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Physical Review A

Feynman path summation for the Dirac equation: An underlying one-dimensional aspect of relativistic particle motion

Gerald Rosen

Published Aug 1, 1983• 15 citations
Publisher

Abstract

It is observed that the Feynman path summation for the one-dimensional Dirac equation can be projected into three spatial dimensions to yield a path-summation formula for physical spin-\textonehalf{} particles of nonzero mass. Since the three-space projection matrix is independent of time and does not involve the particle's mass, relativistic motion governed by the Dirac equation has an underlying one-dimensional aspect.

DOI
10.1103/physreva.28.1139

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•Jan 1, 2002•1 citation
It is observed that the Feynman path summation for the one-dimensional Dirac equation can be projected into three spatial dimensions to yield a path-summation formula for physical spin-\textonehalf{} particles of nonzero mass. Since the three-space projection matrix is independent of time and does not involve the particle's mass, relativistic motion governed by the Dirac equation has an underlying one-dimensional aspect.
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.
Rigorous path integral formulas are given which represent, in two space-time dimensions, the fundamental solution of the Cauchy problem for the Dirac equation as well as the retarded and advanced propagators for the Dirac particle. It is also shown that the theory can be applied to a free particle and a particle in a central electric field in four space-time dimensions and reveals some aspects of the path integral. Heuristically discussed is the connection with the phase space path integral or Hamiltonian path integral.
The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mechanics and in QED are neither unified nor directly connected. This suggests the existence of a missing path integral that bridges relativistic action and the Dirac equation at the single-particle level. In this work, we analyze the consistency and completeness of existing path-integral theories and identify a spinor path integral that fills this gap. Starting from a relativistic action written in spinor form, we construct a spacetime path integral whose kernel reproduces the Dirac Hamiltonian. The resulting formulation provides a direct link between the relativistic classical action and the Dirac equation, and it naturally extends the scalar relativistic path integral developed in our earlier work. Beyond establishing this structural connection, the spinor path integral offers a new way to interpret the origin of classical mechanics for the Dirac equation and suggests a spacetime mechanism for spin and quantum nonlocal correlations. These features indicate that the spinor path integral can serve as a unifying framework for existing path-integral approaches and as a starting point for further investigations into the spacetime structure of quantum mechanics.
Dirac particle dynamics is encoded as a unitary path summation rule and implemented on a qubit array, where the qubit array represents both spacetime and the fermions contained therein. The unitary path summation rule gives a quantum algorithm to model a many-body system of Dirac particles in a gauge field with Lorentz invariance down to the grid scale (Planck scale)--the lattice-based model neither suffers the Fermi-sign problem nor breaks Lorentz invariance. Yet, for the Dirac Hamiltonian to generate the unitary evolution of the 4-spinor field at the Planck scale, there is time dilation between the shortest observable time near a single space point and that time measured at long-wavelength scales. We find gravitational time dilation where the model space around each point (with an even number of qubits) is curved like the space around a Schwarzschild black hole.