History state formalism for Dirac's theory

TL;DR

This paper introduces a novel history state formalism for Dirac particles using quantum clocks, deriving Dirac's equation from a timeless perspective and exploring implications for particle evolution and entanglement.

Contribution

It develops a history state formalism for Dirac particles, deriving Dirac's equation from a timeless Wheeler-DeWitt-like equation and introducing a second quantum clock for observable evolution.

Findings

Dirac's equation derived from a timeless global state formalism.

Invariant product ensures standard Dirac norm and Lorentz invariance.

Analytical expressions for space-time density and electron-time entanglement.

Abstract

We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Dirac's equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Dirac's norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter $\tau$ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of an hypothetical superposition of different masses. Analytical expressions for both space-time density and electron-time entanglement are provided for two particular families of electron's states, the former including Pryce localized particles.