Phases of quartic scalar theories and PT symmetry

TL;DR

This paper investigates the relationship between energy eigenvalues in parity symmetric and PT-symmetric phases of quartic scalar theories, confirming the conjecture at weak coupling but not at strong coupling, with implications for higher-dimensional theories.

Contribution

It demonstrates the validity of a conjectured relation between energy eigenvalues in PT-symmetric and parity symmetric phases for weak coupling in scalar theories, and explores its limitations at strong coupling and in higher dimensions.

Findings

Relation holds for weak coupling in quantum mechanical models.

Conjecture fails at strong coupling.

Relation's applicability in higher-dimensional field theories is uncertain.

Abstract

For quantum mechanical anharmonic oscillator-type Hamiltonians, it is shown that there is a relation between the energy eigenvalues of parity symmetric and PT-symmetric phases for weak coupling. The possibility of such a relation was conjectured by Ai, Bender and Sarkar on examining the imaginary part of the ground state energy using path integrals. In the weak coupling limit, we show that the conjecture is true also for the real part of the ground state energy and of the excited state energies. However, the conjecture is false for strong coupling. The analogous relation for partition functions in zero spacetime dimensions is valid for many cases. However $O(N)$ symmetric multi-component scalar fields, with $N>1$ and a quartic interaction, do not satisfy the conjecture for zero and one dimensional spacetime. The possibility that the conjecture is valid, for a single component field theory in higher dimensional spacetimes, is discussed in a simplified model.