Manifestly Covariant Canonical Quantization of the Scalar Field and Particle Localization

TL;DR

This paper develops a Lorentz covariant canonical quantization framework for scalar fields, enabling well-behaved particle localization and covariant position operators, advancing the understanding of relativistic quantum field theory.

Contribution

It introduces a manifestly covariant canonical quantization method for scalar fields, defining a covariant position operator and analyzing its properties under Lorentz transformations.

Findings

The quantum position operator has proper Lorentz transformation properties.

Single-particle wave functions satisfy a covariant Foldy equation.

The covariant center of mass operator's expectation values are computed.

Abstract

Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface $\Sigma$, which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to $\Sigma$. Position on $\Sigma$ is determined by a 4-vector ${\bar x}^\mu$. The corresponding quantum position operator, formed in terms of the operators $a^\dagger ({\bar x})$, $a({\bar x})$, that create/annihilate particles on $\Sigma$, has thus well behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with $a^\dagger ({\bar x})$, the superposition coefficients forming multiparticle wave packet profiles---wave functions, including a single particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center of mass operator is introduced and its expectation values in a generic multiparticle state calculated.