Quantum Simulation of Hyperbolic Equations and the Nonexistence of a Dirac Path Measure

TL;DR

This paper investigates why a proper probability measure cannot represent the Dirac equation in Minkowski space, linking measure-theoretic obstructions to the indefinite metric and distributional issues.

Contribution

It unifies two perspectives on the measure-theoretic obstructions to path integral representations of the Dirac equation in Minkowski space.

Findings

Distributional nature of the Dirac propagator obstructs nonnegative transition kernels.

Indefinite Minkowski metric prevents positivity of the action, causing oscillatory integrals.

Unified view of measure-theoretic obstructions in relativistic quantum equations.

Abstract

We revisit the longstanding issue of why no well defined probability measure exists corresponding to a classical (Kolmogorov) path integral representation of the Dirac equation in Minkowski space. Two complementary perspectives are compared: (i) Zastawniak's observation that the distributional character of the Dirac propagator (presence of derivatives of the delta distribution) obstructs the construction of a nonnegative transition kernel, and (ii) the indefinite signature of the Minkowski metric which prevents positivity of the action and yields oscillatory integrals. We show how these viewpoints can be unified as different manifestations of a single mathematical obstruction from measure theoretical point of view, and we discuss consequences for stochastic representations of relativistic first-order equations.