Solution of Schwinger-Dyson Equations for ${\cal PT}$-Symmetric Quantum Field Theory

TL;DR

This paper introduces a truncation technique for solving Schwinger-Dyson equations in ${\cal PT}$-symmetric quantum field theories, demonstrating properties like renormalizability and asymptotic freedom in a non-Hermitian $-g\phi^4$ model.

Contribution

It presents a novel method for renormalizing and solving ${\cal PT}$-symmetric field theories using truncated Schwinger-Dyson equations, revealing their potential physical relevance.

Findings

The $-g\phi^4$ theory is renormalizable in four dimensions.

The theory exhibits asymptotic freedom.

It has a nonzero vacuum expectation value and a positive spectrum.

Abstract

In recent papers it has been observed that non-Hermitian Hamiltonians, such as those describing $ig\phi^3$ and $-g\phi^4$ field theories, still possess real positive spectra so long as the weaker condition of ${\cal PT}$ symmetry holds. This allows for the possibility of new kinds of quantum field theories that have strange and quite unexpected properties. In this paper a technique based on truncating the Schwinger-Dyson equations is presented for renormalizing and solving such field theories. Using this technique it is argued that a $-g\phi^4$ scalar quantum field theory in four-dimensional space-time is renormalizable, is asymptotically free, has a nonzero value of $<0|\phi|0>$, and has a positive definite spectrum. Such a theory might be useful in describing the Higgs boson.