Ordinary versus PT-symmetric $\phi^3$ quantum field theory

TL;DR

This paper compares the properties of conventional and PT-symmetric $g\,\phi^3$ quantum field theories, showing that the PT-symmetric version is stable, renormalizable, and exhibits different renormalization-group behavior.

Contribution

It provides a detailed analysis of the renormalization-group properties of PT-symmetric $ig\phi^3$ quantum field theory, highlighting its stability and triviality compared to the conventional theory.

Findings

Conventional $g\phi^3$ theory in 6D is asymptotically free.

PT-symmetric $ig\phi^3$ theory is stable, renormalizable, and similar to $g\phi^4$ in 4D.

The PT-symmetric theory is trivial and energetically stable.

Abstract

A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where the operator P performs space reflection and the operator T performs time reversal. A PT-symmetric Hamiltonian often has a parametric region of unbroken PT symmetry in which the energy eigenvalues are all real. There may also be a region of broken PT symmetry in which some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a PT-symmetric $ig\phi^3$ quantum field theory. This quantum field theory is the analog of the PT-symmetric quantum-mechanical theory described by the Hamiltonian $H=p^2+ix^3$, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization-group properties of a conventional Hermitian $g\phi^3$ quantum field theory with those of the PT-symmetric $ig\phi^3$ quantum field theory. It is shown that while the conventional $g\phi^3$ theory in $d=6$ dimensions is asymptotically free, the $ig\phi^3$ theory is like a $g\phi^4$ theory in $d=4$ dimensions; it is energetically stable, perturbatively renormalizable, and trivial.