Moyal products—a new perspective on quasi-Hermitian quantum mechanics

Abstract

The rationale for introducing non-hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to nonperturbative properties and the metric.

Quasi-Hermitian quantum mechanics in phase space

“…We believe this formalism will be helpful in understanding the applications cited, as well as others. Other recent work along these same lines can be found in [16,17].…”

“…This is perhaps a more conventional problem to attack in the framework of quasi-hermitian theories [18,16,17]. That is to say, we seek all solutions to…”

Quantum Liouville Equation

“…In order to obtain the LHS of (21), according to (12), one, firstly, has to multiply symmetrically nq n−1 •p m and ∂ρ ∂p , wherê ρ = |ψ ψ|, then to multiply symmetrically mq n •p m−1 and − ∂ρ ∂q and, finally, to add these two. Due to (9), the LHS of (21) then becomes: n!m! (n + m)!…”

“…For example, in the article by Scholtz and Geyer [9] one can find discussion of Moyal product in relation to quasi-Hermitian operators that are important when one is analyzing dissipative processes. On the other hand, in the paper by Manko et al [10] one can find more formal consideration of so called star product and significance of this approach comes from the measurement of quantum states at microwaves, see [11] and references therein.…”

Classical oscillator with position-dependent mass in a complex domain

“…The consistency of these models as quantum systems was established [11] by constructing a positive definite inner product that generates unitary evolution. Later on there has been a lot of activity [12,13,14,15,16,17,18] in the study of different aspects of P T -symmetric models.…”