecuaciones de Einstein-Hilbert son estudiadas bajo el enfoque quiral y se discute la electrodinámica quiral y la gravedad en la era de Planck. Palabras clave: Unificación, electrodinámica quiral, era de Planck. ABSTRACT An alternative to the theory of quantum gravity, not yet discovered, is the Theory Simplified Total (TTS) proposal, which aims to unify gravity with the EM taking as a corollary essential quantum Dirac equation. Thus, this article proposes a whole program of unification in which electromagnetism chiral plays the main role. TTS is derived from the original equations of Einstein-Hilbert G µv = kTµv, where the Einstein tensor is unchanged. The EM tensor instead is chiral and the mass of the particles is electromagnetic nature. In the case of the electron the consequence of this is that, for the first time, Dirac’s equation is obtained from EM waves with the electric field spatially parallel to the magnetic field. As a model of the universe an interface or membrane separation is proposed as the only location for quantum events. There are two enanciometrics regions in a closed universe, or right and left universe, connected by an element of PCT (parity, charge, time) symmetry along the interface. Einstein-Hilbert equations are studied under the chiral approach and discusses the chiral electrodynamics and gravity in Planch’s era are discussed. Keywords: Unification, chiral electrodynamics, age Planck. INTRODUCCIÓN No es difícil mostrar, sin lugar a dudas, que la física se lleva el honor de ser la disciplina con los mayores adelantos teóricos y con las más grandes aplicaciones tecnológicas. Hoy, a comienzos del siglo XXI, la física sigue el rumbo 1 6 y la impronta marcada por los grandes progresos logrados el siglo pasado. Como un logro no alcanzado se destaca la formulación de una teoría unificada de todas las fuerzas de la naturaleza. Físicos teóricos de todo el mundo, en solitario o en equipo, han dedicado y dedican una enorme cantidad de tiempo y esfuerzo para la consecución de ese sueño. Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación Al mencionar este esfuerzo, en la búsqueda de la unificación, siempre conviene recordar los trabajos pioneros de Albert Einstein (1879-1955), quien se convirtió en el líder de un sueño singular de hacerlo realidad: la teoría geométrica para la unificación de los campos gravitacionales y electromagnéticos (EM). El programa de Einstein para la unificación no tuvo éxito por el hecho de no considerar la existencia de campos eléctricos y magnéticos paralelos en el espacio y desfasados en el tiempo pues la existencia teórica y experimental de dichas ondas se ha verificado solo en los últimos años. A fin de proporcionar una visión global, se ha dividido este trabajo en tres partes. En la primera, se hace una introducción a la electrodinámica quiral (EQ), en la segunda se describen sintéticamente las ideas acerca de la Teoría General de la Relatividad (TGR), los trabajos de Einstein y otros autores para la unificación de los campos, y en la tercera parte se propone una teoría de unificación basada en la electrodinámica quiral para un universo de cuatro dimensiones que da lugar a la Teoría Total Simplificada (TTS). INTRODUCCIÓN A LA QUIRALIDAD Con el avance en la construcción de compuestos artificiales, los materiales quirales han asumido una gran importancia tecnológica en antenas, circuitos de alta frecuencia y en fibras ópticas. Tales materiales, que no tienen la simetría tipo espejo, también se encuentran en la naturaleza desde las galaxias en espiral a moléculas tipo hélices como el DNA, que son ópticamente activas, las cuales muestran birrefringencia a frecuencias ópticas y de microondas. Ya que la quiralidad es un concepto geométrico, es posible concebir la fabricación de materiales quirales artificiales y metamateriales, con aplicaciones en ingeniería electromagnética. El fenómeno de la actividad óptica, en ciertas substancias biológicas y materiales, fue descubierto por Pasteur (1848-1850) interpretando las observaciones de arreglos asimétricos de átomos dentro de un material ópticamente activo, por lo que se tiene una imagen espejo no superpuesta, arreglo definido como “maniobrable” o quiral. Con los avances en la teoría de campos electromagnéticos el campo de la esteroquímica se expande fuertemente y se conocen detalles de la estructura de las moléculas y, además, de la naturaleza de estructuras moleculares indispensables para la vida. Un medio quiral está caracterizado por su maniobrabilidad en su microestructura, ya sea a la izquierda o a la derecha. Esto resulta en un medio quiral polarizado circularmente a la izquierda (LCP) o a la derecha (RCP) y los campos se propagan a diferentes velocidades de fase: el campo con esta última polarización viaja, a través de un medio manipulado a la derecha, más rápido que un campo circularmente polarizado a la izquierda, y viceversa. La actividad óptica, la cual se encuentra en una serie de moléculas orgánicas a frecuencias ópticas, es una manifestación de la quiralidad nativa de estas moléculas. Se observan fenómenos similares al dicroísmo circular (CD) y a la dispersión óptica rotatoria (ORD), con absorción diferencial de las ondas polarizadas circularmente a la izquierda o a la derecha al interior del medio quiral. Estudios giroscópicos tienen tal riqueza de información que se puede decir que “la actividad óptica entrega una ventana hacia el interior de la fábrica del universo”. Los átomos son ahora considerados quirales debido a la débil violación de paridad de la corriente neutral de interacción entre el núcleo y los electrones; la pequeña actividad óptica resultante para los átomos ha hecho más concordante la teoría con la práctica. Los avances tecnológicos en las décadas de los 80-90 han hecho posible la detección de asimetría quiral en dispersión Raman. Aunque han sido estudiados muchos aspectos de mecánica cuántica, la quiralidad ha sido poco estudiada, falta un estudio sistemático de la teoría clásica de campos electromagnéticos considerando la quiralidad. Con los avances en la ciencia de los polímeros (dieléctricos quirales activos a frecuencias milimétricas, por ejemplo), hacen necesario revisar todos los aspectos relacionales de la teoría de campos electromagnéticos. Además, a causa de que la quiralidad es un atributo geométrico específico, el conocimiento recogido del estudio de la estructura molecular debe trasladarse al diseño y manufactura de medios quirales artificiales, los cuales deben exhibir CD y ORD a frecuencias de telecomunicaciones. Sólo en el rango intermedio de frecuencia la quiralidad molecular no desaparece cuando se efectúa una transición desde escala microscópica a macroscópica en teoría electromagnética. Lo anterior significa que es posible la construcción de medios artificiales introduciendo objetos quirales micrométricos en un medio huésped aquiral. En un rango de frecuencias intermedias, la microestructura podría tener una dimensión adecuada (2-5%) respecto de la longitud de onda del material huésped (substrato); consecuentemente, el medio compuesto podría comportarse como efectivamente quiral. La factibilidad de esta idea ha significado un intenso estudio sobre la propagación de ondas electromagnéticas en medios quirales. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 7 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 A modo de ejemplo, el ojo humano contiene dos tipos de estructura activa para distinguir entre proceso fotópico (umbral alto) y proceso escotópico (umbral bajo) que son: el bastón y los conos, respectivamente. Varias partes en el ojo son anisotrópicas; en particular las fibras de la retina son algo uniaxiales. Se puede usar la anisotropía estructural para explicar la diferencia en la sensibilidad de los ojos a la luz polarizada circularmente a la derecha y a la izquierda. Así, cualquier tratamiento de la física del ojo no sólo debe incluir anisotropía debido a la microestructura, bastones, conos, ganglios, etc., sino que debería también considerar la macroestructura, es decir, la helicidad de los componentes moleculares cuyas dimensiones pueden ser una fracción significativa de la longitud de onda óptica. Esto será relevante en futuros sistemas ópticos. recientes indican que valores de T. para compuestos quirales artificiales, en el rango de 8-40 GHz, estarán luego disponibles. Nuestro trabajo específico sobre teoría y simulaciones de la electrodinámica quiral aplicada a solitones, fibras ópticas, sistemas de microondas, sistemas biológicos, etc. están relacionados en [1-18]. La actividad óptica puede ser explicada por la sustitución directa de nuevas relaciones constitutivas en las ecuaciones de Maxwell, es decir, D = E + T ÑxE y B = µ + T µ Ñ × . Aquí y µ son la permitividad y permeabilidad respectivamente, mientras T es el parámetro con dimensión de longitud y es el resultado directo de cualquier quiralidad en la manoestructura del medio. Como tal, las ecuaciones constitutivas quirales son aplicables a cualquier región del espectro electromagnético como radiación. En óptica, hasta muy recientemente, sólo fue posible realizar mediciones de intensidad, es decir, de la magnitud pero no de la fase. Así la literatura sobre actividad óptica está relacionada, generalmente, sólo con la diferencia en la intensidad de la luz dispersada cuando un volumen quiral es irradiado ya sea por una onda plana LCP o RCP. Esto significa que sólo mediciones de (nL –nR) están disponibles, donde nL y nR son los índices de refracción para las ondas LCP y RCP respectivamente. Aunque cada uno de estos índices de refracción puede estar muy relacionado a , µ , T, el conocimiento de (nL –nR) no es suficiente para inferir los valores de los parámetros constitutivos. Con el surgimiento de la Relatividad Especial en 1905, casi de inmediato los físicos llegaron a reconocer la invariancia lorentziana en la teoría de Maxwell; y dada la geometría de Minkowski se tornó claro para Einstein y D. Hilbert que la unificación implicaba, de algún modo, la unificación del espacio tridimensional y el tiempo en un “espacio-tiempo continuo” cuadridimensional. En 1915, Hilbert presentó por primera vez una teoría de campo unificado basado en los primeros trabajos de Einstein (1914) sobre la teoría relativista de la gravitación y en los artículos de G. Mie (1912) sobre la electrodinámica no lineal de la materia. Cuando uno considera este fenómeno a un rango de frecuencias mucho más bajo, 0.5-100 GHz, se podría estar más interesado en , µ, T que en nL y nR. Con el advenimiento de los analizadores vectoriales de redes ahora ha llegado a ser posible realizar mediciones muy exactas de magnitud y fase, pero la generación de ondas circularmente polarizadas requiere de tecnología de punta y se prevén aplicaciones en antenas de polarización circular en sistemas GPS de última generación y en comunicaciones satelitales. El uso del analizador vectorial de redes, en conjunto con experimentos canónicos deseablemente definidos, puede entonces facilitar la medición de T. Investigaciones 8 EL PROGRAMA DE EINSTEIN PARA LA UNIFICACIÓN La historia muestra que la idea de la unificación de las fuerzas de la naturaleza no se origina con los trabajos de Einstein; ni la propuesta de unificación a altas dimensiones tampoco es original de Kaluza [19]. En el trabajo de Hilbert se obtienen las ecuaciones de Euler-Lagrange, derivadas de un principio variacional. Cinco días después de la conferencia de Hilbert sobre su teoría de unificación, Einstein publicó su TGR verificable y verificada que vinculara directamente la distribución y movimiento de materia a la geometría del espaciotiempo. En la TGR, Einstein geometriza la gravitación en el sentido de que toda la información acerca de las interacciones gravitacionales está contenida en el elemento de línea del espacio-tiempo. La poderosa y bella descripción de la gravitación entusiasmó a los físicos y matemáticos a intentar una geometrización para la unificación con el electromagnetismo (EM). La búsqueda de una teoría geométrica y unificada de los campos gravitacionales y electromagnéticos ocupó un rol dominante en los últimos veinte años de la actividad científica de Einstein. Las ideas principales de Einstein para la unificación clásica de las interacciones eran: geometrizar el electromagnetismo, unificar las variables básicas de la Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación gravitación y el electromagnetismo en un único objeto geométrico y obtener las ecuaciones de campos unificados a partir de un principio variacional. Einstein esperaba que las propiedades fundamentales de las partículas elementales y sus respectivos comportamientos cuánticos pudieran ser de algún modo descritos y explicados en el marco de una teoría clásica puramente geométrica. La teoría general de la relatividad (TGR 1915) está basada en dos objetos geométricos fundamentales: un tensor métrico g y una conexión lineal . La métrica es necesaria para medir distancias, intervalos de tiempos, velocidades relativas y ángulos. La conexión basada en la noción de transporte paralelo de Levi-Civita sirve a su vez para comparar direcciones, fuerzas y campos en puntos separados en el espacio-tiempo de Riemann. Todos los intentos iniciales en unificación estaban basados en los objetos geométricos antes mencionados. La idea básica es obtener nuevos grados de libertad en la TGR para describir el electromagnetismo relajando o imponiendo restricciones sobre el tensor métrico (g) y/o , o incrementar el número de dimensiones de la variedad riemanniana. Dos años después de que Einstein postulara la TGR, H. Weyl (discípulo de Hilbert) propone un modelo geométrico de la gravitación y del electromagnetismo. Weyl consideró que la geometrización podría ser generalizada a otras fuerzas de la naturaleza. Así, propuso una conexión general dependiente de la trayectoria cuando se compara la longitud de vectores en diferentes puntos del espaciotiempo. En otras palabras, él notó que la TGR está basada en la “relatividad de la dirección” y propuso extenderla a fin de tomar en cuenta la “relatividad de las magnitudes” al permitir una transformación conforme de la métrica. Esta idea, que llegó a ser conocida como teoría de calibre o de “gauge”, no prosperó porque llegaba a contradecir la escala absoluta de masas del mundo real. A pesar del fracaso Weyl reinterpretó los calibres (gauge en el contexto de la física cuántica) al indicar que podrían actuar en las funciones de onda de las partículas cargadas más bien que sobre g. Esta idea inspiró a las teorías de calibre no abelianas y a la interpretación de los potenciales electromagnéticos y de Yang-Mills como conexiones en fibrados principales. También llegó a constituirse en el germen para el desarrollo de las llamadas “teorías gauge de la gravedad”. En 1919, T. Kaluza propuso una teoría de la gravitación de cinco dimensiones. Esta idea fue trabajada por Einstein y colaboradores en 1923. Einstein vuelve sobre esta teoría pentadimensional en un trabajo publicado en 1927; también lo hace en cuatro trabajos (1931, 1932, 1938 y 1944) con sus colaboradores, pero sin llegar a geometrizar el EM ni a la unificación de los campos gravitacional y EM. A. Eddington propuso considerar a como la cantidad básica de la TGR y derivar de ella tanto el tensor métrico como el campo electromagnético al dividir el tensor de Ricci, R µ, en sus partes simétricas y antisimétricas. Einstein, en cinco trabajos (cuatro en 1923 y uno en 1925), desarrolló esta idea al postular que la densidad lagrangiana debería ser proporcional a la cantidad ½det Rµ ½1/2. Desafortunadamente, todos estos intentos llevaron a ecuaciones incompatibles con los experimentos y Einstein se vio obligado a abandonar este camino. En 1925 Einstein consideró una teoría basada en la conexión y una gµ no simétrica e identificaba a g[µ] (la parte antisimétrica de gµ ) con el campo electromagnético; volvió sobre esta idea en los últimos años de su vida trabajando en una “teoría asimétrica” fundamentada en la métrica y en la conexión. Sobre esta línea de investigación publicó 11 trabajos entre 1925 y 1955. Bajo la influencia de Cartan, Einstein genera una nueva línea de investigación en la que se dota a la variedad del espacio-tiempo de una nueva entidad geométrica llamada torsión inventada por Cartan en 1922. Este esquema fue transformado por un nuevo y poderoso concepto geométrico llamado teleparalelismo, también desarrollado por Cartan. Teleparalelismo significa que la curvatura total es cero, o una suposición más débil: que el tensor total de Ricci es cero. Estas ideas fueron trabajadas por Einstein y publicadas en tres trabajos (uno en 1929 y dos en 1930). En esa área tampoco logró la ansiada unificación. Las teorías de unificación basadas en teleparalelismo han sido reconsideradas en años recientes siguiendo el enfoque de la geometría diferencial moderna [20]. En resumen, Einstein se embarcó en un programa geométrico de unificación de las interacciones clásicas gravitacionales y electromagnéticas en más de cuarenta trabajos. A pesar del fracaso, aun así entreabrió nuevas sendas hacia la búsqueda de la unificación de las fuerzas de la naturaleza, en cuya tarea se han ocupado importantes físicos durante el siglo XX. Pero la mayoría de estos esfuerzos están bajo el enfoque de la teoría cuántica de campos. De hecho, las tres cuartas partes de las fuerzas de la naturaleza conocidas son estudiadas en el marco de la mecánica cuántica; y ya se ha logrado la unificación de las fuerzas débiles con las electromagnéticas. La unificación con la fuerte en este esquema no debe tardar en concretarse. La gravitación, por Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 9 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 su naturaleza misma, se resiste a entrar en ese esquema. Existen por lo menos cuatro programas de cuantización del campo gravitatorio [21], todos los cuales parecen están destinados al fracaso. El más popular de estos fracasados esfuerzos es el programa de las “variables de Asthekar”, físico hindú que pretende cuantizar el campo gravitatorio “a la canónica”, es decir, siguiendo el mismo procedimiento que llevó a la cuantización del campo EM. El punto crucial es que si deseamos cuantizar el campo gravitacional deberíamos, como bien han señalado R. Penrose y R. Wald [21], reconstruir la mecánica cuántica sobre nuevos fundamentos. El otro programa de unificación vía geometría diferencial es también extraordinariamente difícil. Es pertinente mencionar que en esta área de trabajo se distinguen dos líneas de investigación. Una en la que se mezclan conceptos de calibres de la mecánica cuántica con conceptos y herramientas de la topología diferencial. Un ejemplo de ello son las llamadas teorías de Kaluza-Klein, en cinco o más dimensiones. La otra línea de acción proviene de los trabajos de Wheeler, quien partiendo del enfoque de una geometría diferencial pura ha publicado la más elaborada geometrización del electromagnetismo de toda la literatura [20]. MODELO QUIRAL DEL UNIVERSO Einstein en su visión del universo y en su programa de unificación, aun teniendo presente el origen cuántico de la materia, no pudo concretar la unificación GEM. Tal vez el recorrido zigzagueante de Einstein en su programa de unificación fue producto de las numerosas tentativas de modificar el lado izquierdo de su ecuación GµvkTµv, dejando el tensor de materia Tµv sin alterar. En la TTS lo que cambia es el tensor Tµv. Conviene aquí decir algo al respecto de la relatividad general y la mecánica cuántica. En la actualidad, no hay duda de que la teoría de la relatividad y el modelo del Big-Bang son exitosos a la hora de presentarnos un panorama general de cómo el universo que hoy disfrutamos es consecuencia de la evolución bajo ciertas condiciones iniciales del universo que había luego de unas cuantas fracciones de segundo y de la aplicación de leyes conocidas de la física. Insistimos, no es que se conozcan todas las respuestas ni todos los detalles, sino que el modelo brinda la plataforma sobre la cual estas preguntas y estos detalles pueden ser bien planteados y abordados con la estrategia de las ciencias físicas. La relatividad de Einstein permanecerá como una portentosa contribución de la ciencia del siglo XX y un formidable 10 tributo al ingenio humano. Tanto en su versión especial como en la general cuando haya materia que curve el espacio-tiempo, la relatividad será una poderosa herramienta de interpretación de una parte de la realidad física. La revolución iniciada por la relatividad cambió de manera contundente la forma como debemos entender al espacio, al tiempo y a la materia. Nos brinda una imagen más coherente y unificada del mundo físico: la manera por la que brillan las estrellas tiene que ver con el retraso de relojes en movimiento. Entendemos mejor por qué cierta escala el sistema newtoniano da tan buenos resultados. Una buena parte de sus predicciones han sido corroboradas dándole sentido a las observaciones. Otras, como la existencia de ondas gravitatorias, nos permitirán ‘mirar’ el universo con otra mirada, más profunda, que habrá de revelarnos mucho acerca del universo en que vivimos. La ‘flexibilidad’ del tiempo y el espacio permite considerar las seductoras posibilidades de desaparición del tiempo como en los agujeros negros, la aparición del tiempo en el Big Bang, la expansión del espacio a escala cosmológica, que en la rígida perspectiva newtoniana eran impensables. Sin embargo, sabemos que algo importante está faltando. Las dos grandes revoluciones del siglo XX, la relatividad general y la cuántica, son incompatibles ente sí. Cada una es exitosa en su ámbito: la teoría cuántica describiendo el micromundo y la relatividad general, el cosmos a gran escala. Usan estrategias diferentes, imágenes de la realidad diferentes, metáforas diferentes y métodos matemáticos diferentes. La relatividad elude la naturaleza cuántica y la teoría cuántica elude el espacio-tiempo curvo. La primera no acepta el principio de incertidumbre y la segunda no acepta el principio de equivalencia. Para la relatividad general, la constante de Planck h es igual a cero; para la teoría cuántica, la constante, gravitacional de Newton G es igual a cero. Obviamente ambas son aproximaciones. La construcción de una teoría cuántica de la gravitación de la cual obtengamos casos límites apropiados, a la teoría cuántica de campos y a la relatividad, es la parte faltante de la revolución de la física del siglo XX, y es tarea pendiente para la física del nuevo milenio. Únicamente con esta teoría en la mano podremos entender qué ocurre cuando lo muy pequeño pero muy pesado aparecen en la misma situación física. Tan sólo con una teoría cuántica de la gravedad podremos hablar con propiedad de la naturaleza del Big Bang o de la singularidad escondida en el centro de los agujeros negros. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación Los intentos y acercamientos a esa(s) teoría(s) sugieren que en la llamada escala de Planck en tiempo y espacio (L P » 10–33 cm, tp » 10 –43 seg), característica de los fenómenos cuánticos gravitacionales, la naturaleza del espacio y el tiempo es radicalmente distinta de lo que observamos, tal vez cambie el número de dimensiones del espacio. Lo importante, como siempre, estará en las consecuencias y predicciones que una presunta teoría cuántica de la gravedad proponga, y que nos permita entender un poco mejor el universo que nos alberga, y tal vez un poco mejor a nosotros mismos. Una alternativa a la teoría cuántica de la gravedad, aún no descubierta, es la teoría propuesta en los artículos en esta edición de Ingeniare, la Teoría Total Simplificada (TTS) que postula unificar la gravedad con el EM teniendo como corolario fundamental la ecuación cuántica de Dirac. Ver figura 1. Con ello, aquí se propone todo un programa de unificación en el cual el electromagnetismo quiral juega el rol central [22]. La TTS se deriva de las ecuaciones originales de Einstein-Hilbert Gµv = kTµv, donde el tensor de Einstein no se modifica. El tensor EM en cambio es quiral y la masa de las partículas es de naturaleza electromagnética. Para el caso del electrón se tiene como consecuencia que por primera vez se obtiene la ecuación de Dirac a partir de ondas EM con el campo eléctrico paralelo espacialmente al campo magnético [22-26]. En la figura 1 se muestra una interfaz o membrana de separación donde ocurren solamente eventos y sucesos cuánticos. Hay dos regiones enantioméricas de un universo cerrado, o un universo derecho y un universo izquierdo, relacionados por un elemento de simetría PCT (paridad, carga, tiempo) a lo largo de la interfaz. Las características principales de ambas regiones enantioméricas están definidas en la figura y representan un modelo con todos los atributos requeridos por un vacío teórico. Lejos de la membrana de separación son válidas las ecuaciones de EinsteinUniverso derecho: plasma de partículas espacio-tiempo >0, Electrón: espín ±h/2, masa me, carga –e¯, tiempo t>0 Rµv = clgµn Radiación quiral EM y ondas/partículas producidas por la curvatura del factor T líneas de tiempo: Universo diestro líneas de espacio Membrana Espacio Tiempo (cuerdas quirales) µ() [1+ T Ñx] interfaz de vacío: Rµv = cgµv c: constante cosmológica cuántica Superficie espacio-tiempo Universo izquierdo: plasma de antipartículas espacio-tiempo 0 corresponde a la región enanciométrica diestra donde se tiene un plasma de partículas, T < 0 a la región izquierda, donde existe un plasma de antipartículas (ver figura 1 del modelo para el universo). En el apéndice, las ecuaciones de Maxwell derivadas plenamente en forma relativística de LQ son presentadas además de las ecuaciones de onda en régimen quiral. En la siguiente sección se discute esta teoría en los inicios del Big-Bang. 14 LA ELECTRODINÁMICA QUIRAL Y LA GRAVEDAD EN LA ERA DE PLANCK Se sabe que el universo se está expandiendo debido a la oscuridad de la noche. La dinámica dominante del cosmos es, al parecer, una expansión repulsiva de “antigravedad” a gran escala en contraste con la de corto alcance de atracción de la materia en las galaxias y las estrellas. Este fenómeno es la respuesta a la paradoja de Olber, es decir, el hecho de que el cielo, que se llena de un infinito campo de estrellas y galaxias, no brilla como una estrella sólida, sino que es predominantemente oscuro y tiene sólo la radiación de temperatura de 2,7 ºK. El hecho de que el universo se está expandiendo significa que las estrellas y galaxias se alejan entre sí y hay un corrimiento al rojo, por lo que el cielo de la noche es de por sí oscuro y frío, en lugar de ser un campo brillante y caliente de polvo de estrellas. Este hecho permite un universo de baja temperatura donde la vida pueda florecer. Por lo tanto, la expansión del universo puede ser vista como esencial para la vida. El fenómeno que provoca la expansión acelerada del universo se conoce como “energía oscura” y se puede decir que la causa el vacío. La densidad de la energía oscura puede ser identificada como el término “constante cosmológica” en las ecuaciones de la relatividad general. Este término puede ser entendido a través del concepto de un universo de plasma, donde la electrodinámica cósmica desempeña un papel de igualdad con la gravitación que es la conformación del cosmos y de sus estructuras. De hecho, la gravitación ahora puede ser entendida como una manifestación de la electrodinámica de un gran número de partículas cargadas. Este término cosmológico puede estudiarse en el contexto de esta teoría TTS que determina el valor de la constante de la gravitación. La principal hipótesis del universo de plasma es que la electrodinámica desempeña un papel igualitario con la gravedad en la configuración de las estructuras del cosmos. Es posible ampliar este principio incluso a la microescala del cosmos, y considerar la posibilidad de que incluso el vacío en sí mismo puede ser analizado como un “plasma virtual” de partículas cargadas. De esta manera es posible desarrollar un modelo del universo que va desde la longitud de Planck al radio de Hubble para el universo, como un tejido continuo de la electrodinámica, y con los dos límites de longitud que se correlacionan entre sí. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación Esta unificación se basa en dos postulados: el EM se unifica con la gravedad a la longitud de Planck, donde el campo eléctrico Ep es paralelo al campo magnético Bp. El segundo postulado establece que en el tiempo de Planck la gravedad y los campos EM están unificados y que se separaran, y se convierten en diferentes variables a nivel de mesoescala. Aquí encontramos diferencias con [37]. Esta teoría puede predecir campos de gravedad y el valor de la constante de gravedad. Asimismo, se puede predecir el valor del tiempo de Hubble y temperatura de la radiación cósmica de fondo. Según esta teoría, la razón de masa de electrones a protones Rm asume un rol central. Sin embargo, es evidente que el valor de Rm depende de la fuerza fuerte, y los nucleones, que constituyen la mayor parte de la masa visible del universo, se derivan de los quarks. Esto es esencial para cualquier teoría de unificación [38]. Aquí, en la TTS, la EQ es como una extensión de la teoría de Sahkarov para la gravedad y el origen tanto de protones y electrones, pero la unificación nace en la era de Planck con un plasma cosmológico en nuestro universo diestro. La teoría TTS es un intento de crear una teoría geométrica para resolver el problema de Einstein-Dirac de la unificación de la gravedad y el EM. La teoría por ahora se limita a los protones y electrones. La teoría está todavía en un estado temprano de desarrollo, es decir, que se describe como un “modelo de Bohr” de la unificación, por analogía con la mecánica cuántica y el modelo del átomo de hidrógeno, y se basa en una extensión de los trabajos de Einstein y Kaluza. El aspecto separado de la gravedad y el EM relativos a los protones y electrones proviene desde la escala de Planck y se produce con la aparición de una dimensión quiral como un equivalente de la quinta dimensión de Kaluza-Klein. La teoría TTS comienza con el principio de acción de Hilbert, que permite la obtención de las ecuaciones fundamentales de los campos de vacío con la extremización de la acción integral H = (16 G )−1 ò ( R − 2 ) − gdx 4 24) Donde es la curvatura escalar de Ricci de la relatividad general, es la constante cosmológica, G es la constante de la gravitación de Newton. Aquí, la acción integral para un campo cuántico de partículas con espín y masa, m cuando LEM procede de una EQ y Ep = iBp. Este campo produce un tensor de tensión de la forma p = pc2, es decir, una presión negativa, que a su vez impulsa una explosión del cosmos [39, 40]. Si suponemos que el cosmos es una entidad electrodinámica, entonces sería natural que, en un universo en expansión rápida, una especie de reacción de la ley de Lenz se debe haber producido para frenar la expansión cósmica. Esto también puede ser considerado como un “principio cósmico de la mínima acción”. Esta reacción sería la aparición de campos EM y de materia que presentan una densidad de energía positiva que dramáticamente frena la explosión inicial del universo. Este escenario es, de hecho, el escenario inflacionario, que se considera ahora el principal modelo de la cosmología. Sin embargo, puesto que nuestro objetivo final es la modificación tecnológica práctica de la gravedad, vamos a examinar ahora la relación de la gravedad y el EM en detalle. En la teoría de la relatividad general (RG) de Einstein, la gravedad surge de la geometría del espacio tiempo determinado por las propiedades del tensor métrico gij. El límite newtoniano se recupera donde el espacio no es muy curvado y el potencial de Newton es en realidad parte del elemento diagonal principal del tensor métrico gtt = −1 − 2 / c 2 . Desde el punto de vista de la TTS, la gravedad surge de la electrodinámica quiral. Sobre la base de esto, parece ser que una buena generalización para modelar la gravedad es que el tensor métrico de la RG es en realidad un tensor EM normalizado. Es decir, los campos EM no sólo son la curvatura de la métrica, sino que la métrica misma. Esto significa que el espacio tiempo es en realidad un vasto mar de radiación ultrapoderoso de EM. Con el fin de satisfacer los postulados de la TTS debemos tener una generalización covariante y física de campo para el vacío g ij = 4 Fik Fjk / F µ Fµ = 4 Fik Fjk / T0 (25) Donde Fik es el tensor de Faraday y T0, es la covariante generalización normalizada del escalar de tensión escalar. La forma de esta expresión para el tensor métrico es determinada por el primer postulado TTS, esto es, para que un campo EM ultrafuerte llene el cosmos el tensor de Maxwell es Tij = 1 / 4 ( Fij Fjk − 1 / 4 gij F µ Fµ ) (26) que debe desaparecer en todas partes. Así, en la TTS, un poderoso campo EM determina la geometría del espacio, pero en sí no es detectable directamente, sino que por ser tan poderoso se anula a sí mismo. Cabe señalar que Fjk , la forma mixta del tensor de Faraday, es a menudo escrito como gij F ik. Físicamente, sin embargo, el objetivo principal de la TTS es demostrar que la gravedad, los campos EM, y, por tanto, la geometría, son unificadas y son partes de una relación cíclica. Esto significa que en la TTS, EQ implica la geometría y viceversa, de Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 15 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 manera que para empezar con uno en lugar del otro es una cuestión de convención, y para reclamar que uno es superior y más fundamental que el otro no parece razonable. La EQ es dada por LQ = ò gµ − g EQ ⋅ BQ dx 4 , EQ ( BQ ) = (1 + T Ñ×) E ( B ) (27) la densidad de energía neta del espacio puede ser ligeramente negativa con este lagrangeano si T tiene un valor apropiado. Un mecanismo similar fue propuesto por primera vez en [40] y se denomina “Zeldovich reacción”. En este artículo se deriva el valor de G como la “elasticidad de la métrica del espacio” con el supuesto de que la longitud de Planck Tp = (Gh/c3)1/2, donde Tp es el factor quiral de nuestro R-Universo, es la constante de Planck. El valor de la integral se determina por la frecuencia de corte cerca de la frecuencia de Planck p = c/Tp. Numéricamente este valor de T es igual al radio de Planck [39, 40]. Esta conexión entre la gravedad y el campo EQ alienta la posibilidad de que la gravedad pueda un día ser modificada directamente por medios externos. La manera más sencilla de obtener una energía oscura o la constante cosmológica es permitir que EÕiB y esto genera una constante cosmológica y, por tanto, un universo en expansión. Teniendo en cuenta la hipótesis de un universo de plasma donde domina la electrodinámica quiral, suponemos la existencia de modos quirales que componen la energía oscura. En la escala de Planck, la incertidumbre de Heisenberg permite que se generen masas Mp, que tienen una longitud de onda de Compton igual a su radio de Schwartzchild GM P /c 2 = rP = TP = / 2 M P c = G / 2c3 (28) A la longitud de Planck, los horizontes de evento de los agujeros negros aparecen y desaparecen en un período de Planck, y la distinción topológica entre estar dentro y fuera de un evento horizonte desaparece. Por breves instantes de un período de Planck, las partículas de antimateria pueden interaccionar con las partículas ordinarias. Esto produce la cuantización de la carga e = khc/2T. Sin embargo, el tamaño de T debe ser del orden de la longitud de Planck, para dar el correcto valor de la carga y las masas son del orden de la masa de Planck. Sin embargo, la existencia de la escala de masas de protones y electrones significa que el tamaño efectivo de la dimensión compacta debe ser mucho mayor, es 16 decir, del orden de la longitud de onda de Compton de un protón o el radio clásico electrónico. Esto puede ser entendido conceptualmente como un “renormalization”, efecto debido a la energía negativa de la formación de la dimensión quiral. La teoría de TTS así permite, por un pequeño aumento de la dimensionalidad, la aparición explícita de los campos EM, y las partículas con carga y masa, para protones y electrones, junto con la gravedad, desde un principio variacional con = mp/me)1/2 = 42.85003. Esto constituye una descripción muy básica del cosmos como un todo, cuyos principales componentes más conocidos son los protones y electrones. Si se concibe que el principio de la acción asume un campo EQ no masivo, entonces la aparición de la quiralidad permite la captura o la dispersión de quantas no masivos que crean masa en reposo, carga, espín y por lo tanto las partículas. La captura de la energía EM quirales en una dimensión compacta puede ser concebida como una imagen compactada del espacio-tiempo. El tamaño de la dimensión compactada puede considerarse que sea del orden del radio clásico del electrón, re = e2/40 m 0 c 2, que es el tamaño aproximado del protón. El carácter de la dimensión compacta debe ser una imagen global del espacio-tiempo, esto significa que el protón, es decir, el espacio, como partícula, debe tener tres subdimensiones para satisfacer la condición de radio clásico y también para la neutralidad del cosmos. Esta combinación es satisfecha por la actual teoría de los quarks, donde el protón está formado por tres quarks que satisfacen qx = q y = 2e/3 y qz = –e/3, es decir, qx + q y + qz = e(2 / 3 + 2 / 3 − 1 / 3) = e (29) qx 2 + q y 2 + qz 2 = e 2 (4 / 9 + 4 / 9 + 1 / 9) = e 2 (30) Donde qx, qx, qz son las cargas de los quarks que componen los protones. Por lo tanto, una imagen global del espaciotiempo, sujeta a la simetría de rotación, significa que la dimensión compacta (T) tiene el carácter de un radio o un intervalo de tiempo y, por tanto, actúa como un escalar, pero puede tener tres espacios internos –como grados de libertad. Las condiciones: la suma de las cargas de los quarks y la de suma de sus cuadrados significa que deben satisfacer una simetría de rotación SO (3). Los protones, por tanto, compuestos de quarks, son, pues, fundamentales en la TTS para nuestro universo diestro. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación Si uno se concentra por un momento en la acción de Hilbert-Einstein, sobre la expansión de la EQ de Lagrange y de la selección de aquellas partes que desaparecen en el marco de la caída libre en el modelo de TTS, obtenemos (en unidades ESU). R E 2 − B2 + 16 G 8 2 2 S g =− + E = iB 2 G u c 2 (16 G )−1 K field = Donde F es el tensor de campo electromagnético y j es la densidad de corriente en 4 dimensiones. Claramente, en un sistema inercial local las ecuaciones anteriores se reducen a las ecuaciones estándares en el vacío. Para simplificarlas, se introduce un sistema de coordenadas ortogonal-temporal en el cual: g4 = 0. Se introducen los tensores antisimétricos H y B tal E = iB (31) que F = H / − g44 = B y los vectores D y E por F 4 = − D / − g44 = E / g44 . 0 donde S = 0, se asocia a los campos de la gravedad. Esta expresión cumple el principio de equivalencia, porque ambos términos desaparecen en un sistema de caída libre. Las expresiones covariantes correspondientes son: F = g g F = g g B = H = B − g44 APÉNDICE: ELECTRODINÁMICA QUIRAL Comúnmente, el electromagnetismo de Maxwell es tratado en una aproximación lineal que en muchos casos es suficiente para explicar los resultados de los experimentos. La premisa que en este trabajo de unificación se plantea es que una onda electromagnética propagándose a través de un medio es influenciada por este último y viceversa. La idea anterior se parece al principio de Mach. Se le recuerda al lector que dicho principio sugiere que la masa de un cuerpo es debida a la influencia de todas las estrellas del universo en ese cuerpo. Expresada en términos matemáticos, esta dependencia toma la forma m(x) = [1/T(x)] donde m(x) es la masa de partícula, una constante de acoplamiento y 1/T es la masa generada por el campo a través de la quiralidad. La idea central entonces es no considerar la masa de una partícula como una cantidad fija e intrínseca, sino más bien pensarla como una cantidad variable dependiente del campo en el cual se mueve. En este apéndice se construye la electrodinámica en espacio curvo. Para ello se parte con las ecuaciones de Maxwell en un Sistema Arbitrario de Coordenadas. ∂ F + ∂ F + ∂ F = 0 1 g ∂ (( g ) F ) = J

F 4 = g g4 F = g − g44 D = − g44 D = E Esto permite escribir las ecuaciones de Maxwell en la forma 1 −g ∂ ( ∂ B + ∂ B + ∂ B = 0 ) 1 −g 1 − gH − ∂ −g ( ∂t ( ) − gD = u ) − gD = Una manera de escribirlas en una forma más familiar, se introducen los vectores duales correspondientes a los tensores respectivos por la prescripción estándar, dando como resultado B1 = H1 = 1 −g 1 −g 1 B23 , B 2 = −g H 23 , H 2 = B3 1, B3 = 1 −g H 31 , H3 = 1 −g B12 1 −g H 12 (1) Finalmente haciendo las sustituciones respectivas se tienen las ecuaciones vectoriales de Maxwell sin cargas (2) Ñ×E =− Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 1 −g ∂t ( ) −gB , Ñ ⋅ B = 0 (3) 17 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Ñ×H = 1 −g ∂t ( ) −g D , Ñ ⋅ D = (4) La conexión entre las ondas electromagnéticas en un medio y las mismas ondas en el vacío pero en un espacio curvo emerge claramente. Si g no depende del tiempo y si se tiene µ = = –(–g44)-1, entonces el campo, responsable por la curvatura del espacio, se comporta como un medio con la permitividad dieléctrica y susceptibilidad magnética dada por la ecuación anterior. La conjetura que se plantea en esta investigación es que , 0 y µ0 en el espacio curvado son definidos como los operadores: 0 (1 + T Ñ×) y µ0 (1 + T Ñ×) donde T es el factor quiral que permite la torsión del campo dando lugar a la creación de las partículas. 0) J ( 0 ) = T (−2 B,(00 − T Ñ 2 E,(00 ) ) (11) Las ecuaciones de Maxwell rescritas anteriormente son formalmente similares a las ecuaciones de Maxwell en un medio normal con densidad de corriente J(0) y densidad de carga (0) = 0 que obedece la ecuación de continuidad de la carga Ñ ⋅ J ( 0 ) + ,0 ( 0 ) = 0 . Se llamará J(0) a la corriente quiral en el marco de referencia en reposo. En un sistema arbitrario S, las ecuaciones de Maxwell pueden ser escritas ∂a = a , [41]. , ∂x F, − ( µ − 1) F, u u = µ J (12) Las ecuaciones de campo F , + F , + F , = 0 (13) En un sistema de referencia, donde el medio está en reposo, las relaciones constitutivas quirales de BornFederov son: respectivamente. Aquí u es la velocidad uniforme del medio, F es el tensor del campo electromagnético con componentes F0 i = Ei Fij = − ijk Bk . También se introduce el dual del tensor de campo D(0) = E (0) + T Ñ × E (0) B ( 0 ) = µ H ( 0 ) + µT Ñ × H ( 0 ) (5) en el sistema en reposo So del medio. Se restringe el estudio a medios homogéneos y no dispersivos donde , µ y T son constantes. Las ecuaciones de Maxwell en este marco, en ausencia de cargas son Ñ ⋅ D(0) = 0 Ñ ⋅ B (0) = 0 Ñ × H ( 0 ) = D,(00 ) Ñ × E ( 0 ) = − B,(00 ) (6) (7) ∂a donde a,0 = . Las ecuaciones de Maxwell pueden ser ∂t escritas en términos de los campos E(0) y B(0) Ñ ⋅ E ( 0 ) = 0, Ñ × B ( 0 ) = µ J ( 0 ) + µ E,(00 ) (8) Ñ ⋅ B ( 0 ) = 0, ÑxE ( 0 ) = – B,0(0) (9) 0 ) (10) K ( 0 ) = 2 E,(00 ) + T Ñ × E,(00 ) = 2 E,(00 ) − T B,(00 También se tiene 18 1 G = F 2 con componentes G0 i = Bi Gij = + ijk Ek , de modo que la corriente quiral puede ser escrita como J = T u K , . (14) .. con 0123 = +1 y K = 2 F − T G . Aquí usamos la definición: F = F u ; G = G u y donde la operación punto (.) es definida por a = u a , , la cual se reduce a la derivada temporal ordinaria en el medio en reposo. En este marco Fi se reduce a las componentes del campo eléctrico (y F0 desaparece) y Gi a las del campo magnético (y Go desaparece). La corriente quiral se puede escribir en la forma .. donde J ( 0 ) = T Ñ × K ( 0 ) con . J = T (−2 G + Th µ F,µ ) (15) donde h µv está relacionado al tensor métrico gµv por h µ = g µ − u µ u Se hace notar que la ecuación de continuidad es considerada. J, = 0. Efectuando una contracción con u y notando que Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación u J = 0 se tiene F , = F , u = 0. Este resultado no es nada más que Ñ. E ( 0 ) = 0, Ñ × B ( 0 ) = µ J ( 0 ) + µ E,0 ( 0 ) en el medio en reposo. 1 F = − G 2 Si se introduce esta relación en F , − (µ − 1) F , y haciendo la contracción con se u u = µ J . . T 2 G = Th µ F,µ En un marco de referencia en reposo Sº, esta ecuación se simplifica a la ecuación de Beltrami si hacemos Tº , mc obtiene .. J = T ( −2 G + Th µ F,µ ) =0 .. El inverso de G es o sea Ñ × E (0) + 2 (0) E =0 T (18) . G , + G , + G , − ( µ − 1) F = µ J donde se ha usado la identidad µ = − é µ ù + éµ ù − éµ ù ë û ë û ë û c on éë ùû = − . L a e cu a c ión homogéne a F , + F , + F , = 0 se transforma en G, = 0 que inmediatamente sigue de la contracción de la ecuación homogénea con . Para obtener la ecuación de onda del tensor de campo se diferencia F , + F , + F , = 0 con respecto a , y se usa F , − (µ − 1) F , u u = µ J para obtener .. F , − ( µ − 1) F = − µ ( J , − J , ) (16) en. este result ado f ina l se ut il iza la relación F = u F , = F.. , − F , . La ecuación de onda F , + (µ − 1) F = − µ ( J , − J , ) es el resultado fundamental de este trabajo de investigación que permite la unificación del electromagnetismo con la gravitación. Esta ecuación de onda de segundo orden en tiempo y espacio permite obtener la propagación de gravitones si ( J , − J , ) = 0 y de fotones si T º 0 respectivamente. Para examinar el caso de gravitones con spin 2 el primer miembro de la ecuación de onda (16) es .. F , + (µ − 1) F = 0 (17) que corresponde al caso de corriente quiral igual a cero, esto es es trivial obtener de la ecuación de onda (17) con ∂ / ∂t = i , −1/ 2 y la velocidad de la luz dada por c = (µ ) la expresión k0 = tal que mcT = 2 . Esto corresponde a partículas c con spin 2. Se observa además que de la ecuación Ñ × E ( 0 ) = − B,(00 ) = −i B ( 0 ) 1 se tiene entonces que E ( 0 ) = i B ( 0 ) , o sea los campos son paralelos en el espacio tridimensional con el vector de Poynting E ( 0 ) × B ( 0 ) = 0. Esto implica una gran dificultad en detectar este tipo de partículas con detectores usuales de radiación. El caso de ondas electromagnéticas normales es analizado con la ecuación Ñ × Ñ × (1 − k02T 2 ) E − 2 k02T Ñ × E − k02 E = 0 (19) Si T = 0 se obtiene la usual ecuación de onda en un medio normal y homogéneo que se encuentra en los textos de electromagnetismo, Ñ × Ñ × E − k02 E = 0 , ondas que al ser tratadas como partículas se tiene que el spin es igual a uno. Si k0 T ≤ 1 se obtienen las ondas quirales circularmente polarizadas que se propagan en medios electromagnéticos complejos y en medios biológicos (por ejemplo, ondas en el tejido cerebral debido a las microondas de teléfonos celulares). Si k0 T ≥ 1 o mucho mayor que uno, se obtiene una ecuación de tipo Beltrami que entre otras situaciones físicas puede modelar las ondas en una estrella de neutrones, explicar las llamaradas solares donde la corriente es paralela al campo magnético. Esta relación también permite obtener el radio del universo si T = / mc y µ es la masa del fotón en el espacio curvado de Einstein. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 19 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Usando un modelo de fluido de fotones, permite obtener rotB = B que puede calcular la distancia del Sol a cada uno de los planetas del Sistema Solar. Además puede explicar: la generación de bolas de luz, la formación de galaxias en espiral y las ondas EM quirales como autoestados de moléculas de ADN. Se puede demostrar rigurosamente que con ondas electromagnéticas donde E es perpendicular a B, ( E ^ B ) la ecuación tensorial de Einstein, con el tensor de Maxwell Tµv es Rµ 1 − gµ R = − Tµ 2 (21) Esta es la razón del porqué Einstein no pudo obtener la anhelada unificación del electromagnetismo y la gravitación. Esta formulación es un enlace entre la Teoría Cuántica y la Relatividad General, siendo clave el concepto de campo Beltrami como fundamental en la creación de partículas. En la Física actual, ambas teorías están profundamente cimentadas en marcos espacio-tiempo distintos. La primera en el espacio-tiempo de Minkowski, y la segunda en el espacio-tiempo curvado. Tal como A. Wheeler lo hizo notar, los intentos de unificación y el desafío han sido la introducción de la mecánica cuántica con spin 1/2 en la Relatividad General, por un lado, y la introducción de la curvatura en Mecánica Cuántica por otro. La ecuación de onda .. F , (T ) + (µ − 1) F (T ) = − µ ( J , (T ) − J , (T )) que es la generalización de la ecuación de Klein-Gordon permite esta conexión, si la transformamos poniendo en evidencia el factor de spin 1/2. 20 (22) . .. Jep = ( éë1 − O ùû F − 2T G ) (23) O º 1 + T 2u (u, ), (24) donde manipulando las ecuaciones (22), (23) y (24), se puede obtener la ecuación de onda gµ = gµ + Fµ ds 2 = gµ dx µ dx = gµ dx µ dx O( F, − ( µ − 1) F, u u ) = µ Jep donde el tensor corriente, Jep , corresponde a la corriente quiral del electrón (T>0) o positrón (T= 2 èc ø

ç 2 − m 2c 2  .

Thus, when E is parallel to H , (15) has the form 1 ± M . 2 It is easy to verify that they are mutually complementary and orthogonal projection operators, and the following equality is valid [8] 46 Where and are solutions of (13) and (14) respectively but in general can be full quaternions not necessarily purely vectorial. In particular, we have that D− = P + D + P − D− . THE RELATION () WHEN E IS PARALLEL TO H 1 f = P + + P − , (13) and satisfies the equation (16) Moreover, as P± commutes with D±, we obtain that any solution of (4) is uniquely represented as follows (11) and D = P + D + P − D− . 1 T 2 =2 = Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 ö 1 æ 2 − m 2c 2  2 ç 2 èc ø (17) H. Torres-Silva: The close relation between the Maxwell system and the dirac equation when the electric field is parallel to the magnetic field REFERENCES or equivalently ( )2 r µr = 2 − m 2c4 . [1] F. Brackx, R. Delanghe and F. Sommen. Clifford analysis. Pitman Res. Notes in Math. 1982. [2] K. Carmody. Circular and hyperbolic quaternions, octonions, and sedenions-further results. Applied Mathematics and Computation. Vol. 84 Nº 1, pp. 27-47. 1997. (18) [3] In general, if in (17) we formally use the de Broglie equality p = = / T , we again obtain the fundamental relation (18). W. Greiner. “Relativistic quantum mechanics”. Springer-Verlag. 1990. [4] K. Gürlebeck and W. Sprößig. “Quaternionic analysis and elliptic boundary value problems”. Akademie-Verlag. 1989. [5] K. Imaeda. “A new formulation of classical electrodynamics”. Nuovo Cimento. Vol. 32 B Nº 1, pp. 138-162. 1976. [6] M. Gogberashvili. “Octonionic version of Dirac equations”. International Journal of Modern Physics A. Vol. 21 Nº 17, pp. 3513-3523. 2006. From this equation in the case r = µr = 1 , that is for a vacuum, using the well known in quantum mechanics relation between the frequency and the impulse: = pc we obtain the equality 2 = p2c 2 + m 2c 4 . Thus relation (15) between the Dirac operator and the Maxwell operators is valid if the condition (17) is fulfilled which quite is in agreement with (18), if and only if E is parallel to H . CONCLUSIONS The main result of this paper is that the Dirac equation can be derived from the Maxwell’s equation under a chiral approach (equation 15). The suggested theory is the new quantum mechanics (QM) interpretation. The below research proves that the QM represents the electrodynamics of the curvilinear closed (non-linear) waves. It is entirely according to the modern interpretation and explains the particularities and the results of the quantum field theory. Chiral approach means that our Universe is observable area of basic space-time where temporal coordinate is positive and all particles bear positive masses (electrons). The mirror Universe is an area of the basic space-time, where from viewpoint of regular observer temporal coordinate is negative and all particles bear negative masses (positrons). Also, from viewpoint of our-world observer the mirror Universe is a world with reverse flow of time, where particles travel from future into past in respect to us. The two worlds are separated with the membrane - an area of space-time inhabited by light-like particles that travel along light-like right or left-handed (isotropic) spirals (chiral photons). [7] V.V. Kravchenko. “On a biquaternionic bag model”. Zeitschrift für Analysis und ihre Anwendungen. Vol. 14 Nº 1, pp. 3-14. 1995. [8] V.V. Kravchenko and M.V. Shapiro. “Integral representations for spatial models of mathematical physics”. Addison Wesley Longman Ltd., Pitman Res. Notes in Math. Series. Vol. 351. 1996. [9] I. Yu. Krivsky, V.M. Simulik. “Unitary connection in Maxwell-Dirac isomorphism and the Clifford algebra”. Advances in Applied Clifford Algebras. Vol. 6 Nº 2, pp. 249-259. 1996. [10] A. Lakhtakia. Beltrami fields in chiral media. World Scientific. 1994. [11] J. Vaz, Jr., W. Rodrigues, Jr. “Equivalence of Dirac and Maxwell equations and quantum mechanics”. International Journal of Theoretical Physics. Vol. 32 Nº 6, pp. 945-959. 1993. [12] H. Torres-Silva and M. Zamorano Lucero. Chiral Electrodynamic. URLs: http://www.chiral.cl Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 47 Ingeniare. Revista chilena de ingeniería, vol. 16 vol. 16 Nº 1, Nº 1, 2008, 2008 pp. 48-52 DIRAC MATRICES IN CHIRAL REPRESENTATION AND THE CONNECTION WITH THE ELECTRIC FIELD PARALLEL TO THE MAGNETIC FIELD MATRICES DE DIRAC EN REPRESENTACIÓN QUIRAL Y LA CONEXIÓN CON EL CAMPO ELÉCTRICO PARALELO AL CAMPO MAGNÉTICO H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 5 de diciembre de 2007 Received: September 5, 2007Accepted: December 5, 2007 RESUMEN En este trabajo se presenta una expresión de la transformación general de Foldy-Wouthuysen a la representación quiral de las matrices de Dirac interactuando con un campo de fermión. La hipótesis es que a través de la multiplicación de la matriz de Pauli por las ecuaciones quirales de Maxwell en el caso de E = i H , se obtiene la ecuación quiral de Dirac. Esta es la prueba del teorema de que la mecánica de ondas de partícula cuántica representa una electrodinámica especializada. Palabras clave: Transformación de Foldy-Wouthuysen, ecuación quiral de Dirac, electrodinámica. ABSTRACT In this paper we offer an expression of the general Foldy-Wouthuysen transformation in the chiral representation of Dirac matrices interacting with fermion field.Our hypothesis is that through the multiplication of the Pauli matrix and Maxwell’s chiral equations in the case of E = i H , one obtains the Dirac’s chiral equation. This is the proof of the theorem that the wave mechanics of quantum particles represent a specialized electrodynamic. Keywords: Foldy-Wouthuysen transformation, chiral Dirac equation, electrodynamics. CHIRAL DIRAC MATRICES The paper offers an expression of the general FoldyWouthuysen transformation in the chiral representation of Dirac matrices interacting with fermion field ( x , t . The paper [1, 2] discuss the theory of interacting quantum fields in the Foldy-Wouthuysen representation [3]. These papers offer, in particular, the relativistic nonlocal Hamiltonian HFW in the form of a series in terms of powers of charge e. Quantum electrodynamics in the Foldy-Wouthuysen (FW) representation has been formulated using Halmitonian HFW and some quantum electrodynamics processes have been calculated within the lowest-order perturbation theory. As a result, the conclusion has been made that the FW representation describes some quasi-classic states in the quantum field theories. Both particles and antiparticles are available in these states. Particles, as well as antiparticles, interact ) with each other. However, there is no interaction of real particles with antiparticles – such interaction is possible only in intermediate (virtual) states. The FW representation modification is required to take into a account real particle/antiparticle interactions. In the papers [1, 2] such modification has been made using the symmetry identical to the isotropic spin symmetry owing to invariance of final physical results under change of sings in the mass terms of Dirac Hamiltonian HD and Hamiltonian HFW. In the modified Foldy-Wouthuysen representation, real fermions and antifermions can be in two states characterized by the values of the third 1 component of the isotropic spin S 3f = ± ; real fermions 2 and antifermions interacting with each other must have 3 opposite signs of S f . Quantum electrodynamics in the modified FW representation is invariant under P–, C–, T– transformations. Violations of the introduced 1 Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: 48 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Dirac matrices in chiral representation and the connection with the electric… symmetry of the isotropic spin lead to the corresponding violation of CP– invariance. The standard model in the modified FW representation was formulated in the papers [1, 4]. It has been shown that formulation of the theory in the modified FW representation doesn’t require that Higgs bosons should obligatory interact with fermions to preserve the SU (2)– invariance, whereas all the rest theoretical and experimental implications of the Standard model obtained in the Dirac representation are preserved. In such a case, Higgs bosons are responsible only for the gauge invariance of the boson sector of the ± theory and interact only with gauge bosons Wµ , Z µ , gluons and photons. ) æ0 i = ç è i æ0 Iö i æI 0 ö 0 , 5 = ç , = 0 i (1)  , = = ç  0 − I è ø è I 0ø 0ø Here we propose to change the Foldy-Wouthuysen transformation form by using the chiral representation of Dirac matrices. æ i i c = ç è0 æ0 Iö æI 0 ö i 0 ö 0 , = , = c0 ci (2)  , c = c = ç è I 0ø 5 çè 0 − I ø c − i ø The chiral representation (2) is commonly used in the modern gauge field theories and in the Standard Model, in particular. First consider the structure of equations describing the components of the wave functions D (x) for the two representation of Dirac matrices considered in the paper. In relations (1), (2) and below the system of units with = c = 1 is used; x, p, are 4-vectors; the inner product is taken as ∂ ; k xy = xµyµ = x0y0 – xkyk µ = 0,1,2,3, k = 1,2,3; p µ = i ∂x µ ìï 1, µ = 0 are Pauli matrices; µ = í i ; D (x) is the îï , µ = k , k = 1, 2, 3 ) ) ) four-component wave function, ( x , ( x , R ( x , L ( x ) ) ) ) ) ) ) ) )ö ; )ø (3) With representation (2), relation (3) looks like ) ) æ ( x ö p0 D ( x = ( ⋅ p + m ; D ( x = ç R  ; è L ( x ø ìï p0 R ( x = ⋅ p R ( x + m L ( x üï ý; í îï p0 L ( x = − ⋅ p L ( x + m R ( x þï (4) p0 − ⋅ p −1 L (x = R ( x = ( p0 + ⋅ p m R ( x ; m p + ⋅ p −1 R (x = 0 L ( x = ( p0 − ⋅ p m L ( x ; m ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Relations (4) use the operador equality: p02 = E 2 = p2 + m 2 Comparison between relations (3) and (4) shows that with the substitution below, m Ö ⋅ p, Ö 5 (5) These relations transform into each other. The Foldy-Wouthuysen transformations for the energy and chiral representations of Dirac matrices also transform into each other if the substitution (5) is made. Thus, the general Foldy-Wouthuysen transformation with Dirac matrices in the chiral representation ( ) (1 + chir 0 U FW = U FW chir chir 1 ) + 2chir + + 3chir + .... , as well as the fermion Hamiltonian in the Foldy-Wouthuysen representation are the two-component wave functions. The following operator relations are valid for the free Dirac equation with representation (1): ) ) ) In the papers mentioned above, the energy representation of Dirac matrices derived by Dirac himself is used: iö æ (x p0 D ( x = ( ⋅ p + m ; D ( x = ç è (x ìï p0 ( x = ⋅ p ( x + m ( x üï ý; í îï p0 ( x = ⋅ p ( x − m ( x þï −1 = ( p0 + m ⋅ p ; −1 = ( p0 − m ⋅ p; chir H FW = 5 E + qK1chir + q 2 K 2chir + + q3 K 3chir + ... can be obtained. From the corresponding expressions for en en with Dirac matrices in the energy representation U FW , H FW (see [1, 2]) with substitution m Ö ⋅ p, Ö 5 , we have Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 49 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 the relations ìï p0 R ( x = ⋅ p R ( x + m L ( x í ïî p0 L ( x = − ⋅ p L ( x + m R ( x ) ) ) ) ) üï )ýïþ (6) Also the relations (6) can be obtained under the chiral approach of Maxwell’s Equations where the electric field

E is parallel to the magnetic field H [5], that is E = i H , where = µ0 / 0 . If one wanted to describe the hydrogen gas by means of electrodynamics one should start from the firmly established experience that the hydrogen gas may absorb and reemit electromagnetic energy, and that without external intervention there is no indication that the gas to contain electric charges [6-11]. Thus we consider the hypothesis witch visualizes the gas as charge free electromagnetic field as the starting point with the lest number assumptions; and so we try characterize the field by the covariant chiral Maxwell system [5] ∂ rotE + µ (1 + Tm Ñ×) H = 0, div (1 + Te Ñ×) E = 0 (7) c∂t ∂ rotH − (1 + Te Ñ×) E = 0, div µ (1 + Tm Ñ×) H = 0 (8) c∂t Here, T is the chiral scalar factor with divTe ,m Ñ × E ( H ) = Te ,m divÑ × E ( H ) = 0, and the condition of charge-free ( ) 1 mc rotE ( H ) = E ( H ) ± E(H ) T (10) Our hypothesis is that through the multiplication of the matrix Pauli for chiral Maxwell’s equations with E = i H , one obtains the chiral Dirac equation (6). Using the algebraic relation [12] 50 (11) æ ö mc çè ˆ ⋅ Ñ ± i c ø E ( H ) = − H ( E ) (12) Below the system of units with = c = 1 equation (12) is exactly equal to the chiral Dirac equation (6), if E ( H ) = R ( L ) . To probe this close connection we can obtain the well known normal Dirac equation, we get for (7, 8) the equations ∂ rotE + µ0 (1 + Tm Ñ×) H = 0, divE = 0 c∂t (13) ∂ rotH − 0 (1 + Te Ñ×) E = 0, divH = 0 c∂t (14) with E ^ grad , H ^ grad µ . Equations (13) and (14) can be transformed as: µ0 (1 + Tm Ñ×) Õ µ ( , mc 2 ), 0 (1 + Tm Ñ×) Õ ( , mc 2 ). So, scalar multiplication of the rot equations in (10, 11) by the Pauli-vector, and using the algebraic relation [12]

(ˆ ⋅ Ñ ˆ ⋅ A = divA + iˆ ⋅ rotA we have )( (9) Solv i ng t h e wave e q u a t io n fo r E H w it h Te = Tm = T = / 2mc , and by considering we have

where E ( H ) = ˆ by means of

⋅ E (ˆ ⋅ H ) .

divE = 0 besides divH = 0 in equation (10) together with the two div equations (9), transform that system (10) in to CHIRAL APPROACH OF MAXWELL’S EQUATIONS (ˆ ⋅ Ñ ) (ˆ ⋅ A) = divA + iˆ ⋅ rotA ) ∂ ü ì ï( ⋅ Ñ ⋅ H − c ∂t i ⋅ E = 0 ï ï ï µ ∂ ï ï ⋅H = 0 ý í( ⋅ Ñ ⋅ E + ∂ c t ï ï ï ï E ^ grad , H ^ grad µ ï ï þ î )( ) ( ) )( ) ( ) (15) Equation (15) can be expressed in terms of in matriz notation this reads éæ 0 ö æ 1 0 ö 1 ∂ ù é i ⋅ E êç  ⋅ Ñ − çè 0 µ1ø c ∂t ú ê ⋅ H êëè 0 ø úû êë E ^ grad , H ^ grad µ Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 ( ( )ùú = 0 üï ) úû ïý (16) ï ïþ H. Torres-Silva: Dirac matrices in chiral representation and the connection with the electric… Denoting the quantity on witch the differential operators act by D, that is éi ⋅ E ê êë ⋅ H ) ) ( ( æ iE3 ù ç iE ú=ç + úû ç H3 ç è H+ ' iH − ö æ 1 ç −iE3  ç '2 = H−  ç '  ç 3 − H3 ø ç ' è 4 ''1 ö  ''2   ''3  ''4 ø æ 0 ö çè 0 ø = º (17) (18) between the Pauli and Dirac matrices, we get for (13) system ü ìé æ 1 0 ö 1 ∂ ù ï ê ⋅ Ñ − ç ú D = 0 ï  è 0 µ1ø c ∂t úû ý í êë ï ï þ î E ^ grad , H ^ grad µ (19) D = D e − i t (21) æ 1 0 ö üï ïì ý = 0 í ⋅ Ñ − i ç c è 0 µ1ø þï D îï ⋅ Ñ + i æ + mc ç c è 0 2 ö  D = 0 (23) − mc 2 ø 0 (23’) And the transformation to a chiral Dirac equation is trivial by using relation (5). The equations (20) as well as (21) or (22), show in addition that the electrodinamical and the wave mechanical field component are connected by simple linear relation, the same holding true for the refraction (, µ) in relation to the scalar T. This isomorphism can be checked easily and directly because the eight Eq. (10, 11) may be combined into two systems of four equations each, in the following way: ü ì±i rotH ic −1 E + divH = 0 3 3 ï ï ï±i rotH ic −1 E − rotH + c −1 E = 0 1 2 ï ï 2 1 ý (24)

ï

í −1 ï±idivE − rotE 3 − c µ H3 = 0 ï ï ï −1 −1 ïî±i rotE 2 ic µ H 2 − rotE 1 + c µ H 1 = 0 ïþ ( ( ) ) ( ) ) ) ( ( ) Inserting here the first or second wave function of (21) into the first system (upper signs) or the second one (lower signs), respectively, the wave functions of (20) ends up immediately, in both cases and we are back to Dirac again Using a chiral representation of the Foldy-Wouthuysen transformation for the Dirac equation we show that the same result can be obtained with a chiral electrodynamics using the matrix Pauli. With this we proof the theorem that waves mechanic of quantum particle represents a specialized electrodynamics. The result seems unambiguous and incompatible with the current doctrine which rest on a particle interpretation. (22) If we use equation (13) in (22), it’s agreement with the Dirac amplitude equation ) CONCLUSION finally yields the amplitude equation ) (20) Independently represent a system of functions solving (16). From this, a separation of the time dependence according to ) ( Here one has to bear in mind that each of both columns matrix (14) that is æ iE1 + E2 ö æ iE3 ö ç −iE  ç iE − E  3  2 D = ç 1 and D = ç ç H1 − iH 2  ç H3  ç  ç  è H1 + iH 2 ø è − H3 ø p0 D ( x = ( ⋅ p + m ; D ( x with X ± = X1 ± iX 2 and considering the well-known connection is complete. Now nor ma l izing eq. (23) wit h = 1, c = 1, ⋅ Ñ = ⋅ p , we can write as REFERENCES [1] V.P. Neznamov. Physics of Elementary Particles and Atomic Nuclei (EPAN). Vol. 37 Nº 1. 2006. [2] V.P. Neznamov. Voprosy Atomnoi Nauki I Tekhniki. Ser: Teoreticheskaya I Prikladnaya Fizika. Issues 1-2, p. 41, hep-th/0411050. 2004. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 51 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 [3] L.L. Foldy and S. A. Wouthuysen, Phys. Tev 78, 29. 1950. [4] V.P. Neznamov. Hep-th/0412047. 2005. [5] H. Torres-Silva and M. Zamorano Lucero. Chiral Electrodynamic. URLs: http://www.chiral.cl [6] J.R. Oppenheimer. Phys. Rev. Vol. 38, p. 725. 1931. [7] H.E. Moses. Sup. Nuovo Cimento. Serie X. Vol. 7. Nº 1. 1958. [8] T. Ohmura Prog. Theor. Phys. Vol. 16, p. 684. 1956. 52 [9] S.N. Gupta. Theory of longitudinal photons in quantum electrodynamics. Proc. Phys. Soc. Vol. 63, pp. 681-691. 1950. [10] F. Reines and W. H. Sobel, Test of the Pauli Exclusion Principle for Atomic Electrons, Phys. Rev. Lett. Vol. 32, pp. 954. 1974. [11] W. Heitler Quantum Theory of Radiation, 2nd Ed., Oxford University Press, Oxford, p. 1. 1944. [12] H. Sallhofer. “Maxwell Dirac isomorphism”. Z. Naturforsch. Vol. 41 a, p 1067. 1986. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008, pp. 53-59 H. Torres-Silva: Maxwell equations for a generalised lagrangian functional MAXWELL EQUATIONS FOR A GENERALISED LAGRANGIAN FUNCTIONAL ECUACIONES DE MAXWELL PARA UNA FUNCIONAL DE LAGRANGE GENERALIZADA H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 29 de noviembre de 2007 Received: September 5, 2007Accepted: November 29, 2007 RESUMEN En este trabajo se aborda el problema de la construcción de la funcional de Lagrange de un campo electromagnético. Se introducen las ecuaciones generalizadas de Maxwell de un campo electromagnético en el espacio libre. La idea principal se basa en el cambio de función de Lagrange en virtud de la acción integral. Por lo general, la funcional de Lagrange, que describe el campo electromagnético, se construye con el cuadrado del tensor de campo electromagnético. Ese término cuadrático es la razón, desde un punto de vista matemático, de la forma lineal de las ecuaciones de Maxwell en el espacio libre. Se obtienen las ecuaciones no lineales de Maxwell sin considerar esta suposición. Las ecuaciones obtenidas son bastante similares a las conocidas ecuaciones de Maxwell. Se analiza el tensor de energía del campo electromagnético en un enfoque quiral de la Lagrangiana de Born Infeld en relación con la constante cosmológica. Palabras clave: Lagrange, acción, ecuaciones de Maxwell, Born Infeld. ABSTRACT This work deals with the problem of the construction of the Lagrange functional for an electromagnetic field. The generalised Maxwell equations for an electromagnetic field in free space are introduced. The main idea relies on the change of Lagrange function under the integral action. Usually, the Lagrange functional which describes the electromagnetic field is built with the quadrate of the electromagnetic field tensor Fik . Such a quadrate term is the reason, from a mathematical point of view, for the linear form of the Maxwell equations in free space. The author does not make this assumption and nonlinear Maxwell equations are obtained. New material parameters of free space are established. The equations obtained are quite similar to the well-known Maxwell equations. The energy tensor of the electromagnetic field from a chiral approach to the Born Infeld Lagrangian is discussed in connection with the cosmological constant. Keywords: Lagrange, action, Maxwell equations, Born Infeld. INTRODUCTION The action integral (built to formulate the least-square principle [1]) for a process in an electromagnetic field has the following form: I= æ ò ò çè ( S ) − ( V − j ⋅ A) − 2 ( µ 1 v T V −1 0 ö B2 − 0 E 2  dVdt (1) ø ) All physical phenomena in the electromagnetic field take place so the action integral has the minimal value = 0. The theory of the electromagnetic field [1] leads to the 1 Lagrange motion equations for an electric charge in the electromagnetic field and defines the electromagnetic field tensor: Fik = ∂Ak ∂x i ∂Ai − ∂x k (2) Eqns. (2) are equivalent to the first pair of Maxwell equations: ∂Fik ∂x l + ∂Fkl ∂x i + ∂Fli ∂x k =0 for Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 i≠k ≠l ≠i (3) 53 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Or, in three-dimensional notation: curlE = − B speed in free space for these new Maxwell equations will also be equal to “c”. divB = 0 and (4) Making calculations of variation for the functional I with respect to the four-dimensional potential Ai one obtain the second pair of Maxwell equations: ∂F ik ∂x k = − µ0 j i (5) Or in equivalent vector notation: ) div ( 0 E = and curl ( I= The second pair of Maxwell equations in the case of the generalised action integral eqn. (8) is obtained after evaluating the variation of the action with respect to the four-dimensional potential Ai. We can denote µ0−1 B ) = j + 0 E (6) From the mathematical point of view, the demand for a linear form of Maxwell equations for free space compels one to assume that the field term (the third in integral eqn. (1)) must be built with the electromagnetic field tensor Fik second power (the exponent of any power function under the action integral is one less after calculating the variation). So, the equations obtained are linear with respect to the electromagnetic field tensor Fik. The action integral I could be rewritten in the following form: MAXWELL EQUATIONS FOR THE GENERALISED FUNCTIONAL ò ò ( Sv − ( V − j ⋅ A) − ΔeV ) dVdt (7) TV 1 1 1 Δ = ΔeV = µ0−1 B2 − 0 E 2 = µ0−1F ik Fik 2 2 4 The linear form of the Maxwell equations, from the mathematical point of view, is arbitrarily assumed by eqn. (7). In addition the linear character of Maxwell equations for free space (as well as for air) has been confirmed by many experiments. There is no doubt that linear Maxwell equations, within experimental precision, are satisfied, however, we could not reject other mathematical forms of the electromagnetic field equations. Is we assume, more generally, that the action integral is built with the help of a function f (·), it could be written: æ æ1 öö 1 I = ò ò ç Sv − ( V − jA − f ç µ0−1 B 2 − 0 E 2   dVdt (8) 2 2 è øø TVè ) (the codomain for the function f (·) is the real set). Under this assumption new Maxwell equations having the same mathematical structure are obtained. The wave 54 ) ( ) æ1 ö = ò ò − j i Ai − f ¢ ( Δ ç 2 µ0−1F ik Fik  dVdt è4 ø TV ) According to eqn. (2) we could write 0 = I æ1 æ ∂ Ak ∂ Ai ö ö = − ò ò j i Ai + f ¢ ( Δ ç µ0−1F ik ç − k   dVdt è ∂x i ∂x ø ø è2 TV )) ( After interchanging the indices ‘i’ and ‘k’ in the final term we obtain: Where ( æ1 ö I = ò ò − j i Ai − f ç µ0−1F ik Fik  dVdt è4 ø TV ( )) ( )) æ1 ∂ A ∂ Ai ö 0 = ò ò j i Ai + f ¢ ( Δ ç µ0−1F ik ( i k − F ik ) dVdt è2 ∂x ∂x k ø TV æ ∂ Ak ö = ò ò j k Ak + f ¢ ( Δ ç µ0−1F ik  dVdt è ∂x i ø TV Hence: ) ) æ ö ∂f ¢ ( Δ µ0−1F ik Ak ∂f ¢ ( Δ µ0−1F ik − 0 = ò ò ç j k Ak + Ak  dVdt i i çè ø ∂x ∂x TV Using the Gauss theorem and taking into account the fact of the disappearance of the four-dimensional potential at the boundary of the four-dimensional space we obtain ) ∂f ¢ ( Δ F ik ∂x k = − µ0 j i (9) Eqns. (9) include the second pair of Maxwell equations in the form eqns. (5) and (6). Whereas, on using the function f (·) as the identify function, its derivative will be equal to one; i.e. æ1 ö f ¢ ç µ0−1F ik Fik  = f ¢ ( Δ = 1 è4 ø Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 ) H. Torres-Silva: Maxwell equations for a generalised lagrangian functional This means that the Maxwell equations in the form eqn. (9) include the classical Maxwell equations in the form of eqn. (5) and (6). INTERPRETATION OF THE ESTABLISHED EQUATIONS In three-dimensional notation, eqn. (9) has the following form: æ æ1 ö ö 1 div ç 0 f ¢ ç µ0−1 B2 − 0 E 2  E  = 2 è2 ø ø è æ æ1 ö ö 1 curl ç v0 f ¢ ç µ0−1 B2 − 0 E 2  B 2 è2 ø ø è æ1 ö ö 1 ∂æ 0 f ¢ ç µ0−1 B2 − 0 E 2  E  ç 2 ∂t è è2 ø ø The second pair of Maxwell equations could be rewritten in the same form as the well-known Maxwell equations (eqns. (5) and (6)): ) ) div ( E = curl ( vB = j + ∂ ( E ∂t ) (10) Where it was denoted: æ1 ö 1 = 0 f ¢ ç µ0−1 B2 − 0 E 2  2 è2 ø æ1 ö 1 µ 0−1 = µ0−1 f ¢ ç µ0−1 B2 − 0 E 2  2 è2 ø (11) ) ) ) f (⋅ = (⋅ + (− k 0 2 T 2 ) (⋅ ) ) Þ f ¢ (⋅ = 1 + ( ⋅ Thus: ( = 0 1 + µ0−1 B2 − 0 E 2 µ 0−1 = µ0−1 ( 1 + µ0−1 B2 ( ) curl µ 0−1 B = j + curlE = − B divB = 0 ∂ ( E (13) ∂t ) With modern levels of measurement accuracy, we are able to use laboratory devices that enable determination of the value of magnetic flux density (or electric field strength) with very high accuracy (0.01%), and the material parameters with the same relative error. In a magnetic field B=2T the variation of this material parameter will not be observed according to eqn. if: < × 10 −11 é J −1 ù ë û (14) The constant ‘’ is so small only in the case of strong magnetic or electric fields may the linear Maxwell equations deformation be observed and detected. GENERALIZATIONS OF MAXWELL THEORY FROM BORN-INFELD THEORY There are nonlinear electromagnetic field theories, e.g. Born-Infeld theory of the charged particle [2, 3]. In this Born-Infeld theory the nonlinear Maxwell equations are obtained from the following action integral: æ ö c2 c2 I = ò ò b 2 ç 1 − 1 + 2 I1 − 4 I 22 ,  dVdt = ò ò b 2 (1 − R dVdt ç ø b b è TV TV ) ) − 0 E ) div ( E = The derivative f’(·) which appears in the Maxwell equations is the reason for the nonlinear character of the generalised Maxwell equations with respect to electric field strength and magnetic flux density. The level of ‘deformation’ of the Maxwell equations in comparison with the linear Maxwell equations is determined by the constant ‘’. The less is the value of constant ‘’, the less is the influence of the nonlinear term in eqn. (12). We may evaluate (roughly) the value of this unknown constant. Lets us assume that the function f (·) can be almost linear or linear. Many experiments confirm that if the nonlinear character of the electromagnetic field equations exists, it cannot be strong; it must be weak. It seems to be reasonable to consider only the first nonlinear term of the Taylor series. Here we propose that: = 0 µ0 µ The Maxwell equations (in spite of their nonlinear character) still have the same form: And = j+ According to eqn. (11) and (12) the permittivity and reluctivity (or permeability) for free space have been changed with respect to the strong electric or magnetic field. The strong electromagnetic field causes a change of the free space coefficients. The multiplication ‘’ by ‘µ’ is independent of the function f (·) and equal to 0µ0: 2 ) (12) (15) On this basis, the equations obtained are supposed to be valid inside the electric particle. For fields that are weak compared to the critical strength b, the BornIngeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 55 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Infeld Lagrangian becomes the Lagrangian of classical Maxwell theory. So the energy, the momentum, and the Poynting vector, are now given, respectively, by The well-known Born-Infeld Lagrangian is usually written as é ù E 2 + ( E ⋅ B / a) 2 ê ú 2 2 2 ú ê 1 + B − E − ( E ⋅ B) ú (21) 3 ê 2 4 = òd xê a a ú ê 2 2 2 ú − ( ⋅ ) ê+a2 ( 1 + B E − E B ) ú êë úû a2 a4 L BI = b2 µ0 c (1 − R ) , 2 R = 1+ c2 b 2 I1 − c2 b I 22 , (15’) 4 field Where I1 = B2 − 1 c2 E2 = c 1 F F ik , I 2 = B ⋅ E = Fik F ik , b i s a 2 ik 4 maximum electric field strength (in the absence of magnetic field). If b2 is very much larger than E2 and c2 B2, then LBI » − (1 / 2 µ0 I1 and we recover linear Maxwell theory. We remark here that in the limit as c Õ ∞, LBI tend to zero, while cLBI approach a well-defined, non-zero limit. Pfield = ò d 3 x E×B 1+ ) F= 1 F F ik = B2 − E 2 2 ik 1 G 2 = ( Fik* F ik )2 = ( B ⋅ E )2 4 LBI = a 2 (1 − R ) , R = 1+ F a − 2 G , a4 é F ik - * F ik G/a 2 ù ∂ ê ú =0, R êë úû ∂ j Fik + ∂ k Fji + ∂i Fkj = 0 T ik = F µ j Fjk + G 2ik / a 2 R + ik a 2 R 1+ B − E 2 ( E ⋅ B)2 − a2 a4 2 (23) The volume density of the Lagrangian function in a region outside the electrical charges and currents is equal to (15) ( ) f Δ (24) Where Δ=− v0 4 F F = − v0 F F g g 4 = a0 F F g g (25) And gik means the second-order metric tensor of space [1, 3, 4]. (18) Let us evaluate the energy tensor Tik by the definition [1, 5] in the following form: (19) (20) In deriving this result, use has been made of the identity * Fµ F = −G . 56 a4 (17) Here, we find that the symmetric energy-momentum tensor for that theory is given by (22) ENERGY TENSOR OF THE ELECTROMAGNETIC FIELD The field equations for Born-Infeld theory are ( E ⋅ B)2 (16) 1 ik * F is the dual field strength tensor, Where Fik = 2 Making c=1 we have that equation (15) can be expressed as 2 a2 − E×B S= Since the Lagrangian density must be a Lorentz scalar, the electromagnetic field has only two gauge invariant Lorentz scalars, namely B2 − E 2 1 − gTik = 2 ( ∂ f (Δ ) −g ∂gik ) æ ç ∂ ∂ f (Δ −g − l çç æ gik ö ∂x ç ∂ç l  çè è ∂x ø ( ) ) ö   (26)   ø The tensor satisfies the energy conservation law in the tour-dimensional form: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Ti k, k = 0 (27) H. Torres-Silva: Maxwell equations for a generalised lagrangian functional Substituting eqn. (20) into eqn. (26) one obtains: ( 2 ∂ f (Δ Tik = ) −g ∂gik −g ) Because function f (Δ) is independent of the derivatives of the metric tensor. Thus one obtains: Tik = 4 a0 ∂f F F − gik f ( Δ ∂Δ i k ) (28) For the function f (·) given by the first two Taylor series terms we could write: ) f ( Δ = Δ + Δ Thus the energy tensor is equal to: ( Tki = Tki | = 0 +2Δ Tki | = 0 + ki Δ T = T = T =0 ( + T =0 c2 Δ (32) The sign of constant k thus, take into account an electrostatic field forced by one charged particle. Such a field has spherical symmetry. The Riemannian curvature scalar for two-dimensional space, where only one external charge is situated, must be non-negative, R ≥ 0. Is not, the Riemannian curvature scalar tensor is negative, the space would have two radii of curvature, one positive and the other negative. This is impossible with respect to the assumed spherical symmetry of the electric field (forced by one electric charge), therefore: ≤ 0 (33) We can obtain an especial result. Und er a chiral approach, using equations (21, 22) with E = iB ,we obtain S = 0 and an electromagnetic term which correspond to a cosmological constant given by 8 G 0 / c 4 = 1.8382 ⋅ 10 −54 Volt −2 . This allows the close connection between the electromagnetism and the gravitation (see annex). ) The trace of this energy tensor is equal to: 32 G In the case of a nonlinear electromagnetic field theories, e.g. Born-Infeld theory of the electromagnetic particle [5]. ) f ¢(Δ = 1+ Þ R=− ) + 4 Δ = 4Δ (29) because, in the case = 0, the trace of the covariantcontravariant energy tensor disappears: CONCLUSIONS According to the main Einstein equations, for energy field for which one can introduce the energy tensor [1, 3, 5] it could be written: Generalised Maxwell equations include the classical Maxwell equations of the electromagnetic field for weak fields. The reluctivity and permittivity of free space are changed. If the constant ‘’ cannot be omitted, the Riemannian-Christoffel curvature tensor is not equal to zero. The constant ‘’ is not positive: ≤ 0. In the case of a nonlinear electromagnetic field theory, e.g. BornInfeld theory of the electromagnetic particle [5], we can obtain an especial result. Under a chiral approach, with E = iB, we obtain S = 0 and an electromagnetic term which correspond to a cosmological constant. 1 8 G Rik − gik R = − 2 Tik 2 c ANNEX T | = 0 = 4 a0 F ij Fij − Δ = 0 The trace of the energy tensor is not negative. The trace is equal to zero if and only if the constant ‘’ vanishes. (30) Contraction with respect to the indices ‘i’ and ‘k’ gives the Riemannian curvatura scalar of the electromagnetic field: R=− 8 G c2 T (31) Substituting eqn. (29) into eqn. (31) one finally obtains This work discovers the space-time curvature carried by the electromagnetic field and provides a new unification of geometry and classical electromagnetism. The new unification contains the Einstein equations to handle the mechanics and permits the derivation of the Maxwell equations from the full second Bianchi identities. This is a purely classical work and quantum considerations are merely mentioned. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 57 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Central to this work are the requirements that the electromagnetic field be expressed as a two form F and fit into general relativity under the demand that the total stress-energy tensor used in the Einstein equations contain the Maxwell stress-energy tensor T Max. In the notation with the conventions of [1] and in S.I. units T Max is i TMax k = 0 ji ( F Fjk + *F ji * Fjk ) 2 −12 where 0 = 8.85418782 ⋅ 10 farad/meter is the electric vacuum permittivity. Originally [2] general relativity was conceived as a unification of mechanics and geometry that explained gravitation. It was just a bonus [3] that electromagnetism also entered the unification via equation. If the Maxwell stress-energy tensor carried all the properties of the electromagnetic field, showing electromagnetism to be entirely reducible to mechanics, that would have been the end of the story. However, the electromagnetic field has polarization or phase information that is not contained in the Maxwell stress-energy tensor [4]. Since Weyl’s conformal tensor, the totally traceless piece of Riemann curvature, is supposed to contain the phase or polarization information carried by gravitational radiation, one should expect it to do the same for electromagnetic radiation. This is born out by the discovery of a piece of the Weyl conformal tensor that depends explicitly on the electromagnetic field and contains this polarization or phase information. It is denoted by TMax CF, called “the local gravitational field of the electromagnetic field’’, and given by: C ikjl = 8 G 0 ì 3 ik 1 ( F Fjl − *F ik * Fjl ) − ikjl F ik Fik + 4 í2 4 c î ü 1 + ikjl F ik Fik ý 4 þ where G = 6.6726 ⋅ 10 −11 Newton-meter2/kilogram 2 is Newton’s gravitational constant, c = 2.99792458 ⋅ 108 meter/second is the speed of light, is a fully antisymmetric tensor. The traces in the expression for CF are the Lorentz invariants of the electromagnetic field FikFik = –2 (E2– c2B2) and * FikFik = 4c (E·B), where E is the electric field strength in Volt/meter and B is the magnetic field strength in Tesla. 58 The major discovery of this work is the expression for CF. The arguments that led to that expression are quite general and should defeat the criticism that CF was built on algebraically special black holes and will fail elsewhere. It would be useful to have a physical solution to the Einstein-Maxwell equations with non-zero currents that were not overwhelmed by symmetry. Then one could extend this analysis into the currents and see how the full second Bianchi identity works there. Further successful examples will give knowledge and comfort; but will not prove the generality for CF that is claimed here. However, a single credible counterexample or the observation of a magnetic monopole will vitiate this work. The small coupling constant required by the Einstein G equations, 8 40 = 1.8382 ⋅ 10 −54 Volt −2 , permits the c superposition of electromagnetic fields. It has also led many to believe that the gravitational consequences of electromagnetism are insignificant. Nothing could be further from the truth. It is a matter of principle to unify classical electromagnetism and gravitation and the curvature-based unification presented here allows the electromagnetic field to appear as an algebraically special piece of curvature. This fulfills the nineteenth century speculation that gravity and electromagnetism are both aspects of Riemann curvature. This theory is not experimentally vacuous. The smallness of the coupling constant merely means that it could be along time before curvature detectors are sufficiently sensitive while withstanding an intense electromagnetic field; or sufficiently sensitive over very long distances having less intense fields. One wonders about the consequences of CF in the environment around very strongly magnetised neutron stars [7]. Further, what are its consequences in the Jacobi equation for geodesic separation that might apply to trans galactic travel? When two electromagnetic fields are superposed could the interaction terms in the curvature have any bearing on the problem of emission or absorption? The physical geometry of space-time is determined by specifying the metric tensor or the full curvature tensor [6]. The Einstein equations, which link classical mechanics to physical geometry, may be written as M1 = 8 G æ 1 ì 1 üö ç − íT − 4 T ýø c4 è 2 þ î and M 2 = Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 8 G æ 1 ö ç − T ø c 4 è 12 H. Torres-Silva: Maxwell equations for a generalised lagrangian functional where T is the total stress-energy tensor and T its trace. There is no mention of Weyl’s conformal tensor that would complete the specification of the physical geometry. Placing constraints on Weyl’s conformal tensor is the novel feature of this work. Such constraints are meant to limit the solutions to those with a physical gravitational field. If the constraints are too limiting and they forbid physical solutions, then they will have to be altered. Similar constraints might deal with the embarrassing number of Ricci flat universes, which may or may not describe gravitational radiation. It is an open question whether the Einstein equations will have to be extended to the full curvature to handle gravitational radiation. [2] A. Einstein. “The Principle of Relativity”. Dover Publications, New York. 1952. [3] D. Jackson. Classical Electrodynamics. 3rd ed, pp. 273280, John Wiley & Sons, New York. 1998 [4] T.T. Wu and C.N. Yang. Concept of Non integrable Phase Factors and Global Formulation of Gauge Fields. Phys. Rev. D. Vol. 12, pp. 3845-3857. 1975. [5] H. Torres-Silva. “A new relativistic field theory of the electron”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 111-118. 2008. [6] H. Torres-Silva. “Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 6-23. 2008. [7] W. E. Thirring. “An alternative approach to the theory of gravitation”. Ann. Phys. USA. Vol. 16, pp. 96-117. 1961. REFERENCES [1] C. W. Misner, K. S. Thorne, and J.A. Wheeler. Gravitation. W. H. Freeman, San Francisco. 1973. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 59 Ingeniare. Revista chilena de ingeniería, vol. 16 vol. 16 Nº 1, Nº 1, 2008, 2008 pp. 60-64 ASYMMETRICAL CHIRAL GAUGING TO INCREASE THE COEFFICIENT OF PERFORMANCE OF MAGNETIC MOTORS CALIBRE QUIRAL PARA AUMENTAR EL COEFICIENTE DE RENDIMIENTO DE MOTORES MAGNÉTICOS H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 29 de noviembre de 2007 Received: September 5, 2007Accepted: November 29, 2007 RESUMEN Este trabajo introduce un recalibre físico quiral asimétrico usado para aumentar el coeficiente de rendimiento de un motor eléctrico. Se presenta una revisión de la teoría de calibres y se examina el descarte de la condición de Lorentz para obtener el recalibrado quiral. Se introduce el coeficiente de rendimiento y se analiza un motor magnético bajo el enfoque quiral que permite un proceso Beltrami. Palabras clave: Calibre quiral, motor magnético, Lorentz. ABSTRACT This paper introduces a physical chiral asymmetrical regauging to increase the coefficient of performance of an electric motor. A review of gauge theory and a consideration of the disposal of the Lorentz condition to achieve the chiral regauging are presented. The coefficient of performance terminology is introduced. A magnetic motor is discussed under a chiral approach which gives a Beltrami process. Keywords: Chiral gauge, magnetic motor, Lorentz. INTRODUCTION In this paper we investigate a process referenced in recent permanent magnet (PM) motor patents [1]. These specially designed PM motors claim to capture and use environmental energy as an additional energy input. The technique that allows this energy transfer to occur is called asymmetrical regauging (ASR). The physics behind the ASR process will be examined by reviewing gauge theory, the Lorentz gauge, and the effect of discarding the Lorentz gauge to include the vacuum chiral current density. The term coefficient of performance (COP) is introduced to adequately describe the energy transfer of these motors. REVIEW OF THE LORENTZ GAUGE To understand how environmental energy may be utilized in a motor, to theoretically gain a COP >1, a review of the Lorentz gauge is first presented. The equations used in standard practice to design motors are derived from 1 Maxwell’s equations. It has been accepted practice, to apply the Lorentz gauge to these equations to make them simpler. In abbreviated steps, we start with Maxwell’s equations [14]. All the information in Maxwell’s four equations can be reduced to the following equation: æ 2 æ ∂V ö ∂2 A ö ç Ñ A − µ0 0 2  − Ñ ç Ñ • A + µ0 0 ∂t  = − µ0 0 J (1) è ø ∂t ø è The Lorentz gauge is then applied to reduce the complexity of these two equations. Mathematically, applying any gauge, is represented by (2,3) where gamma is an arbitrary, differentiable scalar function called the gauge function [5]. ) V (t , x ) V ¢(t , x ) = V ( t , x − (2) ) (3) A(t , x ) A'(t , x ) = A(t , x ) + Ñ ( t , x Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 60 ∂ (t , x ) ∂t H. Torres-Silva: Asymmetrical chiral gauging to increase the coefficient of performance of magnetic motors To specifically apply the Lorentz gauge, the “Lorentz Condition” is imposed by choosing a set of potentials (A, V) such that Ñ • A = µ0 0 ∂V ∂t (4) Equations (4) are the ones on which all the equations for motor design are currently based. Since the magnetic vector field and the voltage scalar field are both changed at the same time, this can be referred to as symmetrical gauging, so 2 Ñ A − µ0 0 Ñ 2V − µ0 0 ∂2 A = − µ0 J ∂t 2 ∂2V ∂t 2 =− 1 0 (5) ) −Ñ ( Ñ • A + µ0 0 (6) ∂V ∂t (7) ASYMMETRICAL REGAUGING ∂V −Ñ ( Ñ • A + µ0 0 = µ0 j ∂t (8) and JA = EA (9) Asymmetrical regauging is the equivalent of discarding the Lorentz condition. Further ASR is any process that changes the potential energy of a system and also produces a net force in the process [6]. Understanding the vacuum and its polarization are essential steps to utilizing energy from the environment. According to T.D. Lee, he define the vacuum state as the lowest energy state of the system [7]. Hence, the vacuum is considered to be the worst case model of the environment. Maxwell’s equations must be modified, in the vacuum, since and J vanish. Classically, this causes the Ampere-Maxwell law to be revised. Ñ × B = µ0 0 ∂E ∂t ∂D , ∂t (11) where D = 0E + PA and B = µ0H + µ0M. (12) This leads to the result that ∂PA ∂( T Ñ × E) = jA = . ∂t ∂t (13) (10) D = D = 0 E + PA = 0 E + T Ñ × E (14) B = µ0 H + µ0 M = µ0 H + µ0 T Ñ × H. (15) and Hence, T is the chiral factor which allows to extract vacuum energy, Thus discarding the Lorentz condition in classical electrodynamics leads to new equations that include the effect of the vacuum polarization. Invoking the Lorentz condition in classical electromagnetics discards the vacuum polarization component that exists in quantum electrodynamics [6] since ) Ñ × H = jA + Here we are considered for D, B a chiral term so, [14] Notice that (5) is (1) with the middle term, (7), eliminated. In [6, 8], the authors show that if the vacuum current density factor is included, the above equation changes to COEFFICIENT OF PERFORMANCE The energy transfer of electrical machinery is generally described using the term “efficiency”. Efficiency is defined as the power output divided by the total power input from all sources. The underlying assumption when defining the energy of any system is that all the energy input is from an identifiable and measurable energy sources(s). In an ideal system the efficiency would be one. The equation for efficiency () is normally stated [2] as = POut [ Watts]. PIn (16) Coefficient of performance is a broader energy transfer term that defines the measure of energy output divided by the operator’s energy input. COP is used to describe any machinery that has additional energy input from the environment. For example, COP is commonly used to describe the energy exchange of heat pumps [3] or solar collectors. Unlike the term “efficiency”, the COP can be greater than one. See figure 1 for the energy flow diagram. The following equation defines COP mathematically. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 61 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 COP º POut PIn (Operator ) [ Watts] (17) B= Ñ× A Energy Input from the Environment Energy Input from Operator Heaviside theory the magnetic field B is connected with its generating vector potential A by the relation (18) In TTS theory this law has to be replaced in the simplest case by [14-16]. Dissipative Elements Energy Output Figure 1. Energy flow for machines described by COP. CHIRAL MAGNETIC RESONANCE EFFECTS Besides electrical spacetime devices, self-running magnetic motors have been constructed in a repeatable and reproducible way ([4], see also figure 2, [1, 2]). The functioning of these devices cannot be explained by Maxwell-Heaviside electrodynamics but it can be explained with the chiral electrodynamics. B = Ñ × (1 + T Ñ×) A (19) where is the spin connection vector again. In the following we discern between the magnetic field of the assembly and the magnetic field of the surrounding spacetime itself, denoted by Bs. The torque T acting on the magnetic dipolo moment m of the assembly due to the external field Bs is T = m × Bs (20) Under normal conditions there is no resulting torque because of Bs = 0. Spacetime is force free and does not bear a magnetic field. So there is no rotation of stationary magnets. The situation becomes different if it were possible to create a magnetic field from spacetime. In order to understand how this can be achieved we have first to look closer on the fields of the surrounding spacetime. In case of Bs = 0 it follows from Eq. (19) that Ñ× A=− 1 A T (21) where A is the vector potential of the spacetime itself. In contrast at Maxwell-Heaviside theory, this is no gaugable quantity but is uniquely defined and has a physical meaning. In case of A consisting of plane waves, takes a special form and Eq. (21) can be expressed as Ñ × A = kBA with a wavelength kB = 1/T, [14]. Figure 2. Johnson magnetic motor [1] Schematic representation of spacetime vector potential for a magnetic assembly: magnet stator including rotor magnets, flow with vortices (force field). As was described in the preceding section of this edition, the Cartan torsion of spacetime introduces the spin connection as an additional quantity occurring in the laws of nature so that they take a generally covariant form. In particular this holds for the magnetic field. In Maxwell 62 This equation is known as Beltrami equation in the literature [7]. It describes a flow with longitudinal vortices where streamlines have a helical form. In the case of chiral potential this means that there is no force field present, in accordance with our prior assumption. Taking the curl at both sides of Eq. (21) gives Ñ × (Ñ × A) = Ñ × (− 1 A) Þ (Ñ 2 + k B2 ) A = 0 (22) T This is a Helmholtz equation for the spacetime surrounding the magnetic assembly. Because of the assumption of Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Asymmetrical chiral gauging to increase the coefficient of performance of magnetic motors Bs = 0 there is no torque on the magnets, they remain at rest. Torque can be created by disturbing the Beltrami flow. For the Helmholtz equation this means that the balance to zero is no more fulfilled. Assuming a periodic imbalance leads to (Ñ 2 + 1 T2 ) A = Rcos( k ⋅ r ) (23) with a vector R having units of inverse square meters, therefore it can be interpreted as a curvature. is a wave vector and can be interpreted as the frequency of a driving force which the right hand side of the equation constitutes. If restricted to one coordinate (x) the equation reads ( ∂2 ∂x 2 + 1 T2 ) Ax = Rx cos( kr ) (24) It can be seen that this is a differential equation for a resonance without damping ( = 0). The resonant oscillation occurs in case k = 1/T with A x going to infinity. Because of violating the Beltrami condition, A creates a force field according to Eq. (24), which creates a torque being big enough to spin the magnetic assembly and to maintain the rotation. This is the mechanism how spacetime is able to do work via a resonance mechanism. In total we have shown qualitatively how energy can be obtained from spacetime via magnetic assemblies. This could be the basis for development of an engineering for such devices. It has already been shown in quantum electrodynamics that the vacuum behaves like a dielectric [9]. The vacuum sprouts positron-electron pairs as shown in the Feynman diagram. It has been shown that by discarding the Lorentz gauge, the Ampere-Maxwell law equation evolves to include the current density of the vacuum. Also, the task remains to develop the equation and determine the process to apply it to magnetic motors. Future work is planned to study the magnetic motor to ascertain the exact mechanism involved that allows this motor to exchange energy with the vacuum. CONCLUSION It has been suggested in at least one recent patent that it is possible to make use of energy from the environment as an extra source in permanent magnet motors. This paper presents a new term “coefficient of performance’ which may be used to more adequately describe the energy transfer of such an electromechanical system. This paper also shows the physics behind one possible explanation for this phenomenon. The physics is explained by first considering how the Lorentz gauge is used to give us the design equations used today. The Lorentz gauge is then discarded to show how the current density of the vacuum may be included in the Maxwell-Ampere equation. They term asymmetrical regauging is introduced for this procedure. The particle physics explaining the vacuum polarity is introduced. A thorough investigation for practical application of this new equation is encouraged by suggestion that further study be applied to the “Wankel motor”. Future work is planned to study the magnetic motor to ascertain the exact mechanism involved that allows this motor to exchange energy with the vacuum. REFERENCES [1] J. Bedini. “Device and Method of a Back EMF Permanent Electromagnetic Motor Generator”. US: Bedini Technology, Inc. 2002. [2] A. Trzynadlowski. “Introduction to Modern Power Electronics”. New York: John Wiley & Sons. Inc. 1998. [3] K. Annamalai and I. Puri, “Advanced Thermodynamics Engineering”. New York. CRC Press. 2002. [4] D. Griffiths. “Introduction to Electrodynamics”. New York. Prentice-Hall. 1999. [5] B. Thide. “Electromagnetic Field Theory”. Uppsala: Upsilon Books. 2004. [6] P.K. Anastasovski. “Classical Electrodynamics without the Lorentz Condition: Extracting Energy from the Vacuum”, Physica Scripta. Vol. 61, p. 513. 1999. [7] T.D. Lee. “Particle Physics and Introduction to user Field Theory”. New York: Harwood Academic Publishers. 1981. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 63 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 [8] B. Lehnert and S. Roy. “Extended Electromagnetic Theory”. Singapore: World Scientific. 1998. [9] D. Griffiths. “Introduction to Elementary Particles”. New York. John Wiley & Sons, Inc. 1987. [10] [11] [12] 64 M.W. Evans and H. Eckardt. “Spin connection resonance in magnetic motors”. Documento 74 en la serie sobre ECE. URLs: www.aias.us D. Reed. “Beltrami vector fields in electrodynamics – a reason for reexamining the structural foundations of classical field physics?” Modern Nonlinear Optics, Part 3. Second Edition. Advances in Chemical Physics. Vol. 119. Recopilado por Myron W. Evans. John Wiley & Sons. 2001. G. Kasyanov. Phenomenon of electrical current rotation in nonlinear electric systems, Comment on the Violation of the law of charge conservation in the system, New Energy Technologies. Vol. 2 Nº 21, pp. 28-30. 2005. [13] “Motor de Johnson de imanes permanentes”. US Patent 4151431. 1979. [14] H. Torres-Silva. “Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 6-23. 2008. [15] H. Torres-Silva. “Chiral field ideas for a theory of matter”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 36-42. 2008. [16] H. Torres-Silva. “A metric for a chiral potential field”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 91-98. 2008. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008, pp. 65-71 H. Torres-Silva: Podolsky’s electrodynamics under a chiral approach PODOLSKY’S ELECTRODYNAMICS UNDER A CHIRAL APPROACH ELECTRODINÁMICA DE PODOLSKY BAJO UN ENFOQUE QUIRAL H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 12 de diciembre de 2007 Received: September 5, 2007Accepted: December 12, 2007 RESUMEN En este trabajo se muestra que un nuevo esquema conduce a la electrodinámica de Maxwell y a la electrodinámica de Podolsky, partiendo con relaciones constitutivas quirales en lugar de la usual ley de Coulomb. Palabras clave: Electrodinámica de Podolsky, ecuaciones de Maxwell. ABSTRACT In this paper we show that a new approach leads to Maxwell’s and Podolsky’s electrodynamics, provided we start from chiral constitutive relations instead of the usual Coulomb’s law. Keywords: Podolsky’s electrodynamics, Maxwell’s equations. INTRODUCTION “On the Question of Obtaining the Magnetic Field, Magnetic Force, and the Maxwell Equations from Coulomb’s Law and Special Relativity”, where it can be shown that any attempt to derive Maxwell equations from Coulomb’s law of electrostatics and the laws of special relativity ends in failure unless one makes use of additional assumptions. Kobe [1] gave the answer: all one needs to arrive at Maxwell equations is (i) Coulomb’s law; (ii) the principle of superposition; (iii) the assumption that electric charge is a conserved scalar (which amounts to assuming the independence of the observed charge of a particle on its speed [2]; (iv) the requirement of form invariance of the electrostatic field equations under Lorentz transformations, i.e. the electrostatic field equations are thought as covariant space-space components of covariant field equations. Neuenschwander and Turner [3] obtained Maxwell equations by generalizing the laws of magnetostatics, which follow from the Biot-Savart law and magnetostatics, to be consistent with special relativity. 1 The preceding considerations leads us to the interesting question: what would happen if we followed the same route as Kobe did, using an electrostatic force law other than the usual Coulomb’s one? We shall show that if we start from the force law proposed by Podolsky [4], i.e., F( r) = QQ ' 4 0 æ 1 − er / a e − r / a ö r − ç ra ø r è r2 (1) where a is a positive parameter with dimension of length, Q and Q’ are the charges at r and r = 0, respectively, and F(r) is the force on the particle with charge Q due to the particle with charge Q’ and if we follow the steps previously outlined, we arrive at the outstanding electrodynamics derived by Podolsky in the early 40 s. In other words, we shall show that the same route that leads to Maxwell equations leads also to Podolsky equations. A notable feature of Podolsky’s generalized electrodynamics is that it is free of those infinities which are usually associated with a point source. For instance, (1) approaches a finite value QQ’/80 a2 as r approaches zero. Thus, unlike Coulomb’s law, Podolsky’s electrostatic force law is finite in the whole space. Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 65 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 In Sec. II we derive the equations that make up Podolsky’s electrodynamic under the chiral approach [5, 7, 11]. In Sec. III we arrive at Podolsky’s field equations by generalizing the equations of Sec. II, so that they are form invariant under Lorentz transformations. For consistency, we show in Sec. IV that (1) is indeed the electrostatic force law related to Podolsky’s theory. The conclusions are presented in Sec. V. Natural units = c = 1,are used throughout. As far as the electromagnetic theories are concerned, we will use the Heaviside-Lorentz units with c = 1. To begin with let us establish some conventions and notations to be used from now on. We use the metric tensor æ1 0 0 0 ö ç 0 −1 0 0   =ç ç 0 0 −1 0  ç  è 0 0 0 −1ø with Greek indices running over 0, 1, 2, 3. Roman indices i, j etc, - denote only the three spatial components. Repeated indices are summed in all cases. The space-time four vectors (contravariant vectors) are x µ = (t , x ) , and the covariant vectors, as a consequences are x µ = (t , − x ) . The four-velocities are found, according to dx µ = (1, v ) d uµ = (1, − v ) uµ = where is the proper time (d2 = dt 2 – dx2), and denotes dt / d = (1–v2) –1/2. Let us then generalize (6) so that it satisfies the requirement of form invariance under Lorentz transformations. To do that, we write the mentioned equation in terms of the Levi-Civita density nml, which equals +1 (–1) if n, m, l is an even(odd) permutation of 1, 2, 3, and vanish if two indices are equal. The curl equation becomes jkl∂El = 0 (2) It we define the quantities 66 F0i = -F0i = Ei = –Ei jkl∂kF0l = 0 We imagine now the curl law to be the space-space components of a manifestly covariant field equation (invariance under Lorentz transformations). As a result, we get µv ∂vF = 0 (4) where F is a completely antisymmetric tensor of rank four with 0123 = + 1. CHIRAL FIELD EQUATIONS µ = µ Equation (10) can be rewritten as (3) Of course, this generalization introduces the components F00, F01, and Flk, for which at this point we lack a physical interpretation. Note that the F0i are not necessarily static anymore. On the other hand, as is well-known, the charge density is defined as the charge per unit of volume, which has as a consequence that the charge dq in an element of volume d 3 x is dq = d 3x. Since dq is a Lorentz scalar [3], transforms as the time-component of a four-vector, namely, the time-component of the charge-current four-vector u = ( , j ) . The electric charge, in turn, is conserved locally [3], which implies that it obeys a continuity equation ∂µ j µ = 0 (5) Assuming e j t time dependence, Maxwell’s time-harmonic equations [1] for isotropic, homogeneous, linear media are Ñ × E = –jB Ñ • B = 0 (6) Ñ × H = –jD + J Ñ • D = (7) Chirality is introduced into the theory by defining the following constitutive relations to describe the isotropic chiral medium [5, 7] D = E + T Ñ × E (8) B = µ H + µT Ñ × H (9) Where the chirality admittance T indicates the degree of chirality of the medium, and the y µ are permittivity and permeability of the chiral medium, respectively. In natural units = 1, µ = 1 (the factor 1/4 is absorved in Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Podolsky’s electrodynamics under a chiral approach the current value). Since D and E are polar vectors and B and H are axial vectors, it follows that and µ are true scalars and T is a pseudoscalar. This means that when the axes of a right-handed Cartesian coordinate system are reversed to form a left-handed Cartesian coordinate system, T changes in sign whereas and µ remain unchanged. Since Ñ • B = 0 always, this conditions will hold identically if B is expressed as the curl of a vector potential A since the divergence of the curl of a vector is identically zero. Thus by rearranging equation (7) we have (1 − ko2T 2 )Ñ × B = j E + ( J + T Ñ × J ) (1 + T 2 )∂ F µ = j µ 0 0 0 + jordinary » jordinary Now if j 0 = jchiral , we can imagine now a particle of mass m and charge Q at rest in a lab frame where there is an electrostatic field E. Newton’s second law allows us to write dp = QE dt dp = Q E = Qu 0 E d where u 0 is the time part of the velocity four-vector u µ. For the component along de xi direction, we have (1 + T 2 ∂t 2 )Ñ × B = j E + ( J + T Ñ × J ) dpi = Qu 0 F 0 i d or (1 + T 2 ∂t 2 )(Ñ × B − ∂E ∂ ∂2 ) = (−T 2 2 E + 2T Ñ × E ) + ( J + T Ñ × J) ∂t ∂t ∂t (11) In order that the right-hand side of this equation transforms like a space-component of a four-vector, it must be rewritten as dpi = Qu F i d In relativistic form we have (1 + T 2 ∂i ∂i )∂ j E j = j 0 (12) 0 0 where j 0 = jchiral is given by a chiral current + jordinary plus a ordinary current of electrons and protons, the chiral current is given in ref [1], ∂i = ∂ / ∂x i and ∂i = ∂ / ∂xi . Note 2 i 0j 0 that ∂i = −∂i Using (11), yields (1 + T ∂i ∂ )∂ j F = j In order that the left-hand side of the preceding equation transforms as the time-component of a four-vector, we must write it as (1 + T 2 )∂ j F 0 j = j 0 whose covariant generalization is dp µ (16) = Qu F µ d If (16) is multiplied by p µ = muµ, where m is the rest mass, the result is 1 d ( p p µ ) = Qmuµ u µ F µ 2 d µ However, 2 pµ p µ = m 2 2 (1 − v ) = m 2 2 −2 = m 2 where (15) In terms of the proper time this becomes (10) In terms of (3) can now be rewritten as (14) Therefore, we come to the conclusion that i µ 2 2 2 = ∂i ∂ = ∂ µ ∂ = ∂ / ∂t − Ñ (13) The requirement of form invariance of this equation under Lorentz transformations leads then to the following result uµ u µ F µ = 0 Using this result Kobe [1] and Neuenschwander and Turner [3] showed that Fvµ is an antisymmetric tensor Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 67 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 (Fvµ = –Fµv). Since Fvµ is an antisymmetric tensor of second rank, it has only six independent components, three of which have already been specified. We name therefore the remaining components containing the Lorentz force. For v = 0, (15) assumes the form dU = Qv ⋅ E dt 1 B = ilm Flm 2 (23) i (17) Note that F kl = −klj B j . Writing out the components of (17) explicitly, B1 = F23 = F 23 = − B1 B 2 = − F13 = − F 13 = − B2 B 3 = F12 = F 12 = − B3 Hence, a clever physicist who were only familiar with Podolsky’s electrostatics and special relativity could predict the existence of the magnetic field B , which naturally still lacks physical interpretation. The content of (12) and (14) can now be seen. For µ = 0, (12) gives Ñ ⋅ B = 0 (18) where U = p 0 is the particle’s energy. Accordingly, our smart physicist, who was able to predict the B field only from its knowledge of electrostatics and special relativity, can now-by making judicious use of (22) and (23) - observe, measure and distinguish the B field from the E field of (15). The new field couples to moving electric charge, does not act on a static charged particle, and, unlike the electrostatic field, is capable only of changing the particle’s momentum direction. Equations (18-21) make up Podolsky’s higher-order field equations. Of course, in the limit T = 0, all the preceding arguments apply equally well to Maxwell’s theory. Two comments fit in here: (1) Equation (14) is consistent with the continuity equation (13). In fact, if the divergence of (14) is taken, we obtain (1 + T 2 )∂ µ ∂ F µ = ∂ µ j µ showing that there are no magnetic monopoles in Podolsky’s electrodynamics, while for µ = i we obtain ∂B Ñ×E=− ∂t (19) which says that time-varying magnetic fields can be produced be B fields with circulation. The components µ = 0 and µ = i of (14) give, respectively, (1 + T 2 ) Ñ ⋅ E = (1 + T 2 )( Ñ × B − ∂E )= j dt (20) (21) For v = i, (16) becomes 68 (2) As was recently shown [8], it is not necessary to introduce a formula for the force density fµ representing the action of the field on a text particle. We have only to assume that (– fµ) is the simplest contravariant vector constructed with the current jµ and a suitable derivative of the field Fµv. Applying this simplicity criterion to Podolsky’s electrodynamics, we promptly obtain f µ = − F µ j which are nothing but a generalization of Gauss and Ampère- Maxwell laws in this order. Since Fµv is an antisymmetric tensor ∂ µ ∂ F µ is identically µ zero. On the other hand, according to (14) ∂ µ j . Thus, the equation in hand is identically zero; where, as we have already mentioned, j µ = ( , j ) . Therefore, f 0 = − F 0 i ji = E ⋅ j and dp = Q( E + v × B) dt (22) f k = − F k j = F 0 k j0 + F ik ji = ( E + j × B) k Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Podolsky’s electrodynamics under a chiral approach Thus, the force density for Podolsky’s electrodynamics is the same as that for Maxwell’s electrodynamics, namely, the well-known Lorentz force density. THE FORCE LAW FOR PODOLSKY’S ELECTROSTATICS V ( k ) = (2 )3 / 2 T 2 k 2 ( k 2 + So, V (r ) = We show now that (1) is indeed the force law for Podolsky’s electrostatics. It follows That F = QE E = −ÑV V (r ) = (1 − T 2 Ñ 2 )Ñ 2V (r ) = − (r ) For a charge Q at the origin of the radius vector this equation reduces to (1 − T 2 Ñ 2 )Ñ 2V (r ) = −Q 3 (r ) V ( k ) = − i k ⋅r ò d ke V (k ) 3 (2 )3 / 2 1 (2 )3 / 2 3 ik ⋅r d re V (r ) ò (26) (27) where d 3 k and d 3r , respectively, stands for volumes in the three-dimensional k-space and the coordinate space. If we substitute (26) into (25) and take into account that 3 (r ) = we obtain 1 (2 )3 / 2 ) e − i k ⋅r k 2 (k 2 + 1 T2 ) 1 − e−r / T ) r (2 )2 r Q ( 1 − e−r / T e−r / T r E = −ÑV (r ) = Q / 4 ( − ) rT r (28) r2 It follows then the force law for Podolsky’s electrostatics is QQ ' 1 − er / a e − r / a r ( − ) 4 0 ra r r2 F( r) = (25) We solve this equation using the Fourier transform method. First we define V ( k ) as follows: 1 T2 Accordingly, the electric field due to a charge Q at the origin is given by Eq. (20) can then be rewritten as V (r ) = òd k 3 (2 )3 T 2 (24) (r ')(1 − e − R / T ) V (r ) = ò d 3r ' 4 R Q 1 Integral (28) may be found in any textbook on the theory of functions of a complex variable [8]. As a result, where Q which is nothing but the force law for which we were looking (see Eq. (1)). Recently an algorithm was devised which allows one to obtain the energy and momentum related to a given field in a simple way [8]. Using this prescription we can show that in the framework of Podolsky’s electrostatics the energy is given by field = 1 3 é 2 d x E + T 2 (Ñ ⋅ E ) 2 ù ë û 2ò Making use of the expression for the electrostatic field we have just found, we promptly obtain 3 − i k ⋅r d ke ò field = Q2 2T which tells us that the energy for the field of a point charge has a finite value in the whole space. This is indeed a important feature of Podolsky’s generalized electrodynamics. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 69 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Equations (28) and Ñ × E = 0 are the fundamental laws of Podolsky’s electrostatics. We will slightly analyze an interesting feature of Podolsky’s electrostatics by computing the flux of the electrostatic field across a spherical surface of radius R with a charge Q at its center. Using (28) we arrive at the result ò E ⋅ dS = Q(1 − (1 + R / T )e R/T ) Rs = 2mGN / c 2 = T which tells us that ò E ⋅ dS = 0, ò E ⋅ dS = Q, R > T Therefore, a sphere of radius R || whereas an antiparticle state like (1.3) contains its hidden particle field under the condition |c|>|c|. Fourth, to clarify further why we prefer the new postulate (1.6) instead of C transformation and CPT theorem, we wish to emphasize an important difference between a postulate (law) and a theorem. All quantities in a theorem must be defined in advance separately and unambiguously and the outcome of the theorem is actually contained in its premise implicitly. For example, the definitions of C, P and T are all clear in mathematics and the validity of CPT theorem is ensured by the basic principles of SR and QFT. Once C, P and T are not conserved in experiments, they cease to be meaningful as physical transformations. In this situation, the CPT theorem immediately exhibits itself as a new postulate (1.12) in which the definition of transformation of particle to antiparticle is just contained in the same equation. In general, a postulate or law can often (not always) accommodate a definition of physical quantity, and the validity of the postulate (law) together with the definition must be verified by experiments. Hence the establishment of a law is a process “from particular to general” or an outcome of “analysis and induction method”. By contrast, to prove a theorem from well-established theories is a process “from general to particular” or a consequence of “deduction method”. As is well known, the EP served as a starting point in establishing the theory of general relativity (GR). The possible invalidity of EP in the presence of antimatter implies that GR is dealing with the gravitation of pure matter without the coexistence of antimatter. Indeed, let us look at the Einstein field equation [9]: For example, the definition of gravitational mass m is contained in the gravitational law. The definition of inertial mass m is contained in Newton’s law. What we have done is a similar thing - the definition of particle-antiparticle transformation C is contained naturally in a new postulate (1.6) - not one that comes from elsewhere. (where the superscript c means antimatter,) since under c c Õ −Tµ . the mass inversion, Tµ Õ −Tµ and Tµ Notice that the form of the energy-momentum tensor is the same for both matter and antimatter. We stress once again that the distinction between m and –m is merely relative, not absolute. So in the whole universe eff c (matter+antimatter) we have Tµ = Tµ − Tµ º 0 and the Einstein equation is Generalization of Einstein field equation in general relativity Consider a positronium and an atom of matter. If Newton Equation is correct, there will be no gravitational force between them. This means that the gravitational mass m (grav.) of positronium is zero! However, the energy or the relevant inertial mass m (inert.) of positronium is definitely nonzero. Hence we see that in the case of coexistence of particles and antiparticles, the equivalence principle (EP) in the (weak) sense that [8] m (grav.) = m (inert.) 106 (1.13) On the left side, the Ricci tensor Rµv, curvature scalar R and the metric tensor gµv are all functions of coordinates xµ. While on the right side, the energy-momentum tensor Tµv is introduced to describe the existence of matter in the vicinity (a macroscopic small volume) of xµ. Then under a transformation of mass inversion mÕ–m to reflect that of matter to antimatter, Tµv should change its sign due to its proportionality to the mass m. Hence Eq. (1.13) changes sign on the right side whereas not on the left side. This reflects the fact that GR is a classical field theory and so cannot treat the matter and antimatter on an equal footing. To keep Eq. (1.13) invariant under the mass inversion, we manage to modify its right side by a generalization as: eff c Tµ Õ Tµ = Tµ − Tµ , Rµ − 1 g Rº 0 2 µ (1.14) (1.15) APPENDIX 2: THE COSMOLOGICAL CONSTANT PROBLEM Let’s outline shortly the cosmological constant problem. Consider Einstein equation with Δ -term ( = c = 1 ): (1.12) cannot be valid. 1 Rµ − gµ R = −8 GTµ 2 1 Rµ − gµ R = 8 GTµ + gµ 2 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 (2.1) H. Torres-Silva: Chiral universes and quantum effects produced by electromagnetic fields Here Tµv is energy-momentum tensor of the matter and is some constant parameter having the dimension [cm-2]. In the used unit system the Newtonian gravitational constant G lP2 2, 5 ⋅ 10 −66 cm 2 (2.2) and according to experimental data the mean energy density today is of the order Tµ 1 108 cm −4 Õ 8 GTµ 5 ⋅ 10 −57 cm −2 (2.3) In the vacuum interface (see figure 2), the Einstein equation is described by em Rµ = 8 GTµ + P gµ (2.8) It is clear that the interaction of fields doesn’t changes qualitatively the estimation (2.7). From (2.7) and (2.3) we see that the contribution to the right hand side of Eq. (2.1) estimated in the framework of canonical quantum field theory is larger about 10120 times in comparison with the experimental estimations. and 10 −56 cm −2 (2.4) Thus, if Einstein equation (1) is used for description of the today dynamics of our R-Universe, the quantities in its right hand side are of the same order indicated in (2.3) and (2.4). In our model, for the L-Universe, we have Tµv < 0, < 0. Now let us estimate the possible value of the right hand side of Eq. (2.1) in the framework of canonical quantum field theory. For simplicity consider energy-momentum tensor in quantum electrodynamics in flat spacetime: 1 1 i Tµ = − ( Fµ F − µ F 2 ) + ( ( µ Ñ ) − Ñ( µ ) ) (2.5) 4 4 2 Casimir effect, predicted in [1] and experimentally verified in [2], shows for reality of zero-point energies. Moreover, the attempts to drop out zero-point energies by appropriate normal ordering of creating and annihilating operators in energy-momentum tensor fail for many of reasons (the discussion of this problem see, for example, in [3]). Thus, at estimating vacuum expectation value of energy-momentum tensor (2.5), it should not be performed normal ordering of creating and annihilating operators in (2.5). Thus we obtain for vacuum expectation value of tensor (5) in free theory: Tµ 0 = æ d 3 k ç kµ k 3 ç k0 èç ò (2 ) −2 k0 = k kµ k k o k 0 = m2 + k 2 ö   ø (2.6) Here m is the electron mass. The first item in (2.6) gives the positive contribution but the second item gives the negative contribution since these items give the boson and fermion contributions to vacuum energy, respectively. If integration in (2.6) is restricted by Planck scale, kmax ~ lP−1 , then from (2.6) and (2.2) it follows: 8 G Tµ 0 lP−2 1066 cm −2 P (2.7) APPENDIX 3: COSMOLOGICAL IMPLICATIONS It is instructive to examine some of the cosmological implications of the present proposal. When light traverses intergalactic space it displays a Doppler frequency shift, invariably interpreted in terms of receding sources, and therefore assumed to imply an expanding universe. This interpretation is not inconsistent with the present model, but neither is it a necessary consequence. Curvature of three-dimensional space along a time coordinate implies that a distant source is separated from an observer in both space and time. During transit, the photon moves towards an observer ahead of it in time and therefore appears to lag as if its source was receding. The observed red shift, as before, is a function of separation, but the proportionality constant does not necessarily relate distance to rate of recession, but rather to a time separation interval, Δt, ) Δt, zc = rH = r éë1 t − 1 ( t + Δt ùû , where t=r/c. Hence: ) z = 1 − t ( t + Δt . This formulation allows the calculation of a Hubble radius rather than a Hubble age of the universe. As Δt Õ ∞ , H Õ 1 t 0 = c r0 , where the maximum value of the interval r0 = ct0 is interpreted as an upper bound to some radius of the universe. This effective radius corresponds to 4x109 parsec. Converted to a dimensionless distance, one has N1 = r0 m e c 2 e 2 = t 0 m e c3 e 2 » 10 40 . Using this value with the dimensionless mass, or number of baryons, of the universe, N3 = 1080, a mass density of = N 3 N13 = 10 −40 is calculated. This matches the third large number N 2 = e 2 / (4 0 Gm p me ) = 2.3 × 1039 , Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 107 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 the gravitational coupling constant, as required by Mach’s principle, without invoking anthropic principles. There is no reason why any of these large numbers should change with time. The foregoing does not demand a static universe. As time flows, observers move along the interface and therefore experience a gradually changing curvature. Time evolution amounts to the threading of the interface through the universe, and the consequent changes in configuration are like pseudo-rotation, which increases the isotropic appearance of the universe. The resulting relative motion would probably be not unlike that of an expanding universe. In-so-far as an abstract surface like P2 does not exist in less than four dimensions, the vacuum interface must likewise be four-dimensional. Its three-dimensional aspect differentiates between the chiral forms of matter, and by analogy, the four-dimensional vacuum could conceivably differentiate between opposite directions in time. A closed journey along the interface would gradually turn positive into negative time flow as was shown for the chirality of matter, suggesting that time has no unique beginning. The same conclusion is reached by modern quantum cosmologies. Hawking’s idea that the universe is finite but has no boundary in imaginary time, may indeed be fully consistent with the four-dimensional chiral structure arrived at here. The model of the universe with two chiral spaces The model of the homogenous and isotropic universe with two spaces is considered. The background space is a coordinate system of reference and defines the behaviour of the universe. The other space characterizes the gravity of the matter of the universe produced by electromagnetic waves. In the presented model, the first derivative of the scale factor of the universe with respect to time is equal to the velocity of light. The density of the matter of the universe changes from the Plankian value at the Planck time to the modern value at the modern time. The model under consideration describes the whole universe from the Planck time to the modern time and avoids the problems of the Friedmann model such as the flatness problem and the horizon problem. As known [14, 15], the Friedmann model of the universe has fundamental difficulties such as the flatness problem and the horizon problem. These appear to be a consequence of that the space of the Friedmann universe, on the one hand, is defined by the gravity of the matter of the universe, and on the other hand, 108 is a coordinate system of reference. The solution of the problem is to introduce the background space as a coordinate system of reference. In this case, the background space defines the behaviour of the universe, and the other space characterizes the gravity of the matter of the universe. Let us consider the model of the homogenous and isotropic universe with two chiral spaces. Let us introduce the background space as a coordinate system of reference. Then the evolution of the universe is described as a deformation of the background space. Let us take the homogenous and isotropic background space, with the spatial interval of its metric is given by ds 2 = a 2 dl 2 2 é1 + kl 2 ù 4 ú êë û (3.1) Suppose that the background space is defined by the total mass of the universe including the mass of the matter and the energy of gravity produced by electromagnetic waves Gik = Tiktot = Tikem = Tik + tik . (3.2) Let us consider the case when the total mass of the universe is equal to zero. It means that the Maxwell tensor vanishes, Tikem = 0, but with det F k ≠ 0 . i Tiktot = Tik + tik = 0 (3.3) Then eq. (3.2) take the form Gik = 0. (3.4) The solution of the equations (3.4) gives Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 d 2a =0 (3.5) da = c. dt (3.6) dt 2 H. Torres-Silva: Chiral universes and quantum effects produced by electromagnetic fields Thus the second derivative of the scale factor of the universe with respect to time is equal to zero, and the first derivative of the scale factor is equal to the velocity of light. It should be noted that the scale factor of the universe coincides with the size of the horizon a = ct. (3.7) In the model (3.4), the laboratory coordinate system is synchronous. In the laboratory coordinate system, the background space is described by the flat metric ds 2 = c 2 dt 2 − a 2 dl 2 . (3.8) Thus we arrive at the Milne model [16] in which the size of the universe being the maximum distance between the particles coincides with the scale factor of the universe and coincides with the size of the horizon. In the universe with one space, the Milne model is derived from the condition that the density of the matter tends to zero Õ 0. Here the Milne model describes the background space of the universe, with the total mass of the universe being equal to zero mtot = 0. Let us determine the relationship between the lifetime of the universe and the Hubble constant. Since the Hubble constant is H= 1 da , a dt (3.9) so from (3.6), (3.7), (3.9) one can obtain t= 1 . H (3.10) Allowing for (3.7) and (3.10), from (3.11) it follows that the mass of the matter changes with time as m= c 2 a c 3t c3 = = , G G GH (3.12) and the density of the matter, as = 3c 2 3 3H 2 = = 2 2 4 G 4 Ga 4 Gt (3.13) According to (3.12), growth of the mass of the matter takes place during all the evolution of the universe. At the Planck time tp1, the mass of the matter is equal to 1/ 2 the Planck mass mPl = (c / G ) . At present, the mass 56 of the matter is m0 » 1.4 ⋅ 10 g , and the density of the -29 -3 matter is 0 » 3.2 ⋅ 10 g cm . Thus the model of the universe (3.3)-(3.6) provides growth of the mass of the matter from the Plankian value to the modern one. We have considered the model of the homogenous and isotropic universe with two spaces, with the behaviour of the universe is defined by the background space. Unlike the Friedmann model, the presented model gets rid off the flatness and horizon problems. Remind [14, 15] that the horizon problem in the Friedmann universe is that two particles situated within the horizon at present were situated beyond the horizon in the past. In the universe under consideration, all the particles are situated within the horizon during all the evolution of the universe, since the size of the universe being the maximum distance between the particles coincides with the size of the horizon. Hence the presented model avoids the horizon problem. Let us estimate the size of the universe at the Planck time and at present. Remind that the size of the universe coincides with the scale factor of the universe. According 5 1/ 2 to (3.7), at the Planck time t Pl = (G / c ) , the scale factor of the universe is equal to the Planck length TPl = lPl = (G / c3 )1/ 2 . According to (3.7), (3.9), for the modern Hubble constant H 0 » 3 ⋅ 10 −18 c-1 , the modern scale factor of the universe is a0 » 10 28 cm . Remind [14, 15] that the essence of the flatness problem in the Friedmann universe is impossibility to get the modern density of the matter starting from the Planck density of the matter at the Planck time. In the presented theory, the density of the matter of the universe changes from the Planckian value at the Planck time to the modern value at the modern time. Hence the flatness problem is absent in the presented theory. Let us determine the relationship between the mass of the matter and the scale factor of the universe at t = const. The total mass of the universe is equal to zero, given the mass of the matter is equal to the energy of its gravity In order to resolve the above problems of the Friedmann universe an inflationary episode is introduced in the early universe [14, 15]. Since the presented model describes the universe from the Planck time to the modern time and avoids the above problems of the Friedmann universe, there is no necessity to introduce the inflationary model. m= Gm c2 a . (3.11) Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 109 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 REFERENCES [9] N.A. Bahcall, J.P. Ostriker, S. Perimutter and P.J. Steinhardt. “The cosmic triangle: revealing the state of the universe”. Science. Vol. 284, pp. 1481-1488. May 28 1999. [10] W. Freedman. “The Hubble constant and the expanding universe”. American Scientist, Vol. 91, pp. 36-43. Jan-Fe, 2003. [1] H. Bondi. “Negative mass in general relativity”. Rev. Mod. Phys. Vol. 29, pp. 423-428. 1957. [2] J.D. Bjorken and S.D. Drell. “Relativistic Quantum Mechanics”. McGraw-Hill Book Company. 1964. [3] J.J. Sakurai. “Modern Quantum Mechanics”. John Wiley & Sons Inc. 1994. [11] M. Livio. “Moving right along”. Astronomy. pp. 34-39. July 2002. [4] G.J. Ni and S.Q. Chen. “Advanced Quantum Mechanics”. Chinese Ed. Press of Fudan University. 2000. English Ed. Rinton Press. 2002. [12] P. Davies. “Seven wonders”. New Scientist. September 21 2002. [5] G.J. Ni, H. Guan, W.M. Zhou and J. Yan. “Antiparticle in the light of Einstein-PodolskyRosen paradox and Klien paradox”. Chin. Phys. Lett. Vol. 17, pp. 393-395. 2000. [13] H. Torres-Silva. “Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación”. Ingeniare. Rev. chil. ing. Vol. 16 1, pp. 6-23. 2008. [6] G.J. Ni. “Ten arguments for the essence of special relativity”. Proceedings of the 23rd Workshop on High-energy Physics and Field Theory, pp. 275-292. Edit: I.V. Filimonova and V.A. Petrov. Protvino, Russia. June 2000. [14] A.D. Dolgov, Ya.B. Zeldovich and M.V. Sazhin. “Cosmology of the early universe”. Moscow Univ. Press. Moscow. 1988. [15] A.D. Linde. “Elementary particle physics and inflationary cosmology”. Nauka. Moscow. 1990. [7] L. Smolin. “Three Roads to Quantum Gravity”. Basic Books, p. 149. 2001. [16] Ya. B. Zeldovich and I.D. Novikov. “Structure and evolution of the universe”. Nauka. Moscow. 1975. [8] S. Weinberg. “Gravitation and cosmology”. John Wiley. 1972. [17] L. Landau and E.M. Lifshitz. “The classical theory of fields”. 4th Ed. Pergamon. Oxford. 1976. 110 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: A new relativistic field theory of the electron Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008, pp. 111-118 A NEW RELATIVISTIC FIELD THEORY OF THE ELECTRON UNA NUEVA TEORÍA RELATIVÍSTICA DE CAMPO PARA EL ELECTRÓN H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 21 de diciembre de 2007 Received: September 5, 2007Accepted: December 21, 2007 RESUMEN En este trabajo se presenta un examen cualitativo sobre una nueva Teoría General Relativística para el electrón, con la obtención de la ecuación de Dirac a partir de los campos electromagnéticos con el campo eléctrico paralelo al campo magnético. El principio rector es el de la relatividad general, y la principal hipótesis es que de las ecuaciones fundamentales se desprende la teoría de Dirac y la teoría de Maxwell - Lorentz como de dos casos especiales cuidando la coherencia y compatibilidad entre las condiciones en las que las ecuaciones fundamentales se reducen a la ecuación de Dirac y las ecuaciones de Maxwell - Lorentz. Se espera que la presente investigación arroje alguna luz sobre las desconcertantes dificultades a las que nos encontramos en la comprensión del comportamiento de un electrón exclusivamente en función de la ecuación de Dirac y las ecuaciones de Maxwell - Lorentz. Más allá de esto, se puede investigar la posibilidad de que otras partículas elementales se puedan regir por las mismas ecuaciones fundamentales bajo variadas condiciones restrictivas. Palabras clave: Ecuación de Dirac, tensor de materia, sistema Einstein-Maxwell. ABSTRACT In this paper we present a qualitative discussion of a new General Relativistic Field Theory for the electron, obtaining the Dirac equation from electromagnetic fields with the electric field parallel to the magnetic field. The guiding principle is that of general relativity, and the main hypothesis is that the fundamental equations embrace the Dirac theory and the Maxwell-Lorentz theory as of two special cases respectively. We concern ourselves with the consistency and compatibility among those conditions under which the fundamental equations are reduced to the Dirac equation and the MaxwellLorentz equations. We expect that the present inves encounter in comprehending the behavior of an electron solely according to the Dirac equation and the Maxwell-Lorentz equations. Beyond this, we aim to investigate the possibility that other elementary particles are governed by the same fundamental equations under varied restrictive conditions. Keywords: Dirac equation, matter tensor, Einstein-Maxwell system. INTRODUCTION Einstein’s Dream Albert Einstein spent several years of his life trying to develop a theory which would relate electromagnetism and gravity to a common “unified field”. Hence the name unified field theory. 1 After Einstein finished his first article on the unified field theory in 1922, despite criticism he spent much of the second half of his life pursuing the development of the unified field theory besides the discussion of completeness of quantum mechanics. In the first several years, he was very optimistic, thought success would come soon, but he found it was full of difficulties afterwards. He considered mathematical tools in being was not sufficient, then turned Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 111 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 to study mathematics, but never obtained any result with real physical sense. Because Einstein wanted to found an encompassing mathematical construct that would unite not only gravitational field but also electromagnetic field under a single set of equations, but then the task has become even more difficult, with the discovery of two other basic field: the weak interaction field and strong interaction field. Most physicists thought Einstein’s quest was hopeless, and in fact he never succeeded. But Einstein was convinced such a basic harmony and simplicity existed in nature, he kept his chin up, went ahead along his own road. Because he was apart from the mainstream of physical research - quantum field theory, he was very alone in his old age, but he was fearless. He still prepared to keep on his mathematical calculation of unified field theory on his sickbed until the day before his death. He said with a sigh before his death: I cannot finish this work, it will be forgotten, but it will be rediscovered in the future. Einstein did manage to develop a theory which “wrapped” electromagnetism and gravitation into a common metric tensor. In one of his formulations of a unified field theory (called Einstein-Schrodinger Theory), gravitation was wrapped into the symmetric part of the metric tensor, while electromagnetism was wrapped into the antisymmetric part of the metric tensor. This wrapping is possible because electromagnetism and gravity share some mathematical similarities. They both have a stressenergy tensor. The electric charge is analogous to the gravitational mass. The magnetic moment is analogous to the angular momentum moment. The electric potential and electric field are analogous to the gravitational potential and gravitational field, respectively. Finally, the magnetic field is analogous to the magneto-gravitic field. The mathematical wrapper which Einstein developed exploits this analogy. However, the analogy between electromagnetism and gravity breaks down at higher field strengths when nonlinear field effects set in. As a result, Einstein-Schrödinger theory correctly describes electromagnetism and gravity at low field strengths where they are not coupled to each other. However, it does not describe the interactions between electromagnetism and gravitation which occur at higher field strengths. Thus, Einstein-Schrödinger theory achieved an approximate mathematical unification, but no real physical unification of electromagnetism and gravity. In this sense, it did not really achieve its objective. Kaluza and Klein developed an alternative wrapper for electromagnetism and gravitation. Instead of wrapping electromagnetism into the antisymmetric part of the metric tensor, they retained a symmetric metric tensor but added a fifth dimension. They were able to show that 112 Maxwell’s Laws and General Relativity can be expressed in terms of their five-dimensional metric tensor. Again, this exploits the analogies between electromagnetism and gravity. The problem with Einstein’s unified field theory and Kaluza-Klein’s unified field theory is that they don’t address the fundamental issue. They still treat gravitation and electromagnetism as two completely separate interactions. Neither theory can tell you how a gravitational field is fundamentally produced by a charged particle (electron). Today, the search for a unified field theory has been replaced by loftier goals. Physicists are now looking for a so-called Theory of Everything (TOE) which will unify not only electromagnetism and gravity, but also the nuclear interactions and other potential physical forces such as inflation and “dark energy”. At the time of Einstein, modern particle physics had not yet been developed and the strong and weak nuclear interactions were not well understood. Within of the unified program a fundamental question was if gravitational fields did play an essential part in the structure of the elementary particles of matter (electron). The first unimodular theory was developed by Einstein in 1919, assuming as source the Maxwell tensor, where the quantum electron theory was not reproduced. Thus, as stated by Einstein in 1919, “we cannot arrive at a theory of the electron [and matter generally] by restricting ourselves to the electromagnetic components of the Maxwell-Lorentz theory, as has long been known” [6]. Motivation In the beginning of this century, Lorentz, Poincaré, Abraham, Mie and others attempted to show that the constitution of an electron be explained as a field of electromagnetic nature. In order to make the motion of the electron special-relativistic, however, it was necessary to consider a mechanical core (Poincaré) or to introduce some nonlinearity (Mie) in the electromagnetic field under consideration [1-3]. To overcome these difficulties appeared to have completely been resolved with the Dirac equation for the electron discovered in 1928. It has conventionally been believed that the information of an electron near its core is fully provided by the Dirac equation. The notion of the electron formed by the conventional interpretation of the Dirac equation is hardly acceptable as rational and feasible. Instead, in an accompanying paper, it was shown that the Dirac equation has a solution that indicates that an electron is a field localized in space and deformable and that the motion of this “elementary field” is determinate and causal. Moreover, is shown, it is possible to regard the field governed by the Dirac equation as if electromagnetic. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: A new relativistic field theory of the electron This possibility is not surprising if we note that the intrinsic magnetic moment of an electron is a notion to be comprehended only in the context of Faraday-MaxwellLorentz’s theory of electricity and magnetism. Thus we are led to infer the following: An electron is a localized field of which some part remote from its center may well be regarded as normal electromagnetic, and some other part near its center is governed by the Dirac equation derived from parallel fields. The connection between the two parts must be continuous and gradual, and there is no clear-cut border between them. A real electron, as a whole, must be a unified field governed by a common set of partial differential equations. It is important to anticipate the possibility that those fundamental equations governing the field be reduced to the Maxwell-Lorentz equations under a restrictive condition and to the Dirac equation under another restrictive condition. Although with the early theories of Einstein and others [6-7], there is no deductive way of giving the fundamental equations, it is not difficult to anticipate the following: a. The electronic mass has its representation in the Dirac equation, but not in the Maxwell-Lorentz equations. On the other hand, the electronic charge is seen in the Maxwell-Lorentz equations, but not in the Dirac equation for a free electron. We infer from these observations that the electronic mass and charge are approximate substitutes of field variables that are functions of time and space in the fundamental equations. Only because the variables are comparatively less variants, they may be replaced with constants as depending on conditions of observation. b. The Maxwell-Lorentz equations are covariant under the Lorentz transformation, however, the covariance of the Dirac equation under the same transformation is conditional. Besides, the field variables in MaxwellLorentz equations and those in the Dirac equation are apparently of different characteristics under the Lorentz transformation. In order to embrace those two sets of equations as of special cases, the fundamental equations must be formed in a geometrical frame less restrictive than the Euclidean, i.e., as covariant for observers in varied conditions [8-9]. These difficulties are overcomes with our Maxwell’s Equations with parallel electromagnetic fields, (see accompanying paper). Considering those observations in the above, it is significant to recall the demonstration of the similarity between the Dirac field and the electromagnetic field. In the demonstration, we see a clue to electing a set of fundamental equations that are reducible to both the Dirac equation and the Maxwell-Lorentz equations. In paper (Physical interpretation of the Dirac equation with electromagnetic mass), we considered the Dirac equation for a free electron derived from Maxwell’s equations when the electric field is parallel to the magnetic field. The Nature of the Investigation If one accepts as valid the principle of relativity, i.e., the principle of covariance of the laws under coordinate transformations, the choice of a proper scheme of geometry is an essential part of the task of constructing the fundamental equations concerned. In this respect, it is significant to recall that the Dirac equation is not completely covariant under the Lorentz transformation. It appears that the range of the meaning implied by the Dirac equation can no longer be confined in the Euclid space. This situation suggests first that the scheme of geometry be properly generalized and then that the Dirac equation be modified accordingly. We expect, in this way, that the fundamental equations thus found will be able to embrace the Dirac equation and the Maxwell-Lorentz equations as of two special cases respectively. In a geometrical scheme more general than the Euclidean, each component of the metric tensor gij is a function of space-time coordinates. Therefore, it seems to be sensible to expect that any matter field, with no exception, is accompanied by a gravity field. The fundamental equations govern simultaneously the matter field and the metric 1 field is the equation Rij − gij R = − kTij proposed earlier 2 by Einstein [10-11] . The left hand side of the equation is sometimes called the Einstein tensor Gij. One might surmise that a matter field determines uniquely the Einstein tensor of the space where the matter field is located. Thus it appears that the Einstein tensor can be the representation or the image of the matter field. But the uniqueness of the relation between a matter field and the resulting Einstein tensor is unknown. We note that the equations to be found must be regarded only as of an approximate means of representing the reality concerned. (None of the equations utilized in physics may escape this fate.) Therefore, even when a field, e.g., of an electron, governed by those equations has a singularity that implies a strong distortion of space curvature, one can not immediately conclude that the real field, expected to be represented by the solution, has the same singularity. The field equations of general relativity are rarely used without simplifying assumptions. The most common application treats of a mass, sufficiently distant from other masses, so as to move uniformly in a straight line. All applications of special relativity are of this type, in order to stay in Minkowski space-time. A body that Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 113 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 moves inertially (or at rest) is thus assumed to have fourdimensionally straight world lines from which they deviate only under acceleration or rotation. The well-known Minkowski diagram of special relativity is a graphical representation of this assumption and therefore refers to a highly idealized situation, only realized in isolated free fall or improbable regions of deep intergalactic space. In the real world the stress tensor never vanishes and so requires a non- vanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Here, a strain field appears in the curved surface. At any point on the curved manifold the gradient of the strain field is perpendicular to the tangent vector and coincides with the axis of the local light cone. To relieve the stress, the natural response of the mass point is displacement along the stress gradient and hence it traces out a time-like world line at constant spatial coordinates. This displacement, along the time coordinate only, is the arrow of time, which appears as a direct consequence of the curvature of space. There is no time in euclidean space. The primary cause of mass generation by space curvature is elimination of the strict orthogonality between time and space coordinates which allows the strain field (mass point) to acquire complementary time-like and space-like attributes. This is the mechanism envisaged by Corben [4] as a model for creating mass through relativistically invariant self-trapping of a free bradyon and a free tachyon, (time-like and space-like waves). The essence of the argument advanced here is that real world-space is not euclidean and that space is generally curved into the time dimension, consistent with the theory of general relativity. The curvature may not be sufficient to become obvious in a local context. However, it is sufficient to break the time-reversal symmetry that seems to characterize the laws of physics. Not only does it cause perpetual time with respect to all mass, but actually identifies a fixed direction for this It creates an arrow of time and thereby eliminates an inconsistency 114 in the logic of physics: how reversible microscopic laws can underpin an irreversible macroscopic world. General curvature of space breaks the time-reversal symmetry and produces chiral space, manifest in the right-hand force rule of electromagnetism. The fact that most other fundamental laws of physics do not refer the chirality of space, nor the arrow of time, confirms that the curvature on a local scale is barely detectable. The one exception to apparent time-reversal symmetry is the law of entropy. It has been stated [1] that “...the second law of thermodynamics is excluded from the classication fundamental due to its statistical nature”. This is an unconvincing explanation and the curved-space argument provides a better mechanism for entropy production. In any curved-space manifold gradient vectors drive time-like displacement of separate particles along non-parallel world lines. Even among pairs of stationary particles three-dimensional line elements therefore do not remain invariant over a period of time. An initially stationary array of non-interacting particles (ideal gas) spontaneously generates relative internal (zero point) motion leading to chaotic distribution in a container, or spontaneous dispersal in the open. Where local interactions constrain dispersal, zero-point vibration develops. This intrinsic microscopic instability, caused by the curvature of space, is the source of entropy. The conclusions reached here are clearly related to those of Prigogine [5] who deduced that the irreversible creation of matter generates cosmological entropy and that the arrow of time is provided by the transformation of gravitational energy into matter. The difference is that Progonine’s result was obtained by incorporating the second law of thermodynamics into the relativistic field equations, whereas the present model makes no assumption about macroscopic behaviour. Theses observations, usual in classical mechanics, are significant in evaluating Einstein’s attempt recollected in the following. Recollection of Einstein’s Attempt It seems that Einstein devoted the last twenty years, at least, of his life to the attempt of materializing his deterministic view of particles. Einstein explicitly used the term “unified field theory” (gravitation-electromagnetism) in the title of a publication for the first time in 1925. Ten more papers appeared in which the term is used in the title, but Einstein had dealt with the topic already in half a dozen publications before 1925 . In total he wrote more than forty technical Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: A new relativistic field theory of the electron papers on the subject. This work represents roughly a fourth of his overall oeuvre of original research articles, and about half of his scientific production published after 1920. As is well known, however, the endeavor was not fruitful. In retrospect, the cause of his difficulty appears to be in his interpretation of Schrödinger’s wave equation. A clue to knowing Einstein’s interpretation in question is found in an essay published by him in 1936 (Einstein, 1936). His interpretation of wave mechanics may be summarized as follows: i) The wave function does not in any way describe the condition of a single system; it relates rather to many systems, an ensemble of systems, in the same sense as of statistical mechanics, so Schrödinger’s equation determines the time variation that is experienced by an ensemble of systems. ii) Quantum mechanics will not be the point of departure in the search of the foundation of quantummechanical phenomena, just as one cannot go from thermodynamics to the foundation of mechanics, so there must be a field theory that results in a way of representing particles and the representation must be free of singularities. The foundation of the theory is given by the differential equations of the field, and the theory leads also to quantum mechanics in the same way as classical mechanics of particles leads to thermodynamics. Einstein emphasized often that the field in question must be free of singularities. His reasoning seems to be based on the following two observations: Conventional wave functions in quantum mechanics are free of singularities. On the other hand, in his general theory of relativity completed in 1916, the differential equations of the metric space completely replace the Newton theory of the motion of celestial bodies, if the masses are substituted with singularities of the field; those equations contain the law of force as well as the law of motion while eliminating inertial systems. His theory with Tij = TijMaxwell , however, does not explain quantum-mechanical phenomena, and is not satisfactory (unimodular theory). Considering these two facts, Einstein had a conjecture that a satisfactory theory be obtained by modifying the general theory of relativity so that the singularities do not arise in a field determined by the differential equations of the metric space. He assumed that the desirable modification be made by eliminating the symmetry condition of the metric tensor from the general theory of relativity completed in 1916. According to Einstein, equations of such complexity as expected can be found only through the discovery of a logically simple mathematical scheme that determines the equations of physics completely or almost completely. Once one has a proper mathematical scheme, one requires only little knowledge of physical facts for setting up a proper theory. In 1948, near the end of his life, Einstein thought that he had success in formulating a satisfactory scheme of geometry in which the metric tensor is no longer symmetric. He hoped that this geometry could provide the framework in which the new theory of physics be established. Unfortunately, however, the result was disappointing; a stationary field free from singularities could never represent a mass different from zero. We thus recognize that Einstein’s view of conventional quantum mechanics is partially right, and a causal and determinative law is underlying conventional quantum-mechanical phenomena of the electron. Considering this, it appears to be a serious misjudgment of Einstein to attribute immediately the cause of singularities to the symmetry condition of the metric tensor in the Riemann geometry. Now, we can say that the general solution of a partial differential equation contains a set of functions whose forms are not determined by the equation but by initial and boundary conditions. A physically significant solution is a particular solution that satisfies proper initial and boundary conditions. It is a significant event in the history of physics that Einstein had persistently failed to recognize the significance of initial and boundary conditions in interpreting physical laws. We see the same failure in Dirac’s interpretation of the Dirac equation for the electron if we not considerer that the Dirac equation is derived from chiral electromagnetic fields with E || B. FUNDAMENTAL EQUATIONS In the following investigation, the variables are in general defined as tensors in a four-dimensional Riemannian space. The mathematical treatment of them follows the ordinary rule of tensor calculus. For the convenience of reference, the mathematical symbols employed are mostly similar to those in (Møller, [9-11]), unless otherwise specified. Those equations are mutually coupled, and the strong tendency of the electron to be a localized and stable field must be effected by the characteristics of those equations and proper boundary and initial conditions. For formulating the fundamental equations, it is customary to rely on Hamilton’s principle of variation of deriving covariant equations from a Lagrangian function. But the choice of the Lagrangian function is arbitrary, and so is of variation methods. There is no assurance of uniqueness of Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 115 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 the result. As Eddington remarked earlier [12], the physical significance of the method is unknown and doubtful, particularly when we have no means of evaluating those resulting equations immediately and directly in comparison with empirical information. Our experience in this field of physics is yet naive; instead of taking any axiomatic approach, it seems to be desirable to continue an effort of reflecting on the physical reality via equations known thus far. The guiding principle is that of general relativity, and the main hypothesis is that the fundamental equations embrace the Dirac equation and the Maxwell-Lorentz equations as of two special cases respectively. Although we do not intend to compare solutions of the fundamental equations directly with empirical information, we concern ourselves with the consistency and compatibility among those conditions under which the fundamental equations are reduced to the Dirac equation and the Maxwell-Lorentz equations. We expect that the present investigation will shed some light on those perplexing difficulties which we encounter in comprehending the behavior of an electron solely according to the Dirac equation and the Maxwell-Lorentz equations. Beyond this, we have an ambition to investigate the possibility that other elementary particles are governed by the same fundamental equations under varied restrictive conditions. We expect that those equations in the above will eventually be reduced to the Dirac equation and also to the MaxwellLorentz equations, and write for Fij 1 ∂ ( ∂x −g 1 ∂ −g − gF ( j )−g ij − gF * ∂x j ij )−g ∂ ∂x ij j ∂ ∂x j =0 =0 (1) (2) In these equations, g is the determinant of the metric tensor gij; Fij is an antisymmetric tensor and F*ij is conjugate to Fij; and are scalars. One might ask why these equations are fundamental. The answer is simple: Firstly, these equations are covariant in the Riemannian sense; secondly, i by considering the current select for gij ∂ ∂x j and by ∂ i assuming magnet for gij , these equations can be as ∂x j the Riemannian generalization of the Maxwell-Lorentz equations. However, we do not immediately relate these equations to the Maxwell-Lorentz equations; a physical consideration is needed prior to doing so. 116 −Q y 0 Qx − Qx 0 Py Pz − Px ö  − Py  . − Pz  0 ø F *ij = g ik g jm F *km (3) = 1 − gg ik g jm kmst F st 2 (4) where kmst is the Levi-Civita symbol, we have The equations for the matter field and those for the metric tensor field are intimately coupled together. In a conventional sense, however, we may call the following the equations for the matter field: Qz Considering The Matter Field ij æ 0 ç ç −Qz ij F =ç ç Qy ç P è x æ 0 ç ç Pz F *ij = ç ç − Py ç è − Qx − Pz Py 0 − Px Px 0 −Q y −Qz Qx ö  Qy  × −g , Qz  0 ø æ 0 − Pz¢ Py¢ −Qx¢ ö ç  0 − Px¢ −Qy¢  ç Pz¢ ij F* =ç . 0 −Qz¢  ç − Py¢ Px¢ ç Q¢ Q¢ Q¢ 0 ø è x y z (5) From here on, we shall often write P for (Px, Py, Pz) and Q for (Qx, Qy, Qz) simply for the sake of convenience, although they are not three-vectors. We note that in general. P ≠ P´, Q ≠ Q´ (6) Q = Q´ = B (7) However, if P = P´ = E, We have a nice approximation. We expect the equivalence between the two sets of equations will be established, if the metric tensor field be properly evaluated in the following. The Metric Tensor Field As is well known, Einstein in 1916 proposed an equation for the metric tensor [6]. 1 Rij − gij R = − kTij 2 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 (8) H. Torres-Silva: A new relativistic field theory of the electron where Rij is the contracted curvature tensor, R is the curvature scalar, and Tij is the energy-momentum tensor of the matter field. Einstein gave this equation by considering that the only fundamental tensors that do not contain derivatives of gij beyond the second order are functions of gij and the Riemann-Christoffel curvature tensor and that the equation is analogous to the Poisson equation for the gravitational field to the non-relativistic limit. It seems that Einstein proposed this equation for the purpose of solving cosmological problems, i.e., the structure of the universe as a whole [6]. Therefore, Tij is expected to be a known tensor supplied from the data of astronomical observation of the average mass distribution. Schwarzschild showed that the equation with Tij = 0 has a particular solution that expresses properly the gravity field induced by a material point [12]. Only when Eq. (8) is considered simultaneously with Eqs. (1) and (2), the equation for an elementary particle may be solved. If it is noticed that Eq. (8) alone consists of ten simultaneous partial differential equations of the second order, the analytical treatment of those equations concerned is an extremely difficult task. Moreover, it was not completely known how Tij is to be constructed in terms of Fij , and in the Einstein epoch. (As noted a short time ago, the variation method is not a decisive one.) Our present purpose is to show that the Dirac equation and the Maxwell-Lorentz equations, which are covariant only in the Euclidean sense, are both attainable by linearization of the same one set of nonlinear equations covariant in a non-Euclidean sense. From this viewpoint, we consider that it may not be necessary that the fundamental equations are immediately covariant in the Riemannian sense. There may be schemes of geometry that are more general than the Euclidean and less than the Riemannian. It is noted that, because of the restrictive conditions, viz., Eq. (8), Einstein’s geometry is less general than the Riemannian [12]. According to Einstein, the Einstein tensor, the left hand side of Eq. (8), should vanish in a space empty of matter. On the other hand, in the Riemann geometry, it does not vanish in general. (In the Riemann geometry, the idea of matter does not exist.) However, the covariant divergence of the Einstein tensor vanishes always in the Riemann geometry as well as in [10]. As noted earlier, Einstein chose Eq. (8) as one of the possibly simplest equations. Our conjecture is even when we adopt Einstein’s equation the obtained equation is adequate completely for describing the field extremely near the center of the core of an electron. Instead of taking an axiomatic approach, it is essential to study carefully the circumstances under which the present investigation is motivated. Later Einstein [6] attempted to investigate the structure of an elementary particle as based on the same equation. There, however, he did not pay much attention to Tij. He simply speculated that the matter field is an electromagnetic field, using a unimodular theory with Tij = TijMaxwell and the magnetic field B perpendicular to electric field E, (E ^ B). Contrary to Einstein conjecture, in our present problem in which an electron is considered to be a small universe, we considerer E••B, ie, we suppose that the electron –positron equation is the Dirac equation if only if it is derived from electromagnetic fields with E••B, inserted 1 in the original Einstein equation Rij − gij R = − kTij , 2 Maxwell ij with Tij = Tij . That means F = iF *ij , where E = iB i i , given by i = −1 , and select , magnet ∂F µ ∂x ∂F µ ∂x = imc µ 4 µ i Je = − E = select c e (9) = 4 µ imc µ i J = B = magnet c m m (10) (For the specific demonstration see the article Maxwell’s theory with chiral current) Thus, contrary to Einstein equation for the electron (unimodular theory) [13], i.e., the equations for the matter field and those for the metric tensor, do not contain Planck’s constant h, the electronic mass m and charge e, our equation (8), (9) and (10) contain h, m, e, which are essential to obtain the Dirac equation. With equations (9-10), it’s possible to show that an electron is like a toroid with E••B, spin ½, without radiation and rp = T = / 2mc (figure 1). rp Figure 1. Electron model Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 117 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 Assumptions as regards the metric tensor field According to the above results and considerations, we assume that the field in question is spatially localized. In the space outside the field, the Riemann-Christoffel tensor is negligibly small. It would be more reasonable to regard a part of the space as outside when the RiemannChristoffel tensor is negligibly small in that part. Also we note that, owing to the other bodies of matter contained in the universe, the tensor in question does not completely vanish at any point of the space. But our interest is in the local field, the electron. Hence, we ignore the curvature of the global scale, and may consider an inertial frame of reference outside the electron. (Einstein, perhaps due to his esteem of Ernst Mach, did not necessarily seem to think that the field of an electron can be completely closed and sustained by itself [12, 13]). If we consider an electron fixed to an inertial frame of reference, the electron appears to be free from the influence of the external universe. Classically, if we consider the internal structure of the electron, the situation is not necessarily so simple. It seems possible that a portion of the electron is in acceleration relative to the inertial frame reference in the same way as a portion of a spinning top resting as a whole on the inertial frame is. Such a classical-mechanical structure is inconceivable. However, it is sensible then to assume that the electron has a stable structure with its own permanent gravity field, as independent of the influence of the external universe. [5] S. Weinberg. “A Unified Physics by 2050”. Sci. Am. Vol. 281 Nº 6, pp. 68-75. December, 1999. [6] A. Einstein. “Do Gravitational Fields Play an Essential Part in The Structure of the Elementary Particles of Matter?” The Principle of Relativity. Dover, pp. 191-198. 1952. [7] P.G. Bergmann and R. Thomson. Spin and Angular Momentum in General Relativity. Phys. Rev. Vol. 2 Nº 89, pp. 400-407. 1953. [8] J. N. Goldberg, Conservation Laws in General Relativity, Phys. Rev. Vol. 2 Nº 111, pp. 315-320. 1958. [9] C. Møller. “On the Localization of the Energy of a Physical System in the General Theory of Relativity”. Ann. Physics. Vol. 4, pp. 347-371. 1958. [10] C. Møller. “Further Remarks on the Localization of the Energy in the General Theory of Relativity”. Ann. Physics. Vol. 12, pp. 118-133. 1961. [11] C. Møller. Conservation Laws and Absolute Parallelism in General Relativity. Mat. Fys. Skr. Danske. Vid. Selsk. Vol. 1 Nº 10, pp. 1-50. 1961. [12] A.S. Eddington. The Mathematical Theory of Relativity. Cambridge University Press, Cambridge. 1921. [13] A. Einstein. “Die Kompatibilität der Feldgleichungen in der einheitlichen Feldtheorie”. Preuss. Akad. Wiss. Berlin, Phys. Math. Klasse, Sitzber, pp. 18-23. 1930. CONCLUSION Thus we are presented a new theory called “Teoría Total Simplificada” (TTS) based on chiral electrodynamic which reproduces at the first time the Dirac equation for the electron unifying the gravity with electromagnetism [14-16]. REFERENCES [1] W.G. Dixon, Special Relativity. University Press, Cambridge. 1978. [2] R. Adler, M. Bazin and M. Schiffer, Introduction to General Relativity. McGraw-Hill, NY. 1965. [3] M. Friedman. Foundations of Space-time theories. Princeton U.P., Princeton, NJ. 1983. [4] E. Witten. “Duality, Spacetime and Quantum Mechanics”. Physics Today. Vol. 50, pp. 28-33. 1997. 118 [14] H. Torres-Silva. “Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 6-23. 2008. [15] H. Torres-Silva. “The close relation between the Maxwell system and the Dirac equation when the electric field is parallel to the magnetic field”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 43-47. 2008. [16] H. Torres-Silva. “Chiral field ideas for a theory of matter”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1, pp. 36-42. 2008. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Chiral waves in a metamaterial medium Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008, pp. 119-122 CHIRAL WAVES IN A METAMATERIAL MEDIUM ONDAS QUIRALES EN UN MEDIO METAMATERIAL H. Torres-Silva1 Recibido el 5 de septiembre de 2007, aceptado el 21 de diciembre de 2007 Received: September 5, 2007Accepted: December 21, 2007 RESUMEN En este trabajo se estudia la refracción anómala en el borde de un medio metamaterial con fuerte quiralidad. El hecho de que para una onda monocromática el vector de Poynting es antiparalelo a la dirección de la velocidad de fase conduce a relevantes propiedades que pueden tener ventajas en el diseño de novedosos dispositivos y componentes a frecuencias de microondas. Palabras clave: Ondas quirales, metamateriales. ABSTRACT In this paper we study the anomalous refraction at the boundary of a metamaterial medium with strong chirality. The fact that for a time-harmonic monochromatic plane wave the direction of the Poynting vector is antiparallel with the direction of phase velocity, leads to exciting features that can be advantageous in the design of novel devices and components at microwaves frequencies. Keywords: Chiral waves, metamat erial. INTRODUCTION Composite materials in which both permittivity and permeability possess negative values at some frequencies has recently gained considerable attention. This idea was originally initiated by Veselago in 1967, who theoretically studied plane wave propagation in a material whose permittivity and permeability were assumed to be simultaneously negative. Recently have been constructed such a composite medium for the microwave regime, and experimentally the presence of anomalous refraction in this medium is verified [1]. Previous theoretical study of electromagnetic wave interaction with omega media using the circuit-model approach had also revealed the possibility of having negative permittivity and permeability in omega media for certain range of frequencies [2]. That is important for design of circulary polarized antennas The anomalous refraction at the boundary of such a medium with a conventional medium, and the fact that for a time-harmonic monochromatic plane wave the direction 1 of the Poynting vector is antiparallel with the direction of phase velocity, can lead to exciting features that can be advantageous in design of novel devices and components. For instance, as a potential application of this material, the idea of compact cavity resonators in which a combination of a slab of conventional material and a slab of metamaterial with negative permittivity and permeability. The problems of radiation, scattering, and guidance of electromagnetic waves in metamaterials with negative permittivity and permeability, and in media in which the combined paired layers of such media together with the conventional media are present, can possess very interesting features leading to various ideas for future potential applications such as phase conjugators, unconventional guided-wave structures, compact thin cavities, thin absorbing layers, high-impedance surfaces, to name a few. In this talk, we will first present a brief overview of electromagnetic properties of the media with negative permittivity and permeability, and we will then discuss some ideas for potential applications of these materials. In this work we discuss the chiral waves in metamaterial media. Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 119 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 CHIRAL WAVES In classical electrodynamics, the response (typically frequency dependent) of a material to electric and magnetic fields is characterized by two fundamental quantities, the permittivity and the permeability µ. The field permittivity relates the electric displacement D to the electric field E through D = E, and the permeability µ relates the magnetic field B and H by B = µ H . If we do not take losses into account and treat and µ as real numbers, according to Maxwell’s equations, electromagnetic waves can propagate through a material only if the index of refraction n, is real. Dissipation will add imaginary components to and and µ cause losses, but for a qualitative picture, one can ignore losses and treat and µ as real numbers. Also, strictly speaking, and µ are second-rank tensors, but they reduce to scalars for isotropic materials. In a medium with and µ both positive, the index of refraction is real and electromagnetic waves can propagate. All our everyday transparent materials are such kind of media. In a medium where one of the and µ is negative but the other is positive, the index of refraction is imaginary and electromagnetic waves cannot propagate. Metals and Earth’s ionosphere are such kind of media. In fact, the electromagnetic response of metals in the visible and near-ultraviolet regions is dominated by the negative epsilon concept [1-3]. Although all our everyday transparent materials have both positive and positive µ, from the theoretical point of view, in a medium with and µ both negative, electromagnetic waves can also propagate through. Moreover, if such media exist, the propagation of waves through them should give rise to several peculiar properties. This was first pointed out by Veselago over 30 years ago when no material with simultaneously negative and was known [4]. For example, the cross product of E and H for a plane wave in regular media gives the direction of both flow, and the electric field propagation and energy E , the magnetic field H , and the wave vector k form a right-handed triplet of vectors. In contrast, in a medium with and µ both negative, E x H for a plane wave still gives the direction of energy flow, but the wave itself that is, the phase velocity propagates in the opposite direction, i.e., wave vector lies in the opposite direction of E x k H for propagating waves. In this case, electric field E , magnetic field H , and wave vector k form a left-handed triplet of vectors. Such a medium is therefore termed left-handed medium [5]. In addition to this ‘‘left-handed’’ characteristic, there 120 are a number of other dramatically different propagation characteristics stemming from a simultaneous change of the signs of and µ, including reversal of both the Doppler shift and the Cerenkov radiation, anomalous refraction, and even reversal of radiation pressure to radiation tension. However, although these counterintuitive properties follow directly from Maxwell’s equations, which still hold in these unusual materials. Such type of left-handed materials have never been found in nature but such media can be prepared artificially, they will offer exciting opportunities to explore new physics and potential applications in the area of radiation-material interactions. Following the suggestion of Pendry, Smith and co-workers reported that a medium made up of an array of conducting nonmagnetic split ring resonators and continuous thin wires can have both an effective negative permittivity and negative permeability µ for electromagnetic waves propagating in some special direction and special polarization at microwave frequencies [5]. This is the first experimental realization of an artificial preparation of a left-handed material, where on the one hand, the permittivity of metallic particles is automatically negative at frequencies less than the plasma frequency, and on the other hand, the effective permeability of ferromagnetic materials for electromagnetic waves propagating in some particular direction and polarization can be negative at frequency in the vicinity of the ferromagnetic resonance frequency, which is usually in the frequency region of microwaves. However this configuration exhibit chirality and a rotation of the polarization so the analysis of metamaterial presented by several authors provides a good but not exact characterization of the metamaterial [6]. The evidence of chirality behavior suggests that if it is included in the conditions to obtain a metamaterial behavior of a medium futher progress will be obtained. In this short paper, we propose to investigate the conditions to obtain a metamaterials having simultaneously negative and negative µ and very low eddy current loss. As a initial point, we considerer a media where the electricpolarization P depends not M only on the electric field E , and the magnetization depends not only on the magnetic field H , and we may have, for example, constitutive relations given by the Born-Federov formalism [7]. D(r , ) = ( )( E (r , ) + T ( ) Ñ × E (r , )) (1) B(r , ) = µ ( )( H (r , ) + T ( ) Ñ × H (r , )) (2) Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 H. Torres-Silva: Chiral waves in a metamaterial medium The pseudoscalar T represents the chirality of the material and it has length units [7]. In the limit T Õ 0, the constitutive relations (1) and (2) for a standard linear isotropic lossless dielectric with permittivity and permeability µ are recovered. According to Maxwell’s equations, electromagnetic waves propagating in the direction of magnetization in a homogeneous magnetic material is either positive or negative transverse circularly polarized. If the composite can truly be treated as a homogeneous magnetic system in the case of grain sizes much smaller than the characteristic wavelength, electric and magnetic fields in the composite should also be either positive or negative circularly polarized and can be expressed as − j ( k z − t ) E ± (r , t ) = Eˆ 0 ( ± )e ± 0 (3) − j ( k z − t ) H ± (r , t ) = Hˆ 0 ( ± )e ± 0 (4) where E0± = E0 ( xˆ ± yˆ ) , and Ñ × E ± (r , t ) = k± E ± , k± ≥ 0 is the chiral wave number. In this case of right polarized wave we can see that the effective permittivity p and the effective permeability µp are obtained from with µ p = µ (1 − k+ T ) and where k eff and are related by keff 2 = 2 ( p µ p ) . Equations (5) and (6) are exact in principle assuming that nonlocal effects can be neglected. This assumption is appropriate in many cases. But in some cases, nonlocal effects can be significant and cannot be neglected, as has been shown in the past. In such cases, Eqs. (5) and (6) shall be not exact. For simplicity, in this paper we have assumed that nonlocal effects can be neglected and hence Eqs. (5) and (6) shall be valid. In Eqs. (3 and (4) the sign of the effective wave number can be positive or negative depending on the product k+T and the energy flow. For convenience we assume that the direction of energy flow is in the positive direction of the z axis, but the sign of k eff still can be positive or negative. In the case of right polarization, if 1 ≥ k+ T ≥ 0, the phase velocity and energy flow are in the same directions, and from E Maxwell’s equation, one can see that the electric and magnetic field and H and the wave vector keff will form a right-handed triplet of vectors. This is the usual case for right-handed materials. In contrast, if k+ T ≥ 1 the phase velocity flow are in opposite and energy directions, and E , H , and keff will form a left-handed triplet of vectors. This is just the peculiar case for left handed materials where the effective permittivity eff and the effective permeability µ eff are simultaneously negative. So, for incident waves of a given frequency v, we can determine whether wave propagation in the composite is right handed or left handed through the relative sign changes of k eff.. Based on Eqs. (5)-(6), we have computed T. and /p or − jk z − jk z ò D(r , )e + f dr = ò (E (r , ) + T ÑxE )e + dr (5) − jk z = ò (1 − k+ T ) Ee + dr with p = (1 − k+ T ) and k+ T ≥ 1. − jk+ z close to µ P P , the value of T. is quite large, indicating a strong spatial dispersion. Hence the singular point is the very point of traditional limitation. However, with / µP P continuously increasing, the spatial dispersion strength falls down very quickly. Therefore, if . is not Similary, we have ò B(r , )e µ/µp. versus / µ P P , as shown in Fig. 1 When . is very − jk z dr = µeff ò H (r , )e + dr (6) − jk z = µ (1− k+ T ) ò H (r , )e + dr around µ P P , e.g. < 0.7 µ P P or > 1.3 µ P P , we need not take nonlinear terms into consideration at all. Hence the strong spatial dispersion and nonlinearity cannot put the upper limitation to chirality parameters either. Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 121 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 dispersion, the Pasteur model is meaningful. Neither spatial dispersion nor energy will hinder chirality to be stronger, but we cannot realize strong chirality only by increasing the spatial dispersion. The necessary condition of strong chiral medium is that the chirality and spatial dispersion are of conjugated types. We remark that strong chiral media have found wide applications in the negative refraction and supporting of backward waves, useful in metamaterial substrates. 30 20 10 0 –10 –20 ACKNOWLEDGEMENT –30 0 1 2 3 Figure 1. The strength relationship of chirality and spatial dispersion. T versus / µ P P The point of / µP P = 1 is singularity, corresponding The author acknowledges discussions with colleagues in the school of electrical and electronic engineering EIEE – Universidad de Tarapacá. This work have been supported by the project N° 8721-06 of the Universidad de Tarapacá. infinite spatial dispersion coefficient T. When / µP P > 1, T becomes negative for keeping the positive rotation term coefficients with negative µ and . REFERENCES [1] S. Tretyakov, A. Sihvola, and L. Jylha. Photonics and Nanostructures - Fundamentals and Applications. Vol. 3, p. 107. 2005. CONCLUSIONS [2] From figure 1, it is clear that enhancing spatial dispersion will not lead to strong chirality and will reach the traditional limitation point. This is why we have never succeeded in realizing strong chirality no matter how to improve the asymmetry and spatial dispersion. M.M.I. Saadoun and N. Engheta. “Theoretical study of electromagnetic properties of non-local omega media”. Chapter 15. Progress in Electromagnetic Research (PIER) Monograph series, vol. 9, A. Priou, (Guest Editor), pp. 351-397. 1994. [3] R.A. Shelby. “Experimental verification of a negative index of refraction”. Science. Vol. 292 Nº 5514, pp. 77-79. 6 April, 2001. [4] V.G. Veselago. “The electrodynamics of substances with simultaneously negative values of epsilon and mu”. Soviet Physics Uspekhi. Vol. 10 Nº 4, pp. 509-514. 1968. [5] D.R. Smith. “Composite medium with simultaneously negative permeability and permittivity”. Phys. Rev. Lett. Vol. 84 Nº 18, pp. 4184-4187. 1 May, 2000. [6] D.R. Smith. Physical Review B. Vol. 65, p. 195104. 2002. [7] H. Torres Silva. “Chiro-Plasma Surface Waves”. Advances in complex Electromagnetic Materials, Kruwer Academic Publishers. Netherlands, p. 249. 1997. Fortunately, as pointed out earlier, the strong chirality does not require strong spatial dispersion. Hence the most important difference between strong and weak chirality is that T. and . have opposite signs, which necessarily leads to negative and µ. Here, . stands for chirality and T. is the chiral coefficient of the first order for spatial dispersion. Strong chirality roots from using one type of spatial dispersion to get the conjugate stereoisomer, or chirality. It is an essential condition for supporting the backward eigenwave in strong chiral medium. In conclusion, a strong chiral medium behaves like Veselago’s medium. Under the weak spatial dispersion, the energy is always positive for chiral medium. We show that strong chirality does not equal strong spatial dispersion, which occurs only around a singular point. Even in this small region with very strong spatial 122 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008 DE TARAPACÁ Arica, Chile MAGÍSTER EN TELECOMUNICACIONES MAGÍSTER EN INGENIERÍA DE SOFTWARE Entrega conceptos avanzados de Telecomunicaciones y de Ingeniería Electromagnética, tal que permita a los participantes formarse en estudios de alto nivel en esta área temática. Corresponde a un ciclo de formación profesional, orientado a mejorar las prácticas en la disciplina de desarrollo de software y la calidad de los productos obtenidos. 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Esta decisión será casi siempre final. Ejemplos de motivos de rechazo: contenido del artículo no apropiado para la revista, violaciones crasas de las normas de publicación, artículos carentes de significación, mérito científico o tecnológico. SELECTION PROCESS FOR ARTICLES SUBMITTED FOR PUBLICATION IN THE INGENIARE. REVISTA CHILENA DE INGENIERÍA 1. Preliminary Selection and Peer Evaluation The editor will first verify that the content of the article is appropriate for the magazine and the manuscript is set in accordance with the journal’s standards. The editor may reject, at his discretion, submissions that do not comply with the general directions, are poorly written or do not posses sufficient scientific or technological merit. Articles that pass the preliminary selection are sent for peer review by specialists and scientists of renown research in areas related to the article. Upon assessing the article, peer reviewers write a report including comments and suggestions recommending its acceptance or rejection. 2. About the Evaluation by Peers Reviewers As part of the evaluation of the article, reviewers will consider the soundness of the experimental design, verifying that the conclusions coincide with the statistical and experimental data. Reviewers will also make sure and that the author(s) consulted all literature pertinent to the topic discussed. Wording and writing quality will also be assessed. During this process the identity of the reviewers will not be known by the author(s), but the latter’s will be available to the reviewers. 3. The Editor’s Decision After assessing the reviewers’ recommendations, the editor will make a decision based on the following options: Accepting the article, subject to minor revisions: The editor will return the article to the author(s) with a list of suggestions for minor corrections, including his own as well as those made by the reviewers. Once the modified manuscript is received by the editor, this will confirm the acceptance of the article, indicating the publication date. Minor revisions in an article include: typographical errors, page numbering, articles cited in the text that are not in the bibliography and vice versa, slight discrepancies between the summaries and the abstract, moderate corrections to the text. Returning the article for major changes: The editor will return the article to the author(s) with a list suggestions for major corrections that must be made before the article is accepted for publication. Major revisions include: analyzing data using other statistical tests, adding or redoing figures and tables, repetition of experiments, rewriting the discussion of the problem using additional literature and substantial changes to the text. Rejection: The editor will return the article with the peer evaluation, reporting the reason(s) for the rejection. This decision will be final (in most cases). Reasons for rejection of an article include: inappropriate content, manuscripts not complying with the norms for publication and works lacking scientific or technological merit. NORMS OF PUBLICATION People interested in publishing papers in “Ingeniare. Revista chilena de ingeniería” must send articles which comply with the publication norms detailed below. The topics must relate to the following areas of engineering: Electronics, Electricity, Computing, Mechanics, Industry, Acoustic, Metallurgy and Engineering Education. The author’s rights may not be granted to third parties. If necessary, the editor reserves the right to carry out minor modifications for editing, in order to achieve a better presentation of the work. Papers may be presented in English (if possible), Portuguese and in Spanish. Four types of contributions are regularly considered: Papers. Presentation of significant research, development, or application. Brief Papers. Concise descriptions of a contribution to a specific aspect of design, realization, or operation. Letters. Significant remarks of interest to engineers, and comments on previously published papers. In addition, special papers (tutorials, surveys, and perspectives on the current trends) are solicited. Authors are encouraged to contact the Editor or Co-Editor before submitting such papers. Papers and Brief Papers go through the same review process. Letters go through a shorter review process to facilitate rapid publication. PUBLICATION CHARACTERISTICS The Advisory Editor Committee will recommend the publication of papers whose content will be the author’s full responsibility. Originals accepted for publication will not be returned. Editing or rejections will be notified in due time. The Advisory Editor Committee may consider papers presented in national and international conferences and scientific meetings. The main title of the article should be in English and Spanish. It must be short and it must clearly state the topic of the work. Abstracts must be brief, containing clear and precise ideas. It must contain keywords which define the content of the paper (minimum of five and maximum of ten). The text should begin with a summary of not more than 250 words. It must briefly state: 1) what has been done in this work, 2) how it was done (only if it is important to be detailed), 3) main results obtained, 4) relevance of the results. A summary is an abbreviated but comprehensive presentation of the article and it must inform about the objective, methodology and the results of the work described in the paper. A translation of the summary into English, headed by the word abstract, must be included immediately after Spanish resumen. The introduction must be presented in about a page and a half and it must guide the reader towards an understanding of the problem presented. It should also include information about the nature of the problem, references to previous works, purpose and meaning of the paper. The body of the paper will contain fundamental information about the work. Information must be clearly presented. Language must be objective and impersonal. The author must see to it that the article complies with the norms of publication, language register and specific terms accepted by the scientific community. The introduction and the body of the article must be written in two 8 cm. columns, with a 1.05 cm. space between columns. Conclusions must be clear, stating what is shown in the work as well as its relevance. They must also state advantages, limitations and application results. References will be written with numbers in parenthesis and they will be listed at the end of the paper as follows: Books: [N°] Author’s First Initial Name, Last Name. “Title”. Editorial name. Number of Edition. City, Country. Volume, pp. Pages number. Date. Articles of magazines: [N°] Author’s First Initial Name, Last Name. “Title”. Magazine name. Volume. Number, pp. Pages number. Date. Electronics references: [N°] Author’s Name. “Title”. Site Update, pp. Pages number. Date of visit. URLs. Individualization: Footnotes, written with arabic numbers, will state: 1) date of reception of the original, people involved in the intellectual work, materialy or financially. If necessary the mention whether the paper is part of a study, a thesis, a project, etc., 2) name of the author, profession, institution, author and institution’s electronic and mail address. Aknowledgements: Will be included before the bibliography, identifying individuals and institutions that gave intellectual or financial support to the research. All equations, pictures and photographs must bear a number that will be used for identification purposes throughout the text. Pictures and photographs must also include an explanatory caption. Periodicity: “Ingeniare. Revista chilena de ingeniería” is published periodically, is printed in three issues per volume annually. Reception of papers: Manuscripts should be in Microsoft Word or LaTeX format and submitted in electronic form, via e-mail or in optical or magnetic media, according to given instructions. Address: Editor Committee, “Ingeniare. Revista chilena de ingeniería”, Casilla 6-D, Arica-Chile, Sud América. E-mail: [email protected] Papers should not be longer than 15 pages and must comply with the following format: Paper: letter size 21,59 cm x 27,94 cm; Margins: top 2 cm, bottom 2 cm, left 2,54 cm and right 2 cm; Title of the article, centered in bold type, Times New Roman 10 pt.; two blank spaces before the initial paragraph; Subtitles should be at the left margin in black letters with Times New Roman 10 pt.; The body of the text must be written with Times New Roman 10pt., one blank space between paragraphs. NORMAS DE PUBLICACIÓN Los autores interesados en publicar artículos en la “Ingeniare. Revista chilena de ingeniería”, deben enviar sus trabajos ajustados a las normas de publicación que se detallan más abajo. Los temas deben enmarcarse dentro de las siguientes áreas de la ingeniería: Electrónica, Eléctrica, Computación, Mecánica, Industrias, Acústica, Metalurgia y Enseñanza de la Ingeniería. El trabajo sometido no debe tener “Derechos de Autor” otorgados a terceros. El editor se reserva el derecho a realizar modificaciones menores de edición, para una mejor presentación del trabajo. Se podrá presentar trabajos en idioma inglés (de preferencia), español y portugués. Se consideran cuatro tipos de contribuciones: Trabajos in extenso (papers). Presentaciones relativas a investigación significativa, desarrollo o aplicación de sistemas tecnológicos. Trabajos condensados (brief papers). Descripciones concisas de contribución específica a investigación significativa, desarrollo o aplicación de sistemas tecnológicos. Comunicaciones (letters). Observaciones de interés para los lectores (ingenieros) y/o comentarios acerca de publicaciones previas. Trabajos especiales. Trabajos tales como tutoriales, encuestas y visiones de las tendencias actuales en ingeniería son bienvenidas. Se invita a los autores de tales trabajos a contactarse con el Editor antes de presentarlos para publicación. Los trabajos in extenso y condensados (papers & brief papers) se someten al mismo procedimiento de revisión. Las comunicaciones (letters) son revisadas en un proceso más breve, para facilitar su pronta publicación. CARACTERISTICAS DE LAS PUBLICACIONES El Comité Editor Asesor será el encargado de autorizar la publicación de los trabajos, cuyo contenido será de responsabilidad exclusiva de los autores. Los originales de los artículos aceptados para publicar no serán devueltos. Las modificaciones o rechazos se indicarán con notas explicativas. El Comité Editor Asesor podrá considerar trabajos presentados en congresos y reuniones científicas nacionales e internacionales. El título principal debe estar escrito en inglés y español, indicando claramente la materia del artículo. Éste deber ser breve, pero preciso en la idea que represente. Además, debe contener un número suficiente de palabras clave que definan el contenido del artículo (mínimo cinco y máximo diez). El texto comienza con un resumen de no más de 250 palabras, donde debe precisarse brevemente: 1) lo que el autor ha hecho, 2) como lo hizo (sólo si es importante detallarlo), 3) los resultados principales, 4) la relevancia de los resultados. El resumen es una representación abreviada, pero comprensiva del artículo y debe informar sobre el objetivo, la metodología y los resultados del trabajo descrito. A continuación del resumen, debe incluirse su traducción al idioma inglés, encabezado por la palabra abstract. La introducción deberá orientar al lector respecto al problema presentado e incluir: la naturaleza del problema, los antecedentes o trabajos previos, el propósito o significancia del artículo. Ésta deberá desarrollarse en una página y media, como máximo. El cuerpo contendrá, en detalle, la información fundamental del artículo. Deberá asimismo considerar el objeto de la información, la que deberá ser entregada en forma clara y eficiente. La redacción de los trabajos será de carácter objetivo e impersonal. El autor cuidará que la forma se ajuste a las normas de presentación, corrección en el lenguaje y uso de terminologías aceptadas por organismos científicos. La introducción y el cuerpo deberán ser escritos en doble columna de 8 cm cada una, con un espacio de 1,5 cm entre columnas. Las conclusiones tendrán que ser claramente definidas y deberán cubrir lo siguiente: lo que se demuestra en el trabajo, su relevancia, ventajas y limitaciones y aplicación de los resultados. Las referencias se indicarán con número entre paréntesis y se listarán al final de la publicación, de la siguiente forma: Libros: [N°] Iniciales del Nombre del autor (es), Apellido del autor (es). “Título”. Nombre de la Editorial. Número de Edición. Ciudad, País. Volumen, pp. Páginas consultadas. Año de publicación. Artículos de revistas: [N°] Iniciales del Nombre del autor (es), Apellido del autor (es). “Título del artículo”. Nombre de la revista. Volumen. Número, pp. páginas entre las que se encuentra el artículo. Mes y año. Textos electrónicos: [N°] Iniciales del Nombre del autor (es), Apellido del autor (es). Título del artículo, pp. Páginas consultadas. Fecha de actualización. Fecha de consulta. Dirección web. Individualización: Al pie de la página, mediante números arábigos, deben señalarse: institución a la que pertenecen los autores, dirección postal y electrónica de él o los autores. Agradecimientos: Se ubicarán antes de las referencias bibliográficas, señalando a las personas o instituciones que colaboraron en el trabajo intelectual, material o financiero. Todas las ecuaciones, figuras y fotografías deberán tener un número que las identifique, al que se hará referencia en el texto. Las figuras y fotografías deberán ser nítidas y tener, además, una leyenda explicativa al pie de las mismas. Periodicidad: “Ingeniare. Revista chilena de ingeniería”, edita tres números al año (cuatrimestral). Recepción de colaboraciones: Los manuscritos deberán estar en formato Microsoft Word o LaTeX, y enviarse por algún medio electrónico (correo electrónico, medio de almacenamiento óptico o magnético), de acuerdo a las instrucciones presentadas. Éstas deben ser enviadas a: Comité Editor, “Ingeniare. Revista chilena de ingeniería”, Casilla 6-D, Arica - Chile, Sudamérica. E-mail: [email protected] Los artículos no podrán tener una extensión mayor de 15 páginas y deberán respetar el siguiente formato: papel tamaño carta, 21,59x27,94 cm; márgenes: superior 2 cm, inferior 2 cm, izquierdo 2,54 cm, derecho 2 cm; título del artículo, centrado en negrita, con letra mayúscula tipo Times New Roman 12 pt.; títulos del texto, centrados en negrita, con letra Times New Roman 10 pt., dejando dos líneas en blanco antes del párrafo; texto del cuerpo con letra Times New Roman 10 pt., dejando una línea en blanco entre párrafos.

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