positive, and the dynamics is governed by unitary time evolution. After a review of $\mathcal{P T}$-symmetric quantum mechanics, new results are presented here in which the $\mathcal{C}$ operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.
address: 'Blackett Laboratory, Imperial College, London SW7 2BZ, UK'
author:
- |
    Carl M. Bender[^1], Joachim Brod, André Refig,\
    and Moritz Reuter
title: |
    The $\mathcal{C}$ Operator in $\mathcal{PT}$-Symmetric\
    Quantum Theories
---

Introduction {#s1}
============

In this paper we present a brief review of some recent work on an alternative way to formulate of quantum mechanical models and present new results concerning the perturbative calculation of what has become known in this theory as the $\mathcal{C}$ operator.

In the conventional formulation of quantum mechanics the Hamiltonian $H$, which incorporates the symmetries and specifies the dynamics of a quantum theory, must be Hermitian: $H=H^\dagger$. The usual meaning of the symbol $\dagger$, which indicates Dirac Hermitian conjugation, is combined transpose and complex conjugation. It is commonly thought that a Hamiltonian must be Hermitian in order to ensure that the energy spectrum (the eigenvalues of $H$) is real and that the time evolution of the theory is unitary (probability is conserved in time). 