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arXiv · hep-th/0612093

The disappearing $Q$ operator

H. F. Jones·R. J. Rivers
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Abstract

In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator $η\equiv e^{-Q}$ 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.

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H. F. Jones, R. J. Rivers. 2006-12-11. The disappearing $Q$ operator. https://doi.org/10.1103/physrevd.75.025023

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