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Frontiers in Physics

The Feynman chessboard model in 3 + 1 dimensions

Garnet N. Ord

Published Nov 28, 2023• 2 citations
Publisher

Abstract

The chessboard model was Feynman’s adaptation of his path integral method to a two-dimensional relativistic domain. It is shown that chessboard paths encode information about the contiguous pairs of paths in a spacetime plane, as required by discrete worldlines in Minkowski space. The application of coding by pairs in a four-dimensional spacetime is then restricted by the requirements of the Lorentz transformation, and the implementation of these restrictions provides an extension of the model to 4D, illuminating the relationship between relativity and quantum propagation.

DOI
10.3389/fphy.2023.1286030

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The chessboard model was Feynman’s adaptation of his path integral method to a two-dimensional relativistic domain. It is shown that chessboard paths encode information about the contiguous pairs of paths in a spacetime plane, as required by discrete worldlines in Minkowski space. The application of coding by pairs in a four-dimensional spacetime is then restricted by the requirements of the Lorentz transformation, and the implementation of these restrictions provides an extension of the model to 4D, illuminating the relationship between relativity and quantum propagation.
The mathematical similarities between non-relativistic wavefunction propagation in quantum mechanics and image propagation in scalar diffraction theory are used to develop a novel understanding of time and paths through spacetime as a whole. It is well known that Feynman’s original derivation of the path integral formulation of non-relativistic quantum mechanics uses time-slicing to calculate amplitudes as sums over all possible paths through space, but along a definite curve through time. Here, a 3+1D spacetime wave distribution and its 4-momentum dual are formally developed which have no external time parameter and therefore cannot change or evolve in the usual sense. Time is thus seen “from the outside”. A given 3+1D momentum representation of a system encodes complete dynamical information, describing the system’s spacetime behavior as a whole. A comparison is made to the mathematics of holograms, and properties of motion for simple systems are derived.
Recently quantum walks have been considered as a possible fundamental description of the dynamics of relativistic quantum fields. Within this scenario we derive the analytical solution of the Weyl walk in dimensions. We present a discrete path-integral formulation of the Feynman propagator based on the binary encoding of paths on the lattice. The derivation exploits a special feature of the Weyl walk, that occurs also in other dimensions, that is closure under multiplication of the set of the walk transition matrices. This result opens the perspective of a similar solution in the case.
A path integral that reduces to Feynman's checkerboard rule in one space dimension is found for the retarded Dirac propagator in three space dimensions. The only variable are two-component spinors and a binary chirality variable. No action functional is employed. Each spinor together with a chirality corresponds to a spacetime displacement during a time epsilon . A sequence of spinors and chiralities determines a polygonal spacetime path, the first and last spinors specify the initial and final spin states. The transition amplitude for a sequence is given by ( nu N mod nu N-1 )( nu N-1 mod . . . mod nu 2 )( nu 2 mod nu 1 )(i epsilon m) R where nu i are the spinors, < mod > is the ordinary inner product in spin space, m is the electron mass and R is the number of times the chirality switches. Integrating over all sequences corresponding to a given displacement yields the Dirac propagator in the limit epsilon to 0. With epsilon >0 this formulation provides an alternative to the point particle model of the electron. In an external electromagnetic potential A mu the amplitude for a path C is multiplied by exp(-ie integral c A mu dx mu ), requiring spacetime coordinates to specify A mu (x). The usual perturbation expansion is derived from this rule. These results are extended to nonAbelian gauge potentials. Quantised interaction of particles is not treated.
In relativistic quantum constraint mechanics the state of a physical system is constrained to a 3-dimensional subspace of Minkowski 4-space. Fourier transformation can be used to relate this state between constraint spaces in 4-position and 4-momentum space. It is shown that integral transforms of this nature can be carried out using Lorentz-invariant 3-dimensional constraint space coordinates such that a complete equivalence class of 4-space representations can be constructed from the transform. This method is further applied to develop a relativistic generalization of the Segal-Bargmann transformation that leads to the representation of quantum systems in a three-dimensional subspace of Bargmann 4-space.