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akhmeteli
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- TL;DR Summary
- My journal article on classical models of quantum gauge theories
My article has been published in Quantum Reports.
Expanded abstract:
There is currently no consensus on the interpretation of quantum theory, so this article may be of interest as it contains a review and new results on some relevant mathematical models emulating well-known quantum theories, such as scalar electrodynamics (Klein-Gordon-Maxwell electrodynamics), spinor electrodynamics (Dirac-Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations, so the models arguably do not require interpretation any more than classical electrodynamics and are classical in this sense.
For example, in the case of scalar electrodynamics, Schrödinger noticed in 1952 that the scalar complex wave function can be made real by a gauge transformation. It turned out that, after that, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe independent evolution of the electromagnetic field.
Similar results were obtained for spinor electrodynamics (which is more realistic) at the expense of introduction of a complex 4-potential of electromagnetic field. There are reasons to believe that this limitation can be removed. One of the reasons is that Schrödinger's observation was extended to the Dirac equation in electromagnetic field: three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. Furthermore, a similar result was obtained for the Dirac equation in the Yang-Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general.
The resulting non-second-quantized theories describing independent evolution of a gauge field can be embedded into quantum field theories using a generalization of the Carleman linearization (Kowalski, nightlight). For a system of nonlinear partial differential equations, this procedure generates a system of linear equations in the Fock space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter.
One real function being enough to describe matter in some well-established theories, such as the Dirac equation in an arbitrary electromagnetic field, suggests some "symmetry" between positive and negative frequencies and, therefore, particles and antiparticles. As a result, one-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. Another motivation for such models is the similarity of dispersion relations for quantum theories of matter, such as the Klein-Gordon equation, and some simple plasma models.
A criterion was developed for approximation of continuous charge density distributions with integer total charge by discrete ones with integer charges based on the equality of partial Fourier sums. An example of such approximation is computed using the homotopy continuation method. It was also proven that for any number of Fourier coefficients one can always find a discrete distribution satisfying the criterion. An example mathematical model is proposed. A modification of the description for composite particles, such as nucleons or large molecules, describes them as collections including a composite particle and a large number of pairs of elementary particles and antiparticles.
While it is not clear if there is some reality behind such a description, it can become a basis of an interesting model of quantum mechanics. For example, it can offer an intuitive picture of the double-slit experiment. It also seems to enable simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative, whereas, according to Feynman, "The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate."
Expanded abstract:
There is currently no consensus on the interpretation of quantum theory, so this article may be of interest as it contains a review and new results on some relevant mathematical models emulating well-known quantum theories, such as scalar electrodynamics (Klein-Gordon-Maxwell electrodynamics), spinor electrodynamics (Dirac-Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations, so the models arguably do not require interpretation any more than classical electrodynamics and are classical in this sense.
For example, in the case of scalar electrodynamics, Schrödinger noticed in 1952 that the scalar complex wave function can be made real by a gauge transformation. It turned out that, after that, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe independent evolution of the electromagnetic field.
Similar results were obtained for spinor electrodynamics (which is more realistic) at the expense of introduction of a complex 4-potential of electromagnetic field. There are reasons to believe that this limitation can be removed. One of the reasons is that Schrödinger's observation was extended to the Dirac equation in electromagnetic field: three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. Furthermore, a similar result was obtained for the Dirac equation in the Yang-Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general.
The resulting non-second-quantized theories describing independent evolution of a gauge field can be embedded into quantum field theories using a generalization of the Carleman linearization (Kowalski, nightlight). For a system of nonlinear partial differential equations, this procedure generates a system of linear equations in the Fock space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter.
One real function being enough to describe matter in some well-established theories, such as the Dirac equation in an arbitrary electromagnetic field, suggests some "symmetry" between positive and negative frequencies and, therefore, particles and antiparticles. As a result, one-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. Another motivation for such models is the similarity of dispersion relations for quantum theories of matter, such as the Klein-Gordon equation, and some simple plasma models.
A criterion was developed for approximation of continuous charge density distributions with integer total charge by discrete ones with integer charges based on the equality of partial Fourier sums. An example of such approximation is computed using the homotopy continuation method. It was also proven that for any number of Fourier coefficients one can always find a discrete distribution satisfying the criterion. An example mathematical model is proposed. A modification of the description for composite particles, such as nucleons or large molecules, describes them as collections including a composite particle and a large number of pairs of elementary particles and antiparticles.
While it is not clear if there is some reality behind such a description, it can become a basis of an interesting model of quantum mechanics. For example, it can offer an intuitive picture of the double-slit experiment. It also seems to enable simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative, whereas, according to Feynman, "The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate."