{PT}-Symmetric Quantum Electrodynamics---{PT}QED
The construction of PT -symmetric quantum electrodynamics is reviewed. In particular, the massless version of the theory in 1 + 1 dimensions (the Schwinger model) is solved. Difficulties with unitarity of the S-matrix are discussed. 1. PT QED Quantum electrodynamics (QED) is by far the most successful physical theory ever devised [1]. However, although its reach includes all of atomic physics, there are a myriad of phenomena that we do not understand. Thus it is essential to explore alternative theories, in the hope that we may be able to describe aspects of the world that are as yet not under our understanding. One very promising new approach to quantum theories are those included under the rubric of non-Hermitian theories, in particular theories in which invariance under the combined operation of space and time reflection PT replaces mathematical Dirac Hermiticity in order to guarantee unitarity of the theory. (For a recent review, see [2].) Little work, however, has been done on applying this idea to quantum field theory. This paper represents our continuing effort to develop a PT -symmetric version of quantum electrodynamics, in the hope that a consistent theory, with a unitary S-matrix, results that may eventually find physical applications in nature. 1.1. Transformation properties At the first International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics [3] a PT -symmetric version of quantum electrodynamics was proposed. A non-Hermitian but PT -symmetric electrodynamics is based on the assumption of novel transformation properties of the electromagnetic fields under parity P transformations, that is, P : E → E, B → −B, A → A, A → −A, (1) PT -Symmetric Quantum Electrodynamics—PT QED 2 just the statement that the four-vector potential is assumed to transform as an axial vector. Under time reversal T , the transformations are assumed to be conventional, T : E → E, B → −B, A → −A, A → A. (2) Fermion fields are also assumed to transform conventionally. We use the metric g = diag(−1, 1, 1, 1). 1.2. Lagrangian and Hamiltonian The Lagrangian of the theory then possesses an imaginary coupling constant in order that it be invariant under the product of these two symmetries: L = − 4 F Fμν − 1 2 ψγγ 1 i ∂μψ − m 2 ψγψ + i 2 ψγγeqψAμ. (3) Here, because we are discarding Hermiticity as a physical requirement, it is most appropriate to use a “real” field formulation, where correspondingly the (antisymmetric, imaginary) charge matrix q = σ2 appears. Furthermore, γ γ is symmetric and γ is antisymmetric. In the radiation (Coulomb) gauge ∇ · A = 0, the dynamical variables are A and ψ, and the canonical momenta are πA = −ET , πψ = i 2ψ, where T denotes the transverse part, and so the relation between the Hamiltonian and Lagrangian densities are H = E + E · ∇A + i 2 ψψ̇ − L. (4) Then, if integrate by parts and use ∇ · E = ij, we find that the corresponding Hamiltonian is H = ∫ (dr) { 1 2 (E + B) + 1 2 ψ [ γγ ( 1 i ∇k − ieqAk )