The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics

Abstract

Most recently it has been observed e.g. by [1]) that the C-operator related to a PT -symmetric non-Hermitian Hamilton operator is not unique.Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th] [2]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT -symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT -symmetric 2 × 2-matrix Hamilton operator it is shown that -by a suitable choice of the norm of its eigenvectors -the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.

Identification of the metric for diagonalizable (anti-)pseudo-Hermitian Hamilton operators represented by two-dimensional matrices

“…• Eq. ( 8) displays and explains -as has been conjectured e. g. in 2009 [43] -clearly the ambiguity of the metric due to its manifest dependence on the undetermined invertible and diagonal matrix N renormalizing the right eigen-vectors of the Hamilton operator H (see also [35][44] [45][46]). 3 • The -to our best knowledge -new aspect of this manuscript is the explicit inclusion and investigation of the matrix Q into the expression Eq.…”

“…Now, 16 years later, I am very grateful to be able to provide hereby my definite answer to the question confirming to great extend my conjectures made in Refs. [43] and [60] which had been developed mainly at the Nucl. Phys.…”

“…As pointed out in the context of Eqs. ( 42) and (43) the "trivial phase" (Q = 1 2 ) and "non-trivial phase" (Q = P ) are related to the pseudo-Hermiticity or anti-pseudo-Hermiticity of the matrix n • σ, respectively, or -equivalently -to the pseudo-Hermiticity or anti-pseudo-Hermiticity of the Hermitian conjugate matrix ( n • σ) + . It is easy to show that there hold the following four identities:…”

Density Matrix Formalism for PT-Symmetric Non-Hermitian Hamiltonians with the Lindblad Equation

“…Furthermore, using these two normalized eigenvectors, one can introduce the metric operator η, as shown in Ref. [43], viz.,…”

Probing non-Hermitian phase transitions in curved space via quench dynamics

“…(27) To define the bi-orthogonal basis appropriately [113,[118][119][120], we need to determine the constants c α(n,m,λ) Γ,R and d α(n,m,λ) Γ,R from of eqs. ( 14) and (27).…”