Relativistic quantum theory without quantized fields

Morgan, Peter

A classical statistical field theory hidden variable model for the quantized Klein-Gordon model is constructed that preserves relativistic signal locality and is relativistically covariant, but is at the same time relativistically nonlocal, paralleling the Hegerfeldt nonlocality of quantum theory. It is argued that the relativistic nonlocality of this model is acceptable to classical physics, but in any case the approach taken here characterizes the nonlocality of the quantized Klein-Gordon model in terms of concepts from classical statistical field theory.

Lawrence Paul Horwitz

Peter Bongaarts

Giacomo Gradenigo, Roberto Livi, Luca Salasnich

Abstract First of all we shortly illustrate how the symplectic quantization scheme (Gradenigo and Livi, Found Phys 51(3):66, 2021) can be applied to a relativistic field theory with self-interaction. Taking inspiration from the stochastic quantization method by Parisi and Wu, this procedure is based on considering explicitly the role of an intrinsic time variable, associated with quantum fluctuations. The major part of this paper is devoted to showing how the symplectic quantization scheme can be extended to the non-relativistic limit for a Schrödinger-like field. Then we also discuss how one can obtain from this non-relativistic theory a linear Schrödinger equation for the single-particle wavefunction. This further passage is based on a suitable coarse-graining procedure, when self-interaction terms can be neglected, with respect to interactions with any external field. In the Appendix we complete our survey on symplectic quantization by discussing how this scheme applies to a non-relativistic particle under the action of a generic external potential.

George Jaroszkiewicz

Introduction The quantized harmonic oscillator has proven to be perhaps the most important system in continuous time (CT) quantum mechanics (QM). It was the atomic oscillators assumed to line the containing walls of a black-body radiation field that Planck referred to when he postulated the quantization of energy in 1900 (Planck, 1900, 1901), thereby triggering the development of QM. Subsequently, the quantized harmonic oscillator resurfaced in many contexts, most notably in quantum optics and relativistic quantum field theory, including string theory. There are two reasons for the importance of the quantized oscillator: (i) it can be solved completely; and (ii) it has ladder operators that generate Planck's quantized energy levels in particle QM and particle-like states in relativistic quantum field theory. Temporal discretization does not destroy the properties of the oscillator. Rather, it enhances them. We shall show that the parabolic barrier encountered in the classical theory of the discrete time (DT) oscillator, which was discussed in the previous chapter, provides a natural upper bound for important physical quantities such as energy and momentum. In other words, the quantized DT oscillator has a natural cutoff. This may turn out to be of fundamental importance in the regularization (renormalization) of relativistic quantum field theories, where there exist immense problems related to the fact that the CT oscillator has no upper bound to its energy spectrum.

W-Y. P. Hwang, Ta‐You Wu