Quantum Mechanics of Proca Fields

TL;DR

This paper develops a comprehensive quantum framework for Proca fields by constructing a Lorentz-invariant Hilbert space, analyzing conserved currents, symmetries, and operators, thus advancing the understanding of massive spin-1 quantum fields.

Contribution

It introduces a five-parameter family of positive-definite inner products for Proca fields, enabling a unitary quantum description without positive-frequency restrictions.

Findings

Constructed a Lorentz-invariant Hilbert space for Proca fields.

Analyzed conserved current densities and global gauge symmetries.

Derived operators for position, momentum, helicity, spin, and angular momentum.

Abstract

We construct the most general physically admissible positive-definite inner product on the space of Proca fields. Up to a trivial scaling this defines a five-parameter family of Lorentz invariant inner products that we use to construct a genuine Hilbert space for the quantum mechanics of Proca fields. If we identify the generator of time-translations with the Hamiltonian, we obtain a unitary quantum system that describes first-quantized Proca fields and does not involve the conventional restriction to the positive-frequency fields. We provide a rather comprehensive analysis of this system. In particular, we examine the conserved current density responsible for the conservation of the probabilities, explore the global gauge symmetry underlying the conservation of the probabilities, obtain a probability current density, construct position, momentum, helicity, spin, and angular momentum operators, and determine the localized Proca fields. We also compute the generalized parity ($\cP$), generalized time-reversal ($\cT$), and generalized charge or chirality ($\cC$) operators for this system and offer a physical interpretation for its $\cP\cT$-, $\cC$-, and $\cC\cP\cT$-symmetries.