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It has recently been shown that a non-Hermitian Hamiltonian $H$ possessing an unbroken $\mathcal{PT}$ symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator $\mathcal{C}$, which was originally defined as a sum over the eigenfunctions of $H$. However, using this definition to calculate $\mathcal{C}$ is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating $\mathcal{C}$ directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the $\mathcal{C}$ operator in quantum field theory. The $\mathcal{C}$ operator is a new time-independent observable in $\mathcal{PT}$-symmetric quantum field theory.

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Updated: April 5, 2004