PT-Symmetric Quantum Electrodynamics

Question

Referee this paper: arXiv:hep-th/0501180v1 by Carl M. Bender, Ines Cavero-Pelaez,

Kimball A. Milton, K. V. Shajesh

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Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

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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.''

Comments

From Abstract: "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 $A_{\mu}$ in such a theory transforms as a pseudovector rather than a vector, then the Hamiltonian becomes PT symmetric. ... 

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."

I think the paper is a great achievement. Indeed, if for some particular reason one wants to make a QED calculation with an imaginary charge (that is not forbidden by any human law), one can take advantage of this construction and be happy with the unitary evolution. The latter provides conservation of probability in the theory. The necessary C-operator $C$ can be constructed perturbatively. In addition, the theory is asymptotically free which is another thing to enjoy.

The only thing I could not understand was whether the perturbative series in this theory became free from Dyson's argument of being asymptotical.