Manifestly Covariant Canonical Quantization of the Scalar Field and Particle Localization

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 $Σ$, which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to $Σ$. Position on $Σ$ is determined by a 4-vector ${\bar x}^μ$. The corresponding quantum position operator, formed in terms of the operators $a^\dagger ({\bar x})$, $a({\bar x})$, that create/annihilate particles on $Σ$, 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.