Preconjugate variables in quantum field theory and their use

TL;DR

This paper explores preconjugate variables in quantum field theory, examining their geometric and group-theoretic properties, representations on Fock space, and applications to spacetime structures, especially in Minkowski space.

Contribution

It provides a detailed study of preconjugate variables, including their geometric and group-theoretic aspects, and introduces wedge variables through a non-trivial massless limit.

Findings

Preconjugate variables have meaningful representations on Fock space.

The massless limit leads to wedge variables with unique properties.

Applications extend to more general spacetimes.

Abstract

Preconjugate variables X have commutation relations with the energy-momentum P of the respective system which are of a more general form than just the Hamiltonian one. Since they have been proven useful in their own right for finding new spacetimes we present here a study of them. Interesting examples can be found via geometry: motions on the mass-shell for massive and massless systems, and via group theory: invariance under special conformal transformations of mass-shell, resp. light-cone -- both find representations on Fock space. We work mainly in ordinary fourdimensional Minkowski space and spin zero. The limit process from non-zero to vanishing mass turns out to be non-trivial and leads naturally to wedge variables. We point out some applications and extension to more general spacetimes. In a companion paper we discuss the transition to conjugate pairs.