# Disappearing Q operator

## Abstract

In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator {eta}{identical_to}e{sup -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.

- Authors:

- Physics Department, Imperial College, London, SW7 2AZ (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 21010936

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.025023; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ASYMMETRY; FIELD OPERATORS; HARMONIC OSCILLATORS; INTEGRALS; PROBABILISTIC ESTIMATION; QUANTUM FIELD THEORY; QUANTUM MECHANICS; SCHROEDINGER EQUATION; TRANSFORMATIONS

### Citation Formats

```
Jones, H. F., and Rivers, R. J..
```*Disappearing Q operator*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.025023.

```
Jones, H. F., & Rivers, R. J..
```*Disappearing Q operator*. United States. doi:10.1103/PHYSREVD.75.025023.

```
Jones, H. F., and Rivers, R. J.. Mon .
"Disappearing Q operator". United States.
doi:10.1103/PHYSREVD.75.025023.
```

```
@article{osti_21010936,
```

title = {Disappearing Q operator},

author = {Jones, H. F. and Rivers, R. J.},

abstractNote = {In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator {eta}{identical_to}e{sup -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.},

doi = {10.1103/PHYSREVD.75.025023},

journal = {Physical Review. D, Particles Fields},

number = 2,

volume = 75,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}