On the Origin of Charge-Asymmetric Matter. I. Geometry of the Dirac Field

Makhlin, A.

doi:10.4236/jmp.2016.77061

Cited by 2 publications

(49 citation statements)

References 17 publications

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AbstractThis paper continues the author's work [1] [2], where a novel framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation was discovered. The previous analysis of solitary waveforms' properties [2] is extended to the four-component Dirac field.

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confidence: 52%

“…In the previous papers of the author [1] [2] a novel framework of the matterinduced affine geometry (MIAG) was developed and the simplest (two-component) autolocalized solutions of the nonlinear Dirac equations were found in explicit form. The solitary autolocalized Dirac field waveforms in free space turned out to be spherically symmetric, and, most importantly, this symmetry is dynamical; it is a consequence of the equations of motion.…”

Section: Introductionmentioning

confidence: 99%

“…Being otherwise unwieldy, calculations are significantly simplified by accounting, ab initio, for the earlier discovered [1] dynamical spherical symmetry; below, we call it the "spherical ansatz". The explicit calculations confirm our previous conjecture that only one of the two major types of isolated localized solutions is genuinely stable (has well-defined energy and satisfies all the consistency conditions).…”

Section: Introductionmentioning

confidence: 99%

“…The earlier developed [1] mathematical background for the present work is based on the following ideas and results. It has been observed long ago [4] that if a physical Dirac field is defined at a point in spacetime continuum (the principal differentiable manifold  ), then such field determines the tetrad of Dirac currents.…”

Section: Introductionmentioning

confidence: 99%

“…The physics was naturally brought into this mathematical picture by the equations of motion of the Dirac field. The 28 primary differential identities that were derived in [1] from equations of motion fully determined all the components of the matterinduced affine connection (the Ricci coefficients of rotation of the tetrad) in  and without resorting to a particular coordinate system. Connections determined this way completely defined an affine geometry (endowed with the connection but with no metric), which was dubbed [1] a matter-induced affine geometry.…”

Section: Introductionmentioning

confidence: 99%

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On the Origin of Charge-Asymmetric Matter. III. Properties of Autolocalized Dirac Waveforms

Makhlin¹

2017

JMP

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This paper continues the author's work [1] [2], where a novel framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation was discovered. The previous analysis of solitary waveforms' properties [2] is extended to the four-component Dirac field. It is found that the internal spherical symmetry of the Dirac waveforms is broken to the axial one. The nonlinear Dirac equation is solved and the localized configurations are found analytically. A strict proof that the proper time slowdown is the major mechanism of autolocalization is presented. The previous qualitative conjecture regarding stability or instability of the two types of the waveforms and the origin of cosmological charge asymmetry is supported by detailed analysis. A solution of the problem of mapping between the matter-induced geometry of autolocalized waveforms and the geometry of an ambient Minkowski space is proposed. These results resolve the longstanding puzzle of how the physical Dirac field of real matter becomes a finite-sized particle.

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confidence: 52%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Origin of Charge-Asymmetric Matter. III. Properties of Autolocalized Dirac Waveforms

Makhlin¹

2017

JMP

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On the Origin of Charge-Asymmetric Matter. II. Localized Dirac Waveforms

Makhlin¹

2016

JMP

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This paper continues the author's work [1], where a new framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation discovered. Here, the nonlinear Dirac equation is solved and the localized configurations are found analytically. Of the two possible types of the potentially stationary localized configurations of the Dirac field, only one is stable with respect to the action of an external field and it corresponds to a positive charge. A connection with the global charge asymmetry of matter in the Universe and with the recently observed excess of the cosmic positrons is discussed. 662Alexander Makhlin space, including the infinitesimal displacements in coordinate space.The Dirac currents are employed as the Cartan's moving frame in spacetime which, in its turn, results in the technique of covariant derivatives for the vector and spinor fields. The physics is naturally brought into this mathematical picture by the equations of motion of the Dirac field, which made unnecessary an artificial tangent (pseudo)Euclidean space. Differential identities derived from equations of motion fully determine all the components of the matter-induced affine connection (the Ricci coefficients of rotation of the tetrad) in M and without resorting to a particular coordinate system. Thus determined connections completely define an affine geometry (endowed with the connection but with no metric). Thus defined connection depends on the Dirac field which makes the Dirac equation nonlinear.With known connections, it became possible to find the coordinate lines and coordinate surfaces of the matter-induced affine geometry, which have a clear physical meaning and quite high degree of symmetry. The congruence of lines of the timelike vector current appeared to be normal, thus determining the family of the hypersurfaces of the constant world time τ . The lines of the spacelike axial current appeared to be straight and their congruence normal. They define the surfaces of the constant distance ρ. The two-dimensional surfaces of constant ρ at a given time τ were proved to be just spherical surfaces.Below, the inevitable localization of the Dirac field into particles observed in real world, but not explained by any theory so far, is confirmed by the analytic solutions of the nonlinear Dirac equation in one-body approximation. One of the solutions has maximum near its center and is clearly associated with a stable localized positive charge. Another one has minimum and is sought to be an intrinsically unstable negative charge, which can be only weakly localized by an external field.The content of the paper is organized as follows. In Sec.2 we use the previously developed [1] tools of the matter-induced affine geometry to write down the Dirac equation in its most general coordinate-independent form. Then, in Sec.3 we derive the formulae that connect the Dirac matrices in the principal manifold M and in arithmetic R 4 . In Sec.4, the Dirac equation in written down in a mixed representation, with deri...

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On the Origin of Charge-Asymmetric Matter. I. Geometry of the Dirac Field

Cited by 2 publications

References 17 publications

On the Origin of Charge-Asymmetric Matter. III. Properties of Autolocalized Dirac Waveforms

On the Origin of Charge-Asymmetric Matter. III. Properties of Autolocalized Dirac Waveforms

On the Origin of Charge-Asymmetric Matter. II. Localized Dirac Waveforms

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