Abstract
Pseudo-Hermitian quantum theories are those in which the Hamiltonian H satisfies H† = ηHη-1, where η = e-Q is a positive-definite Hermitian operator, rather than the usual H† = H. In the operator formulation of such theories the standard Hilbert-space metric must be modified by the inclusion of η in order to ensure their probabilistic interpretation. With possible generalizations to quantum field theory in mind, it is important to ask how the functional integral formalism for pseudo-Hermitian theories differs from that of standard theories. It turns out that here Q plays quite a different role, serving primarily to implement a canonical transformation of the variables. It does not appear explicitly in the expression for the vacuum generating functional. Instead, the relation to the Hermitian theory is encoded via the dependence of Z on the external source j(t). These points are illustrated and amplified in various versions of the Swanson model, a non-Hermitian transform of the simple harmonic oscillator.
Highlights
1.1 The metric operator in the Schrödinger formulation of pseudo-Hermitian quantum mechanicsThe recent interest in non-Hermitian Hamiltonians stems from the work of Bender and Boettcher [1], who showed numerically that the class of HamiltoniansH = 1 p2 - g(ix)N (1)has a completely real spectrum for N 3 2
It is related to the Q operator [4], which provides a positive-definite metric for the quantum mechanics governed by
The fundamental question we are asking here is, what is the corresponding expression in pseudo-Hermitian quantum mechanics? One might perhaps expect something like ò ò Z[ j] =
Summary
1.1 The metric operator in the Schrödinger formulation of pseudo-Hermitian quantum mechanics. A rigorous proof [2] of the reality came a few years later by exploiting the ODE-IM correspondence, i.e. the correspondence between ordinary differential equations in their different Stokes sectors and integrable models. In such cases there exists a similarity transformation from the non-Hermitian H to a Hermitian h:. It is related to the Q operator [4], which provides a positive-definite metric for the quantum mechanics governed by.
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