PT-Symmetric Quantum Electrodynamics and Unitarity

TL;DR

This paper reviews the development of PT-symmetric quantum electrodynamics, discussing its potential for unitarity and probability conservation, and addressing the challenges in establishing its consistency as a quantum field theory.

Contribution

It provides a comprehensive review of PT-symmetric QED, highlighting the difficulties in proving unitarity and the implications for quantum field theory consistency.

Findings

PT-symmetric QED faces challenges in maintaining unitarity.

Unitarity issues relate to the photon propagator and Green's functions.

The review discusses the theoretical status and open questions in PT-symmetric quantum field theories.

Abstract

More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, $\mathcal{PT}$. It was shown that if $\mathcal{PT}$ is unbroken, energies were, in fact, positive, and unitarity was satisifed. Since quantum mechanics is quantum field theory in 1 dimension, time, it was natural to extend this idea to higher-dimensional field theory, and in fact an apparently viable version of $\mathcal{PT}$-invariant quantum electrodynamics was proposed. However, it has proved difficult to establish that the unitarity of the scattering matrix, for example, the K\"all\'en spectral representation for the photon propagator, can be maintained in this theory. This has led to questions of whether, in fact, even quantum mechanical systems are consistent with probability conservation when Green's functions are examined, since the latter have to possess physical requirements of analyticity. The status of $\mathcal{PT}$QED will be reviewed in this report, as well as the general issue of unitarity.