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Path integral representation of the Dirac equation and integration order

Han J.F. Geurdes

Published Sep 5, 2022• 0 citations
Publisher

Abstract

In this paper we ask if there is an interesting twist in mathematics of the Feynman propagator of the present day [1] formalism for the Dirac equation. The case studied is with only a potential function V . If there is no special integration order per step for e.g. dx(0) = dx1dx2dx3 and dp(0) = dp1dp2dp3, then a delta function occurs when c∆t ≈ 0 and −∞ <pj <∞ for j = 1,2,3. The found geoemtry of propagation can be tested in solid state chemistry. A crystal structure where only a two dimensional propagation of quasi particles can be created must be attached to the final step of the path integral. Neutron emission giving chemical isotopes is briefly discussed.

DOI
10.26434/chemrxiv-2022-xzz3r

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In this paper we ask if there is an interesting twist in mathematics of the Feynman propagator of the present day [1] formalism for the Dirac equation. The case studied is with only a potential function V . If there is no special integration order per step for e.g. dx(0) = dx1dx2dx3 and dp(0) = dp1dp2dp3, then a delta function occurs when c∆t ≈ 0 and −∞ <pj <∞ for j = 1,2,3. The found geoemtry of propagation can be tested in solid state chemistry. A crystal structure where only a two dimensional propagation of quasi particles can be created must be attached to the final step of the path integral. Neutron emission giving chemical isotopes is briefly discussed.
The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded Dirac particles is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. The limit of integration techniques for evaluating path integral is only valid for the piecewise constant potential. Finally, the Dirac propagator is expressed in terms of standard special functions.
The feasibility of applying the path-integral formalism for solving the Dirac equation is shown in the case of a free particle for which the Dirac propagator is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. Furthermore, the propagator for a Dirac electron in a constant magnetic field is indirectly obtained from the propagator of an auxiliary wave equation by evaluating a world-line (space-time path) integral. The spectrum of the Dirac equation is also, in this case, extracted from an auxiliary propagator.
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.
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.
(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex