PT-Symmetric Quantum Electrodynamics

TL;DR

This paper explores PT-symmetric non-Hermitian quantum electrodynamics, demonstrating its unitarity and asymptotic freedom by constructing the C operator perturbatively, thus establishing a consistent quantum field theory framework.

Contribution

It constructs the C operator for PT-symmetric QED perturbatively, proving the theory's unitarity and highlighting its asymptotic freedom as a novel quantum field theory model.

Findings

The non-Hermitian PT-symmetric QED has a positive spectrum.

The theory's unitarity is established through the construction of the C operator.

PT-symmetric QED is asymptotically free.

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. The resulting non-Hermitian theory of electrodynamics is the analog of a spinless quantum field theory in which a pseudoscalar field $\phi$ has a cubic self-interaction of the form $i\phi^3$. 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.