A positive-definite scalar product for free Proca particle

Jakubský, Vít; Smejkal, J.

doi:10.1007/s10582-006-0394-x

Cited by 25 publications

(35 citation statements)

References 29 publications

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“…The application of pseudo-Hermitian QM in dealing with the Hilbert space problem in relativistic QM and quantum cosmology [150,151,161,179,114,238], and the removal of ghosts in certain quantum field theories [35,119] relies on the construction of an appropriate (positive-definite) inner product on the space of solutions of the relevant field equation.…”

Section: Relativistic Qm Quantum Cosmology and Qftmentioning

confidence: 99%

Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

834

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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Section: Relativistic Qm Quantum Cosmology and Qftmentioning

confidence: 99%

Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

834

1,210

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“…In our present language this would lead to the modified theoretical arrangement of our three Hilbert spaces, New ways towards old problems could be sought/found in this direction. For example, in the light of some recently obtained new results on the first quantization of relativistic particles with spin [33], an extension of these studies to a scattering arrangement (e.g., along the lines indicated in our present paper) would be a particularly challenging task.…”

Section: Discussionmentioning

confidence: 95%

Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

Znojil¹

2009

SIGMA

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Abstract. One-dimensional unitary scattering controlled by non-Hermitian (typically, PT -symmetric) quantum Hamiltonians H = H † is considered. Treating these operators via Runge-Kutta approximation, our three-Hilbert-space formulation of quantum theory is reviewed as explaining the unitarity of scattering. Our recent paper on bound states [Znojil M., SIGMA 5 (2009), 001, 19 pages, arXiv:0901.0700] is complemented by the text on scattering. An elementary example illustrates the feasibility of the resulting innovative theoretical recipe. A new family of the so called quasilocal inner products in Hilbert space is found to exist. Constructively, these products are all described in terms of certain nonequivalent short-range metric operators Θ = I represented, in Runge-Kutta approximation, by (2R − 1)-diagonal matrices.

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“…Certainly, the latter family is not small. Pars pro toto it contains Hamiltonians of relativistic quantum mechanics [41,42], the well-known P -symmetric imaginary cubic oscillator [43][44][45][46] (which appears, after a more detailed scrutiny, strongly nonlocal [31,47]), its power-law generalizations [10][11][12]48] as well as exactly solvable models [49][50][51][52], models with methodical relevance in the context of supersymmetry [53,54], realistic and computation-friendly interacting-boson models of heavy nuclei [2], benchmark candidates for classification of quantum catastrophes [55][56][57], and so forth.…”

Section: A2 Physical Inner Productsmentioning

confidence: 99%

Hermitian–Non-Hermitian Interfaces in Quantum Theory

Znojil

2018

Advances in High Energy Physics

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In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlaps alias interfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.

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A positive-definite scalar product for free Proca particle

Abstract: We implement recent results of pseudo-Hermitian quantum mechanics to description of relativistic massive particle with spin-one. We derive a one-parameter family of Lorentz invariant positive-definite scalar products on the space of solutions of Proca equation.

Cited by 25 publications

References 29 publications

Pseudo-Hermitian Representation of Quantum Mechanics

Pseudo-Hermitian Representation of Quantum Mechanics

Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

Hermitian–Non-Hermitian Interfaces in Quantum Theory

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