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Register a new userClassical model of discrete QFT: Klein Gordon and electromagnetic fields
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Authors
Roman Sverdlov
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The purpose of this paper is to propose a classical model of quantum fields which is local. Yet it admittedly violates relativity as we know it and, instead, it fits within a bimetric model with one metric corresponding to speed of light and another metric to superlumianl signals whose speed is still finite albeit very large. The key obstacle to such model is the notion of functional in the context of QFT which is inherently non-local. The goal of this paper is to stop viewing functionals as fundamental and instead model their emergence from the deeper processes that are based on functions over $mathbb{R}^4$ alone. The latter are claimed to be local in the above bimetric sense.
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A system of coupled kinetic transport equations for the Wigner distributions of a free variable mass Klein-Gordon field is derived. This set of equations is formally equivalent to the full wave equation for electromagnetic waves in nonlinear dispersive media, thus allowing for the description of broadband radiation-matter interactions and the associated instabilities. The standard results for the classical wave action are recovered in the short wavelength limit of the generalized Wigner-Moyal formalism for the wave equation.
The goal of this paper is to re-express QFT in terms of two classical fields living in ordinary space with single extra dimension. The role of the first classical field is to set up an injection from the set of values of extra dimension into the set of functions, and then said injection will be used in order to convert the second field into a coarse grained functional, thereby approximating QFT state. It turns out that this work also has a side-benefit of modeling ensemble of states in terms of one single state which, in turn, is interpretted in the above way. It is important to clarify that by classical we mean functions over ordinary space rather than configuration, Fock or function space. The classical theory that we propose is still non-local.
In this note, we consider discrete nonlinear Klein-Gordon equations with potential. By the pioneering work of Sigal, it is known that for the continuous nonlinear Klein-Gordon equation, no small time periodic solution exists generically. However, for the discrete nonlinear Klein-Gordon equations, we show that there exist small time periodic solutions.
We study the ladder operator on scalar fields, mapping a solution of the Klein-Gordon equation onto another solution with a different mass, when the operator is at most first order in derivatives. Imposing the commutation relation between the dAlembertian, we obtain the general condition for the ladder operator, which contains a non-trivial case which was not discussed in the previous work [V. Cardoso, T. Houri and M. Kimura, Phys.Rev.D 96, 024044 (2017), arXiv:1706.07339]. We also discuss the relation with supersymmetric quantum mechanics.
Three objections to the canonical analytical treatment of covariant electromagnetic theory are presented: (i) only half of Maxwells equations are present upon variation of the fundamental Lagrangian; (ii) the trace of the canonical energy-momentum tensor is not equivalent to the trace of the observed energy-momentum tensor; (iii) the Belinfante symmetrization procedure exists separate from the analytical approach in order to obtain the known observed result. It is shown that the analytical construction from Noethers theorem is based on manipulations that were developed to obtain the compact forms of the theory presented by Minkowski and Einstein; presentations which were developed before the existence of Noethers theorem. By reformulating the fundamental Lagrangian, all of the objections are simultaneously relieved. Variation of the proposed Lagrangian yields the complete set of Maxwells equations in the Euler-Lagrange equation of motion, and the observed energy-momentum tensor directly follows from Noethers theorem. Previously unavailable symmetries and identities that follow naturally from this procedure are also discussed.
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