An alternative construction of the positive inner product in non-Hermitian quantum mechanics

Abstract

Within the context of non-Hermitian quantum mechanics, we use the generators of eigenvectors of the Hamiltonian to construct a unitary inner product space. Such models have been of interest in recent years, for instance, in the context of PT symmetry, although our construction extends to the larger class of so-called pseudo-Hermitian Operators. We provide a detailed example to illustrate the concept and compare with known results.Let us consider a non-Hermitian quantum mechanical Hamiltonian operator, H, which is related to its adjoint through a similarity transformation(1)An operator with the above relation is sometimes called pseudo-Hermitian [1]. The adjoint, H † , is defined with respect to an auxilary Hilbert space, H, equipped with an inner product ·|· (conventionally taken to be the Dirac inner product), such that φ|Hψ = H † φ|ψ , with |ψ , |φ ∈ H. Relation (1) implies that S can be chosen to be self-adjoint.Since H is not Hermitian, it follows that e −itH (for t ∈ R, = 1) is not unitary and, therefore, it is not possible to construct a unitary quantum theory on H. To construct a unitary quantum theory we must define a new Hilbert space, H ′ , with a modified inner product leading to a modified adjoint, H ‡ , for which the Hamiltonian H is self-adjoint.By choosing H ′ so that H ‡ = H, we can guarantee that e −itH is a unitary time evolution operator in H ′ .

Supersymmetry, PT-symmetry and spectral bifurcation

“…Detailed analysis of the properties of PT-symmetric quantum mechanical systems exhibiting supersymmetry has been carried out by Mostafazadeh [15] and Plyushchay et al [16], including the construction of appropriate norm [15]. The question of norm in general pseudo-Hermitian Hamiltonian systems has been answered recently [17].…”

Entangled quantum state discrimination using a pseudo-Hermitian system

“…In this article we would like to extend the work in Ref. [2] for two entangled quantum states using pseudo-Hermitian system [11]- [14]. We show that like PT-symmetric non-Hermitian system, pseudo-Hermitian quantum system can also be useful for discriminating entangled quantum states.…”

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

“…with E defined in (3). Since the diagonal elements of the zero temperature propagator in (12) have poles at p 0 = ±E where the off-diagonal elements also have nontrivial contributions and since U CT (T, p) depends only on the magnitude |p 0 |, it is clear that the matrix U in (18) arises naturally from the momentum space Bogoliubov transformation matrix (13) when we Fourier transform (14) to the mixed space. We note that the factorizing matrix, U (T, p), is independent of the t coordinate in this case, which we will see in the next section not to be the case for 0 < σ < 1.…”

“…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].…”