An alternative construction of the positive inner product for pseudo-Hermitian Hamiltonians: Examples

Abstract

This paper builds on our earlier proposal for construction of a positive inner product for pseudo-Hermitian Hamiltonians and we give several examples to clarify our method. We show through the example of the harmonic oscillator how our construction applies equally well to Hermitian Hamiltonians which form a subset of pseudo-Hermitian systems. For finite dimensional pseudo-Hermitian matrix Hamiltonians we construct the positive inner product (in the case of 2×2 matrices for both real as well as complex eigenvalues). When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues). We illustrate how this general formalism works out in practice by calculating the inner product for a couple of PT symmetric quantum mechanical theories.

Bogoliubov transformation and the thermal operator representation in the real time formalism

“…The finite temperature propagator in (15) can now be written in a factorized form in terms of the zero temperature propagator in (16) as…”

“…Thermofield dynamics (σ = 1 2 ) has an operator description with the usual Dirac inner product for the doubled Hilbert space [9,10,13]. For any other value of σ, however, the Hilbert space develops a modified inner product (which depends on the value of σ) [12,14,15].…”

Model pseudofermionic systems: Connections with exceptional points

“…This is not a major problem since in the literature, see for instance [5,6], extended versions of PT -symmetry exist, where it is not required that [P, T ] = 0 or that P = P † . One of such an extension has the formP = 0 x 1/x 0 , (2.23) with x = 0.…”

“…The first example we want to consider was originally discussed, in our knowledge, in [5], and, in a slightly different version, by others. The hamiltonian is H DG = r e iθ s e iφ t e −iφ r e −iθ ,…”

A concise review of pseudobosons, pseudofermions, and their relatives

“…where r, s, t, θ and φ are all real, non zero, quantities, see [22]. Another interesting Hamiltonian is, [23] …”