Time Operator in Relativistic Quantum Mechanics

TL;DR

This paper introduces a self-adjoint relativistic time operator for spin-1/2 particles, modifies Dirac's equation to allow discrete time, and explores implications like negative energy states and energon creation, addressing longstanding quantum mechanics debates.

Contribution

It presents the first exact self-adjoint relativistic time operator and demonstrates its implications, resolving Pauli's objection and expanding understanding of time in quantum mechanics.

Findings

Existence of a self-adjoint relativistic time operator.

Particles can occupy negative energy levels with small probability.

Proposal of energons as a new bosonic field.

Abstract

It is first shown that the Dirac's equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint $4\times 4$ relativistic time operator for spin-$\frac{1}{2}$ particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly small but non-zero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli's objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found.