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Abstract
La notion du déplacement parallèle d'un demi-vecteur (fonctions ψ Dirac) permet d'établir, dans la géométrie de Riemann, une équation d'onde invariante qui est une généralisation de celle de Dirac. On peut former un tenseur non symétrique qui joue le rôle du tenseur de l'énergie et satisfait aux équations de mouvements d'Einstein : sa divergence est égale à la force de Lorentz. On en déduit les équations de mouvement t de la mécanique quantique. La théorie donne une interprétation géométrique de l'opérateur pσ = e/c φσ qui figure dans l'équation de Dirac et semble fournir une base pour l'unification de la théorie de l'électricité et de la matière.
... " 268 ([134], p. 1469) In order that gauge-invariance results, í µí¼ must transform with a factor of norm 1, innocuous for observation, i.e., í µí¼ → exp(í µí± 2í µí¼ ℎ í µí± í µí± í µí¼) if í µí°´íµí°´í µí± → í µí°´íµí°´í µí± + í µí¼í µí¼ í µí¼í µí±¥ í µí± . Another note and extended presentations in both a French and a German physics journal by Fock alone followed suit [133, 131, 132]. In the first paper Fock defined an asymmetric matter tensor for the spinor field, í µí± í µí± í µí± = í µí±ℎ 2í µí¼í µí± ...
... In this point, our theory, developed independently, agrees with the new theory by H. Weyl expounded in his memoir 'gravitation and the electron'. " 272 ([132], p. 405) In both of his papers, Fock thus stressed that Einstein's teleparallel theory was not needed for the general covariant formulation of Dirac's equation. In this regard he found himself in accord with Weyl, whose approach to the Dirac equation he nevertheless criticised: " The main subject of this paper is 'Dirac's difficulty' 273 . ...
... This seems to us the most important aspect of Einstein's recent work, and by far the most hopeful portent for a unification of the divergent theories of quanta and gravitational relativity. " [418] The correction of this misjudgement of Wiener and Vallarta by Fock and Ivanenko began only one month later [134], and was complete in the summer of 1929 [134, 133, 131, 132]. In March, Tamm tried to show " that for the new field theory of Einstein [84, 88] certain quantum-mechanical features are characteristic, and that we may hope that the theory will enable one to seize the quantum laws of the microcosm. ...
This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin - with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by short-lived attempts to include the matter field described by Schrödinger's or Dirac's equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included.
... Generalization of the Dirac equation on Riemann geometry is connected with the parallel transport and covariant differentiation of the spinor in curvilinear space. These problems were considered for the first time in the articles [25] and further in the articles [26]. We will use the most important results of this theory below. ...
... For generalization of the Dirac equation on Riemann geometry it is necessary [25,26] to replace the usual derivative ...
... Since, according to general theory [25,26], the increment in spinor µ Γ has the form and the dimension of the 4-vector of the energy-momentum, it is logical to identify µ Γ with 4 -vector of energy-momentum of the photon electromagnetic field: ...
Introduction In the previous paper, [1] we showed that the Dirac electron theory can be presented one-to-one as a complex form of the Maxwell theory. The question arises: why should this be so? Is this representation sheer coincidence, or is it a result of a deep relation between these theories. This question can also be formulated differently: can the vectorial Maxwell theory produce the spinor Dirac theory? To see how this question can be solved, we will consider a brief review of the development of field theory. After the discovery of the Dirac electron equation, the main task of physicists was to find the equations that describe all other particles and fields. Attempts were made, mainly, on the basis of a generalization of the already known equations: the equations of the Maxwell electromagnetic theory, the equation of the Dirac electron theory and of the Einstein equation of gravitation. On one hand, various generalizations of the Maxwell equation (see the review in [2]) have bee
... The generalization of the Dirac equation on the Riemann geometry is connected with the parallel transport and covariant differentiation of the spinor in the curvilinear space. These problems were considered for the first time in the articles [21] and further in the articles [22]. We will use the most important results of this theory below. ...
... For the generalization of the Dirac equation on the Riemann geometry it is necessary [21,22] to replace the usual derivative ...
... are the summing indices and µ Γ is the analogue of Christoffel's symbols in the case of the spinor theory (called Ricci connection coefficients or the coefficient of the parallel transport of the spinor). In the theory it is shown [21] that ...
In the present paper it is shown that the Dirac electron theory is the approximation of the special nonlinear electromagnetic field theory
... Interest in this formulation of electrodynamics significantly grows in recent years. In the present paper the extension of the formalism to curved space-time models is performed on the base of tetrad formalism by Tetrode-Weyl-Fock-Ivanenko [4,5,6,7,8,9,10,11,12,13]; see the list of references on the subject in the recent book [14]. ...
... ] ; (13) it is true if the identity holds ...
The Riemann–Silberstein–Majorana–Oppenheimer approach to the Maxwell electrodynamics in the presence of electrical sources, arbitrary media and curved space-time is investigated within the matrix formalism and tetrad method. Symmetries of the matrix Maxwell equation under transformations of the local gauge complex rotation group SO(3,C) is demonstrated explicitly. Equivalence of the approach to general covariant Proca technique and spinor formalism is shown.
... Interest in this formulation of electrodynamics significantly grows in recent years. In the present paper the extension of the formalism to curved space-time models is performed on the base of tetrad formalism by Tetrode-Weyl-Fock-Ivanenko [4,5,6,7,8,9,10,11,12,13]; see the list of references on the subject in the recent book [14]. ...
... ] ; (13) it is true if the identity holds ...
The Riemann -- Silberstein -- Majorana -- Oppengeimer approach to the Maxwell electrodynamics in presence of electrical sources and arbitrary media is investigated within the matrix formalism. The symmetry of the matrix Maxwell equation under transformations of the complex rotation group SO(3.C) is demonstrated explicitly. In vacuum case, the matrix form includes four real matrices . In presence of media matrix form requires two sets of matrices, and -- simple and symmetrical realization of which is given. Relation of and to the Dirac matrices in spinor basis is found. Minkowski constitutive relations in case of any linear media are given in a short algebraic form based on the use of complex 3-vector fields and complex orthogonal rotations from SO(3.C) group. The matrix complex formulation in the Esposito's form, based on the use of two electromagnetic 4-vector, is studied and discussed. Extension of the 3-vector complex matrix formalism to arbitrary Riemannian space-time in accordance with tetrad method by Tetrode-Weyl-Fock-Ivanenko is performed. Comment: 32pages. Proccedings of the 14th Conference-School "Foundation & Advances in Nonlinear Science", Minsk, September 22-25, 2008. P. 20-49; ed. V.I. Kuvshinov, G.G. Krylov, Minsk, 2009
... Gauge fields, or Yang-Mills fields, were introduced to the wide audience of physicists in 1954 in a short paper by C.N. Yang and R. Mills [7], dedicated to the generalization of the electromagnetic fields and the corresponding principle of gauge invariance. The geometric sense of this principle for the electromagnetic field was made clear as early as in the late 20s due to the papers of V. Fock [8] and H. Weyl [9]. They underlined the analogy of the gauge (or gradient in the terminology of V. Fock) invariance of the electrodynamics and the equivalence principle of the Einstein theory of gravitation. ...
... Both H. Weyl and V. Fock were to use the language of the moving frame with spin connection, associated with local Lorentz rotations. Thus the Lorentz group became the first nonabelian gauge group and one can see in [8] essentially all formulas characteristics of nonabelian gauge fields. However, in contradistinction to the electromagnetic field, the spin connection enters the description of the space-time and not the internal space of electric charge. ...
Personal view of author on goals and content of Mathematical Physics.
... This plays the role of a gauge field. Indeed , the two realizations of the covariant Dirac equation that are got with two different tetrad fields are equivalent, at least locally [8]. However, it has been proved in recent years that the Hamiltonian operator associated, in a given coordinate system, with the covariant Dirac equation, does depend on the choice of the tetrad field. ...
The definition of the Hamiltonian operator H for a general wave equa-tion in
a general spacetime is discussed. We recall that H depends on the coordinate
system merely through the corresponding reference frame. When the wave equation
involves a gauge choice and the gauge change is time-dependent, H as an
operator depends on the gauge choice. This dependence extends to the energy
operator E, which is the Hermitian part of H. We distinguish between this
ambiguity issue of E and the one that occurs due to a mere change of the
"represen-tation" (e.g. transforming the Dirac wave function from the "Dirac
representation" to a "Foldy-Wouthuysen representation"). We also assert that
the energy operator ought to be well defined in a given ref-erence frame at a
given time, e.g. by comparing the situation for this operator with the main
features of the energy for a classical Hamilto-nian particle.
... The generalization of the Dirac equation on the curvilinear (Riemann) geometry is connected with the parallel transport of the spinor in the curvilinear space [5,6,7]. We use below the most important results of this theory. ...
A special non-linear equation of curvilinear electromagnetic wave is presented. The particularity of this equation lies in the fact that in matrix form it is mathematically equivalent to the Dirac electron equation. It is shown that the solution of this is the motion of the plane-polarized electromagnetic wave on a circular trajectory. It is also shown that such twirled wave can be considered as a massive charge particle with a spin of one half, similar to electron. The non-linear equation and its Lagrangian of these EM particles are founded.
In 1938, Einstein, Infeld, and Hoffmann contributed to the problem of motion in general relativity by establishing approximate equations for several particles of comparable masses. A year later, the exact same problem was also solved independently by the Soviet physicist Vladimir Fock. However, his approach proved to be different and laid the foundations for an original interpretation of general relativity which notably defended the existence of privileged frames of reference. This chapter presents the trajectory of the dispute between Fock and Leopold Infeld that arose from their contributions to the problem of motion. It aims at highlighting some specificities of the scientific and sociocultural contexts of the Soviet Union at the dawn of the “renaissance” of general relativity in the mid-1950s. In particular, this chapter shows that the Fock-Infeld dispute received special exposure in philosophical journals. The latter gave the two physicists opportunities to expose and mature their arguments, favoring an entrance with resonance into the international context at the 1955 Bern Conference on relativity. From there, Fock secured a position on the international scene, contributed to the Soviet inclusion into the “renaissance” process, and marked durably the debate on the interpretation of the general relativity.
The purpose of this chapter of nonlinear theory of elementary particles (NTEP) is to describe the mechanism of generation of massive elementary particles. The theory, presented below, indicates the possibility of the particle mass production by means of massive intermediate boson, but without the presence of Higgs's boson. It is shown that nonlinearity is critical for the appearance of particles' masses. PASC: 11.10.Lm, 12.10.En Keywords: nonlinear theory, intermediate boson, nonlinear equation, mass origin, Higgs mechanism 1.0. The postulate of generation of massive elementary particles The purpose of this chapter is to describe the mechanism of generation of massive elementary particles on the basis of the hypotheses, accepted in the chapter (Kyriakos, 2010a). Here, the mass-free particle, the photon, i.e. a quantum of an electromagnetic field, serves as a "billet" for the generation of a massive particle. In this case, the appearance of the mass of a particle is identical to the generation of the massive particle itself. There is a mechanism in the theory of Standard Model (SM) that ensures the generation of currents of elementary particles. This mechanism is known as the gauge (or phase) transformation of particles' field (let's recall that the particle's field in QFT is the particle's wave function). In the SM there is also a mechanism that ensures the generation of masses of elementary particles. Here, all particles do not have a mass at the initial stage. In the SM, the mass-free particles acquire mass because of the spontaneous breakdown of the gauge symmetry of vacuum. This mechanism is called Higgs's mechanism. In the framework of axiomatic nonlinear elementary particle theory we accepted (see (Kyriakos, 2010a)) the following basic postulates, which ensure the generation of massive particles: 1) In the proposed theory, in contrast to SM, only a quantum of an electromagnetic wave (photon) does not have mass. 2) The fields of an electromagnetic wave quantum (photon) can under specified conditions undergo a rotation transformation and initial symmetry breaking, which generate different massive elementary particles. It follows from these hypothesis that the equations of elementary particles must be nonlinear modifications of the equations of quantized electromagnetic (EM) waves. The detailed analysis presented in the following chapters shows that because of the rotation transformation and different types of symmetry breaking of the initial photon, such elementary particles can have mass, electric charge, spin (which is a multiple of 1 and ½), helicity, chirality, and all other characteristics of existing elementary particles.
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