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symmetry and renormalisation in quantum field theory
Authors: Bender Carl M , Felski Alexander , Klevansky Sandra P , Sarkar Sarben 
Published: October 2021

Quantum systems governed by non-Hermitian Hamiltonians with

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symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that

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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

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-symmetric effective Lagrangians. We show how

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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

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-symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the

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norm that occurs in Hamiltonian quantum theory. We discuss examples where

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-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

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-symmetric from the start.