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This 2005 paper by C.M. Bender, I. Cavero-Pelaez, K.A. Milton and K.V. Shajesh considers the situation where the Hamiltonian for quantum electrodynamics becomes non-Hermitian if the unrenormalized electric charge e is taken to be imaginary.
''However, if one also specifies that the potential $A^μ$ in such a theory transforms as a pseudovector rather than a vector, then the Hamiltonian becomes PT symmetric. The resulting non-Hermitian theory of electrodynamics is the analog of a spinless quantum field theory in which a pseudoscalar field $ϕ$ has a cubic self-interaction of the form $iϕ^3$. The Hamiltonian for this cubic scalar field theory has a positive spectrum, and it has recently been demonstrated that the time evolution of this theory is unitary. The proof of unitarity requires the construction of a new operator called C, which is then used to define an inner product with respect to which the Hamiltonian is self-adjoint. In this paper the corresponding C operator for non-Hermitian quantum electrodynamics is constructed perturbatively. This construction demonstrates the unitarity of the theory. Non-Hermitian quantum electrodynamics is a particularly interesting quantum field theory model because it is asymptotically free.''