Non-manifest symmetries in quantum field theory

TL;DR

This paper rigorously analyzes non-manifest symmetries in quantum field theory, revealing the non-uniqueness of associated charge operators and the need for additional physical input to define their action on states.

Contribution

It provides a rigorous axiomatic framework showing that charge operators for non-manifest symmetries are not unique and discusses conditions for their unique determination.

Findings

Charge operators are not unique for non-manifest symmetries.

Vacuum invariance ensures unique charges for translations and Lorentz symmetry.

Supersymmetry charges are not uniquely defined without additional physical input.

Abstract

Non-manifest symmetries are an important feature of quantum field theories (QFTs), and yet their characteristics are not fully understood. In particular, the construction of the charge operators associated with these symmetries is ambiguous. In this paper we adopt a rigorous axiomatic approach in order to address this issue. It turns out that charge operators of non-manifest symmetries are not unique, and that although this does not affect their property as generators of the corresponding symmetry transformations, additional physical input is required in order to determine how they act on states. Applying these results to the examples of spacetime translation and Lorentz symmetry, it follows that the assumption that the vacuum is the unique translationally invariant state is sufficient to uniquely define the charges associated with these symmetries. In the case of supersymmetry though there exists no such physical requirement, and this therefore implies that the supersymmetric charge, and hence the supersymmetric space of states, is not uniquely defined.