A Canonical Decomposition in Collective and Relative Variables of a Klein-Gordon Field in the Rest-Frame Wigner-Covariant Instant Form

TL;DR

This paper reformulates the canonical decomposition of a Klein-Gordon field on spacelike hypersurfaces, deriving a complete canonical reduction in the rest-frame Wigner form and extending it to charged fields and particle-field interactions.

Contribution

It introduces a reformulation of the Klein-Gordon field decomposition on hypersurfaces, providing a complete canonical reduction and new definitions of the center-of-mass and phase variables.

Findings

Derived the canonical center-of-mass variable as a phase of the field.

Extended the formalism to charged Klein-Gordon fields with combined phase centers.

Analyzed the coupling between particles and fields in the reduced phase space.

Abstract

The canonical decomposition of a real Klein-Gordon field in collective and relative variables proposed by Longhi and Materassi is reformulated on spacelike hypersurfaces. This allows to obtain the complete canonical reduction of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant instant form of dynamics. From the study of Dixon's multipoles for the energy-momentum tensor on the Wigner hyperplanes we derive the definition of the canonical center-of-mass variable for a Klein-Gordon field configuration: it turns out that the Longhi-Materassi global variable should be interpreted as a center of phase of the field configuration. A detailed study of the kinematical "external" and "internal" properties of the field configuration on the Wigner hyperplanes is done. The construction is then extended to charged Klein-Gordon fields: the centers of phase of the two real components can be combined to define a global center of phase and a collective relative variable describing the action-reaction between the two Feshbach-Villars components of the field with definite sign of energy and charge. The Dixon multipoles for both the energy-momentum and the electromagnetic current are given. Also the coupling of the Klein-Gordon field to scalar relativistic particles is studied and it is shown that in the reduced phase space, besides the particle and field relative variables, there is also a collective relative variable describing the relative motion of the particle subsytem with respect to the field one.