The Analysis of Time-Space Translations in Quantum Fields

TL;DR

This paper examines the indefinite metric structure of time-space translations in relativistic quantum fields, highlighting how positivity conditions relate to gauge invariance and the structure of asymptotic particle states.

Contribution

It introduces a new perspective on the indefinite unitary representations of translations and their relation to gauge invariance and particle state positivity in quantum field theory.

Findings

Indefinite inner products lead to non-diagonalizable translation representations.

Positive unitarity requires the nilpotent part of translation operators to vanish.

Gauge invariance relates to the projection onto eigenstates of translations.

Abstract

I discuss the indefinite metrical structure of the time-space translations as realized in the indefinite inner products for relativistic quantum fields, familiar in the example of quantum gauge fields. The arising indefinite unitary nondiagonalizable representations of the translations suggest as the positive unitarity condition for the probability interpretable positive definite asymptotic particle state space the requirement of a vanishing nilpotent part in the time-space translations realization. A trivial Becchi-Rouet-Stora charge (classical gauge invariance) for the asymptotics in quantum gauge theories can be interpreted as one special case of this general principle - the asymptotic projection to the eigenstates of the time-space translations.