Borchers' Commutation Relations and Modular Symmetries

TL;DR

This paper explores how certain operators derived from fundamental constructions in quantum field theory commute with translation symmetries, implying that any symmetry implemented by these operators is essentially fixed up to translations.

Contribution

It demonstrates that operators implementing any symmetry in local observable theories are determined up to translations, extending Borchers' commutation relations and modular symmetry results.

Findings

Operators commuting with translations are fixed up to translation.

Symmetries implemented by these operators are constrained to translations.

The results generalize Borchers' relations to broader classes of symmetries.

Abstract

Recently Borchers has shown that in a theory of local observables, certain unitary and antiunitary operators, which are obtained from an elementary construction suggested by Bisognano and Wichmann, commute with the translation operators like Lorentz boosts and \pct-operators, respectively. We conclude from this that as soon as the operators considered implement {\em any} symmetry, this symmetry can be fixed up to at most some translation. As a symmetry, we admit any unitary or antiunitary operator under whose adjoint action any algebra of local observables is mapped onto an algebra which can be localized somewhere in Minkowski space.