Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

Mostafazadeh, Alí; Zamani, Farhad

doi:10.1016/j.aop.2006.02.007

Cited by 59 publications

(97 citation statements)

References 51 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%

“…Therefore, although considering H η+ with a (positive-definite) metric operator η + generally yields an equivalent "pseudo-Hermitian representation" of the conventional QM, a clever choice of η + may be of practical significance in deriving the physical properties of the system under investigation. As we discuss in Subsection 9.2, it turns out that indeed these new representations play a key role in the resolution of one the oldest problems of modern physics, namely the problem of negative probabilities in relativistic QM of Klein-Gordon fields [150,151,161,179] and certain quantum field theories [35].…”

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Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

865

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%

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confidence: 99%

Pseudo-Hermitian Representation of Quantum Mechanics

Mostafazadeh

2010

Int. J. Geom. Methods Mod. Phys.

865

1,210

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confidence: 99%

“…This has led to the discovery of an intriguing structural similarity between quantum mechanics and general theory of relativity [15]. It has also found applications in dealing with the Hilbert-space problem in quantum cosmology [16], the old problem of constructing a unitary first-quantized quantum theory of Klein-Gordon fields [17], bound state scattering [18], and ghosts in certain quantum field theories [19].In [2] the authors consider a class of two-level nonHermitian PT -symmetric Hamiltonians, define H phys using the so-called CPT -inner product, and explore the evolution of state vectors, i.e., elements of H phys . They conclude that for a fixed initial and final state vectors, ψ I and ψ F , one can obtain a Hamiltonian operator that evolves ψ I into ψ F in an arbitrarily short time τ .…”

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confidence: 99%

Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics

2007

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A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT -symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.Pacs numbers: 03.67.Lx, 02.30.Yy, Since the publication of the pioneering work of Bender and Boettcher [1] on non-Hermitian PT -symmetric Hamiltonian operators, there have appeared numerous research articles exploring the mathematical properties of such operators and their possible physical applications. Recently, it has been suggested that one can obtain arbitrarily fast quantum evolutions using a class of such Hamiltonians [2]. If true, this will have drastic consequences in quantum computation, because for example it removes the bound on the time-optimal unitary NOT operations [3] that is obtained within the framework of conventional (Hermitian) quantum mechanics [4,5,6,7]. As pointed out in [8], this seems to contradict the equivalence of the quantum theory based on such non-Hermitian Hamiltonians and the Hermitian quantum mechanics [9,10]. In this article, we offer a comprehensive treatment of this problem that is based on a detailed study of the projective space PH phys of physical states. In particular, we obtain the explicit form of the natural metric tensor on PH phys and unravel the subtleties of the quantum Brachistochrone problem for a general unitary quantum system that is defined by a non-Hermitian Hamiltonian.In general if a linear (possibly non-Hermitian) operator has a complete set of eigenvectors and a real spectrum, then it can serve as the Hamiltonian operator for a unitary quantum system provided that the physical Hilbert space of the system is defined using an appropriate inner product [11,12,13]. This leads to a quantum theory that turns out to be equivalent to the conventional quantum mechanics [9,10]. In other words, this theory, that we refer to as Pseudo-Hermitian Quantum Mechanics [14], is an alternative representation of the conventional quantum mechanics. The key ingredient of this representation is that the inner product of the physical Hilbert space H phys is determined by the Hamiltonian operator of the system. This has led to the discovery of an intriguing structural similarity between quantum mechanics and general theory of relativity [15]. It has also found applications in dealing with the Hilbert-space problem in quantum cosmology [16], the old problem of constructing a unitary first-quantized quantum theory of Klein-Gordon fields [17], bound state scattering [18], and ghosts in certain quantum field theories [19].In [2] the authors co...

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“…A century of experimentation has not accomplished this feat, providing strong evidence that one cannot physically impose the necessary initial conditions to solve 1 See section 6 for a related discussion. 3 the KGE. In other words, for a generic (neutral, spinless) particle, the maximum amount of knowable information can be encoded by the instantaneous values of a single complex scalar field ψ -only half as much information as the instantaneous values of the φ andφ fields needed to solve the KGE for a complex scalar field.…”

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A Novel Interpretation of the Klein-Gordon Equation

Wharton

2009

Found Phys

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The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrödinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an initial boundary condition, this paper considers the possibility that the solutions are determined by both an initial and a final boundary condition. By constructing an invariant joint probability distribution from the size of the solution space, it is shown that the usual measurement probabilities can nearly be recovered in the non-relativistic limit, provided that neither boundary constrains the energy to a precision nearh/t 0 (where t 0 is the time duration between the boundary conditions). Otherwise, deviations from standard quantum mechanics are predicted.

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Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

Cited by 59 publications

References 51 publications

Pseudo-Hermitian Representation of Quantum Mechanics

Pseudo-Hermitian Representation of Quantum Mechanics

Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics

A Novel Interpretation of the Klein-Gordon Equation

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