Thermodynamics of Pseudo-Hermitian Systems in Equilibrium

Jakubský, Vít

doi:10.1142/s0217732307023419

Cited by 21 publications

(29 citation statements)

References 31 publications

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“…This explains the results of [113] pertaining the metric-independence of thermodynamical quantities associated with non-interacting pseudo-Hermitian statistical mechanical models. However, in contrast to the view expressed in [122], the metric-independence of Z[0] does not extend to Z[J] with Z = 0.…”

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confidence: 55%

Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

853

1,210

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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.

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supporting

confidence: 55%

Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

853

1,210

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“…Subsequently, the latter recipe has been used in ref. [1] where operator Ω (Jones) ≡ ̺ was defined as a self-adjoint square root (19) of metric η emphasizing that in terms of physics, "the relevant wave function is not…”

Section: Ambiguity Problemmentioning

confidence: 99%

Scattering theory with localized non-Hermiticities

Znojil

2008

Phys. Rev. D

102

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In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that the well known variability of the ad hoc choice of the metric Θ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable Θ a fully plausible physical picture of the scattering can be recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.

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“…This is indeed necessary for a correct formulation of PT -symmetric field theories. It also provides a straightforward interpretation of the results pertaining the metric-independence of thermodynamical quantities associated with non-interacting pseudo-Hermitian statistical mechanical systems that is discussed in [7]. Consider the definition of the generating functional (partition function) that is the starting point of pathintegral formulation of quantum mechanics and quantum field theory:…”

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confidence: 99%

“…It is in this sense that the metric operator η + or Q = − ln η + "disappears" from the calculations [6]. This phenomenon is the real reason for the metric independence of the thermodynamical quantities for the statistical systems considered in [7]. However, it does not extend to the case where one needs to couple the system to an external source.…”

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confidence: 99%

Path-integral formulation of pseudo-Hermitian quantum mechanics and the role of the metric operator

Mostafazadeh

2007

Phys. Rev. D

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We provide a careful analysis of the generating functional in the path integral formulation of pseudo-Hermitian and in particular PT -symmetric quantum mechanics and show how the metric operator enters the expression for the generating functional.Pacs numbers: 03.65. Ca, 03.65.Db, 11.30.Er In the pseudo-Hermitian representation of quantum mechanics [1], a quantum system is determined by a triplet (H, H, η + ) where H is an auxiliary Hilbert space, H : H → H is a linear (Hamiltonian) operator with a real spectrum and a complete set of eigenvectors, and η + : H → H is a linear, positive-definite, invertible (metric) operator fulfilling the pseudo-Hermiticity condition [2]The physical Hilbert space H phys of the system is defined as the complete extension of the span of the eigenvectors of H endowed with the inner productwhere ·|· is the defining inner product of H, and the observables are identified with the self-adjoint operators acting in H phys , alternatively η + -pseudo-Hermitian operators acting in H, [1]. Pseudo-Hermitian quantum mechanics [1] has primarily developed in an attempt to unravel the mathematical structures responsible for the intriguing spectral properties of PT -symmetric Hamiltonians [3] such asThese properties were originally noticed in dealing with the field theoretical analogues of (3). It has also been conjectured that such models may have important applications in particle physics [4]. This has led a number of researchers to investigate various PT -symmetric field theories [5]. Similarly to standard quantum field theory, the path-integral techniques are indispensable in dealing with PT -symmetric field theories. Yet a careful analysis of the path-integral formulation of PseudoHermitian quantum mechanics has been lacking till recently. The first careful analysis of this problem is given in Ref.[6] where the authors use the Hermitian representation of the corresponding pseudo-Hermitian models to determine the appropriate source term in the generating functional Z [J]. The results of [6] give the impression that the metric operator, which is the central ingredient in the operator formulation of pseudo-Hermitian quantum mechanics, does not play an important role in the path-integral formulation of the theory. The purpose of the present paper is to elucidate the role of the metric operator in the latter formulation. This is indeed necessary for a correct formulation of PT -symmetric field theories. It also provides a straightforward interpretation of the results pertaining the metric-independence of thermodynamical quantities associated with non-interacting pseudo-Hermitian statistical mechanical systems that is discussed in [7]. Consider the definition of the generating functional (partition function) that is the starting point of pathintegral formulation of quantum mechanics and quantum field theory:where "tr" stands for the trace, T exp is the time-ordered (chronological-ordered) exponential [8,9], t 1 and t 2 are initial and final time labels that in field theory are taken to be −∞ and...

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Thermodynamics of Pseudo-Hermitian Systems in Equilibrium

Cited by 21 publications

References 31 publications

Pseudo-Hermitian Representation of Quantum Mechanics

Pseudo-Hermitian Representation of Quantum Mechanics

Scattering theory with localized non-Hermiticities

Path-integral formulation of pseudo-Hermitian quantum mechanics and the role of the metric operator

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