Almost zero-dimensional PT-symmetric quantum field theories

TL;DR

This paper explores PT-symmetric quantum field theories, extending a nonperturbative dimensional expansion technique to complex non-Hermitian models, and demonstrates that the ground-state energy remains real for certain parameters.

Contribution

It applies a 1992 dimensional expansion method to PT-symmetric quantum field theories, showing the ground-state energy is real for psilon0, which was previously uninvestigated.

Findings

Ground-state energy is real for psilon0 in the studied models.

First two coefficients in the dimensional expansion are calculated.

The approach confirms the physical acceptability of these complex theories.

Abstract

In 1992 Bender, Boettcher, and Lipatov proposed in two papers a new and unusual nonperturbative calculational tool in quantum field theory. The objective was to expand the Green's functions of the quantum field theory as Taylor series in powers of the space-time dimension D. In particular, the vacuum energy for a massless \phi^{2N} (N=1,2,3,...) quantum field theory was studied. The first two Taylor coefficients in this dimensional expansion were calculated {\it exactly} and a set of graphical rules were devised that could be used to calculate approximately the higher coefficients in the series. This approach is mathematically valid and gives accurate results, but it has not been actively pursued and investigated. Subsequently, in 1998 Bender and Boettcher discovered that PT-symmetric quantum-mechanical Hamiltonians of the form H=p^2+x^2(ix)^\epsilon, where \epsilon\geq0, have real spectra. These new kinds of complex non-Dirac-Hermitian Hamiltonians define physically acceptable quantum-mechanical theories. This result in quantum mechanics suggests that the corresponding non-Dirac-Hermitian D-dimensional \phi^2(i\phi)^\epsilon quantum field theories might also have real spectra. To examine this hypothesis, we return to the technique devised in 1992 and in this paper we calculate the first two coefficients in the dimensional expansion of the ground-state energy of this complex non-Dirac-Hermitian quantum field theory. We show that to first order in this dimensional approximation the ground-state energy is indeed real for \epsilon\geq0.