The disappearing $Q$ operator

TL;DR

This paper demonstrates that in functional integral formulations of non-Hermitian quantum theories, the $Q$ operator is unnecessary for expectation values, with the connection to Hermitian theories encoded via external sources, contrasting with the operator formalism.

Contribution

It shows that the $Q$ operator is not needed in functional integral approaches, providing a new perspective on non-Hermitian quantum theories and their Hermitian equivalents.

Findings

The $Q$ operator is subliminal in functional integrals.

Expectation values are computed without the $Q$ operator.

The relation to Hermitian theories is via external sources.

Abstract

In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\eta\equiv e^{-Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.