Symmetry and renormalisation in quantum field theory

Abstract

Quantum systems governed by non-Hermitian Hamiltonians with symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to -symmetric effective Lagrangians. We show how symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the norm that occurs in Hamiltonian quantum theory. We discuss examples where -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but -symmetric from the start.