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 $\Sigma$, 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 $a^{†}(\bar{x})$, $a(\bar{x})$, 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 $a^{†}(\bar{x})$, 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.

16 pages; A discussion on the probability and the current is included; more references are added

References in corpus (2)

• Causal evolution of wave packets
• Relativistic dynamics, Green function and pseudodifferential operators

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