Green Functions for the Wrong-Sign Quartic

TL;DR

This paper verifies that approximate Green functions obtained from truncated Schwinger-Dyson equations are highly accurate for the wrong-sign quartic oscillator, supporting the method's extension to more complex non-Hermitian theories.

Contribution

The paper provides an independent numerical validation of Green functions derived from truncated Schwinger-Dyson equations for a non-Hermitian theory with known metric, confirming their accuracy.

Findings

Green functions are accurate within 6-8% at lowest-order truncation.

Validation supports using Schwinger-Dyson truncation for non-Hermitian theories.

Method can be extended to higher dimensions where metric calculation is difficult.

Abstract

It has been shown that the Schwinger-Dyson equations for non-Hermitian theories implicitly include the Hilbert-space metric. Approximate Green functions for such theories may thus be obtained, without having to evaluate the metric explicitly, by truncation of the equations. Such a calculation has recently been carried out for various $PT$-symmetric theories, in both quantum mechanics and quantum field theory, including the wrong-sign quartic oscillator. For this particular theory the metric is known in closed form, making possible an independent check of these approximate results. We do so by numerically evaluating the ground-state wave-function for the equivalent Hermitian Hamiltonian and using this wave-function, in conjunction with the metric operator, to calculate the one- and two-point Green functions. We find that the Green functions evaluated by lowest-order truncation of the Schwinger-Dyson equations are already accurate at the (6-8)% level. This provides a strong justification for the method and a motivation for its extension to higher order and to higher dimensions, where the calculation of the metric is extremely difficult.