Feynman Checkerboard Research Articles

Path integral of the relativistic point particle in Minkowski space

In this article, we analyze the fundamental global and local symmetries involved in the action for the free relativistic point particle in Minkowski space. Moreover, we identify a hidden local symmetry, whose explicit consideration and factorization utilizing of a Fujikawa prescription leads to the construction of relativistic two-point correlation functions that satisfy the Chapman-Kolmogorov identity. By means of a detailed topological analysis, we find three different relativistic correlation functions (orthochronous, spacelike, and Feynman) which are obtained from the exclusive integration of paths within different sectors in Minkowski space. Finally, the connection of this approach to the Feynman checkerboard construction is explored.

Quantum mechanical motion of relativistic particle in non-continuous spacetime

The quantum mechanical motion of a relativistic particle in a non-continuous spacetime is investigated. The spacetime model is a dense, rationale subset of two-dimensional Minkowski spacetime. Solutions of the Dirac equation are calculated using a generalized version of Feynman's checkerboard model. They turn out to be closely related to the continuum propagator.

Discrete physics and the Dirac equation

We rewrite the 1 + 1 Dirac equation in light cone coordinates in two significant forms, and solve them exactly using the classical calculus of finite differences. The complex form yields “Feynman's checkerboard” — a weighted sum over lattice paths. The rational, real form can also be interpreted in terms of bit-strings.

Spinor chain path integral for the Dirac equation

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, 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 Amu the amplitude for a path C is multiplied by exp(-ie integral cAmu dxmu ), requiring spacetime coordinates to specify Amu (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.