Pseudo-Hermitian Representation of Quantum Mechanics

Abstract

A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT , the true meaning and significance of the so-called charge operators C and the CPT -inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between localnon-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.

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

“…[27][28][29], leading to the average of an observable O(t) given by O(t) ≡ tr [ρ N (t)O(t = 0)]. On the other hand, the Lindblad equation with η-pseudo-Hermitian Hamiltonians [30], for which ηHη −1 = H † with η being a Hermitian metric operator in the Hilbert space h, has been derived in Refs. [31,32] for the generalized density matrix ρ G (t) ≡ ρ(t)η, viz.,…”

Classical-quantum correspondence for two-level pseudo-Hermitian systems

“…It was shown that, albeit relevant, the role played by the PT symmetry and the C operator is not a fundamental one. Indeed, it can be seen from PHQM that η = CPT is just an example of a positive-definite metric operator [21]. In fact, the existence of a preferred metric, and its physical meaning, is an open issue in the PHQM.…”

“…In fact, the existence of a preferred metric, and its physical meaning, is an open issue in the PHQM. There are several contexts where pseudo-hermitian operators appear [21]. In special, recent treatments of topological aspects of non-hermitian systems use the framework of PHQM [22][23][24][25][26][27][28][29][30].…”

“…A subtle point regarding quantization in general, and quantization in the PHQM framework in particular, is that canonical transformations, which are transformation on the level of the algebra of operators, do not necessarily translate as isometries or unitary transformations between the Hilbert spaces upon which these operators act [31]. When one is faced with non-unitary canonical transformations, for instance, in the infinite-dimensional case, a physical meaning for these transformations can be established by looking at the classical limit of the theory [19,21,32]. This procedure is called η-pseudo-hermitian canonical quantization.…”

Spectral Singularities in the TE and TM modes of a PT-Symmetric Slab System: Optimal conditions for realizing a CPA-Laser

“…A few years ago [2] it was noticed that this condition coincided with the requirement that the continuous spectrum of the corresponding optical potential includes certain points known to mathematicians as spectral singularities [3,4]. Spectral singularities entered into physics literature as mathematical obstructions [5,6,7] to a Hermitization procedure developed to construct unitary quantum systems using certain non-Hermitian Hamiltonian operators [8,9]. Spectral singularities turn out to have an interesting physical meaning [10]; they correspond to a special class of scattering states with a real and positive energy that behave exactly like resonances.…”