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Crypto-Hermitian Approach to the Klein-Gordon Equation
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Abstract
We explore the Klein-Gordon equation in the framework of crypto-Hermitian quantum mechanics. Solutions to common problems with probability interpretation and indefinite inner product of the Klein-Gordon equation are proposed.
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Generalized parity , time-reversal , and charge-conjugation operators were initially defined in the study of the pseudo-Hermitian Hamiltonians. We construct a concrete realization of these operators for Klein–Gordon fields and show that in this realization {\mathcal{PT} and operators respectively correspond to the ordinary time-reversal and charge-grading operations. Furthermore, we present a complete description of the quantum mechanics of Klein–Gordon fields that is based on the construction of a Hilbert space with a relativistically invariant, positive-definite, and conserved inner product. In particular we offer a natural construction of a position operator and the corresponding localized and coherent states. The restriction of this position operator to the positive-frequency fields coincides with the Newton–Wigner operator. Our approach does not rely on the conventional restriction to positive-frequency fields. Yet it provides a consistent quantum mechanical description of Klein–Gordon fields with a genuine probabilistic interpretation.
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of PT symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories. {copyright} {ital 1998} {ital The American Physical Society}
We formulate the quantum mechanics of the solutions of a Klein-Gordon-type
field equation: (\partial_t^2+D)\psi(t)=0, where D is a positive-definite
operator acting in a Hilbert space \tilde H. We determine all the
positive-definite inner products on the space H of the solutions of such an
equation and establish their physical equivalence. We use a simple realization
of the unique Hilbert space structure on H to construct the observables of the
theory explicitly. In general, there are infinitely many choices for the
Hamiltonian each leading to a different notion of time-evolution in H. Among
these is a particular choice that generates t-translations in H and identifies
t with time whenever D is t-independent. For a t-dependent D, we show that
t-translations never correspond to unitary evolutions in H, and t cannot be
identified with time. We apply these ideas to develop a formulation of quantum
cosmology based on the Wheeler-DeWitt equation for a Friedman-Robertson-Walker
model coupled to a real scalar field. We solve the Hilbert space problem,
construct the observables, introduce a new set of the wave functions of the
universe, reformulate the quantum theory in terms of the latter, reduce the
problem of time to the determination of a Hamiltonian operator acting in
L^2(\R)\oplus L^2(\R), show that the factor-ordering problem is irrelevant for
the kinematics of the quantum theory, and propose a formulation of the
dynamics. Our method allows for a genuine probabilistic interpretation and a
unitary Schreodinger time-evolution.
The relativistic Klein-Gordon system is studied as an illustration of Quantum Mechanics using non-Hermitian operators as observables. A version of the model is considered containing a generic coordinate- and energy-dependent phenomenological mass-term m
2(E,x). We show how similar systems may be assigned a pair of the linear, energy-independent left- and right-acting Hamiltonians with quasi-Hermiticity property and, hence, with the standard probabilistic interpretation.
We use the theory of pseudo-Hermitian operators to address the problem of the construction and classification of positive-definite invariant inner-products on the space of solutions of a Klein-Gordon-type evolution equation. This involves dealing with the peculiarities of formulating a unitary quantum dynamics in a Hilbert space with a time-dependent inner product. We apply our general results to obtain possible Hilbert space structures on the solution space of the equation of motion for a classical simple harmonic oscillator, a free Klein-Gordon equation and the Wheeler-DeWitt equation for the FRW-massive-real-scalar-field models.
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!
Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.
The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics, the book features: Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and perturbation theory Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics and condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.
In der vorliegenden Arbeit wird die konsequente Anwendung des Heisenberg-Pauli’schen Formalismus der Quantisierung der Wellenfelder auf die skalare relativistische Wellengleiehung für Materiefelder im Falle von Einstein-Bose-Statistik der Teilchen durchgeführt. Dabei ergibt sich ohne weitere Hypothese die Existenz von zu einander entgegengesetzt geladenen Teilchen gleicher Kuhmasse, die unter Absorption bzw. Emission von elektromagnetischer Strahlung paarweise erzeugt bzw. vernichtet werden können. Die Häufigkeit dieser Prozesse erweist sich als von derselben Grössenordnung wie die für Teilchen derselben Ladung und Masse aus der Dirac’schen Löchertheorie folgende (§ 4). Die hier untersuchte, ebenfalls den Relativitätsforderungen genügende korrespondenzmassige Möglichkeit von entgegengesetzt geladenen Teilchen ohne Spin mit Einstein-Bose-Statistik hat gegenüber der Löchertheorie den Vorzug, dass die Energie von selbst immer positiv ist. Ebenso aber wie aus der ursprünglichen Fassung der Löchertheorie folgt aus der hier besprochenen Theorie neben den unendlich grossen Selbstenergien auch eine unendliche Polarisierbarkeit des Vakuums.
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 , the true meaning and significance of the so-called charge operators and the -inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-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.
Zusammenfassung Die Schrödingersche Wellengleichung wird auf den Fall der Anwesenheit der (in den Geschwindigkeiten) linearen Glieder in der Lagrangeschen Funktion verallgemeinert, und es werden einige Beispiele der Quantisierung betrachtet.
By combining two irreducible representations of the proper inhomogeneous Lorentz group, certain irreducible unitary representations of the complete Lorentz group including space and time inversion are obtained, together with a Schrödinger equation whose solutions constitute the representation space for these representations. The representations thus define a "canonical" form for covariant particle theories, in which not only the wave equations but the manner in which the wave functions transform under Lorentz transformations is prescribed. It is shown that by a suitable choice of representation, the Dirac, Klein-Gordon, and Proca equations can all be reduced to this canonical form. It is further shown that in the representation space provided, several possibilities exist for the identification of the transformations to be associated with space inversion, time inversion, and charge conjugation, thus suggesting the existence of several distinct relativistic theories for particles of any given spin. Conjectures are made as to the physical significance of these different possibilities when the equations are second-quantized. It is shown that each of the conventional theories employs only one of the available possibilities for these transformations, the choices being different for integral and half-integral spin theories.
An ideal model of a ferromagnet is studied, consisting of a lattice of identical spins with cubic symmetry and with isotropic exchange coupling between nearest neighbors. The aim is to obtain a complete description of the thermodynamic properties of the system at low temperatures, far below the Curie point. In this temperature region the natural description of the states of the system is in terms of Bloch spin waves. The nonorthogonality of spin-wave states raises basic difficulties which are examined and overcome.The following new results are obtained: a practical method for calculating thermodynamic quantities in terms of a nonorthogonal set of basic states; a proof that in 3 dimensions there do not exist states (shown by Bethe to exist in a one-dimensional chain of spins) in which two spins are bound together into a stable complex and travel together through the lattice; a calculation of the scattering cross section of two spin waves, giving a mean free path for spin-spin collisions proportional to T-7/2 at low temperatures; and an exact formula for the free energy of the system, showing explicitly the effects of spin-wave interactions.Quantitative results based on this theory will be published in a second paper.
A unified picture of the relativistic single particle wave mechanics for ; both the spin 0 boson and the spin 1/2 h fermion is presented. The single ; particle equations discussed are the Klein-Gordon equation for spin 0, and the ; Dirac equation for spin 1/2 h. The wave mechanics of these particles is discussed. ; The field theoretic interpretation, in which the wave equations give the ; equations of motion for field operators, is connected with the particle ; interpretation. (M.H.R.);
Die beim Comptoneffekt ausgestrahlten Frequenzen und Intensitten werden nach der Schrdingerschen Theorie berechnet. Die quantentheoretischen Gren ergeben sich als geometrische Mittel aus den klassischen Gren des Anfangs- und Endzustandes des Prozesses.
Die Schrdingersche Wellengleichung wird als invariante Laplacesche Gleichung und die Bewegungsgleichungen als diejenigen einer geodtischen Linie im fnf-dimensionalen Raum geschrieben. Der berzhlige fnfte Koordinatenparameter steht in enger Beziehung zu der linearen Differentialform der elektromagnetischen Potentiale.
Auf den folgenden Seiten mchte ich auf einen einfachen Zusammenhang hinweisen zwischen der von Kaluza vorgeschlagenen Theorie fr den Zusammenhang zwischen Elektromagnetismus und Gravitation einerseits und der von de Broglie und Schrdinger angegebenen Methode zur Behandlung der Quantenprobleme andererseits. Die Theorie von Kaluza geht darauf hinaus, die zehn Einsteinschen Gravitationspotentiale gik und die vier elektromagnetischen Potentiale i in Zusammenhang zu bringen mit den Koeffizienten ik eines Linienelementes von einem Riemannschen Raum, der auer den vier gewhnlichen Dimensionen noch eine fnfte Dimension enthlt. Die Bewegungsgleichungen der elektrischen Teilchen nehmen hierbei auch in elektromagnetischen Feldern die Gestalt von Gleichungen geodtischer Linien an. Wenn dieselben als Strahlengleichungen gedeutet werden, indem die Materie als eine Art Wellenausbreitung betrachtet wird, kommt man fast von selbst zu einer partiellen Differentialgleichung zweiter Ordnung, die als eine Verallgemeinerung der gewhnlichen Wellengleichung angesehen werden kann. Werden nun solche Lsungen dieser Gleichung betrachtet, bei denen die fnfte Dimension rein harmonisch auftritt mit einer bestimmten mit der Planckschen Konstante zusammenhngenden Periode, so kommt man eben zu den obenerwhnten quantentheoretischen Methoden.
A simple derivation of a meaningful, manifestly covariant inner product for real Klein—Gordon (KG) fields with positive semi-definite
norm is provided, which turns out — assuming a symmetric bilinear form — to be the real-KG-field limit of the inner product
for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December 2003, and February 2006 (quant-ph/0312078, quant-ph/0602151,
quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for
real KG fields measures the number of active positive and negative energy Fourier-modes of the real KG field on the relativistic
mass shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral,
KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This
observation sheds some additional light on the meaning of the C operator in the CPT inner product of PT-symmetric quantum
mechanics defined by C.M. Bender, D.C. Brody and H.F. Jones.
We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. This involves the construction of a metric (if it exists) for the given set of non-Hermitian observables. We discuss uniqueness of this metric. We also show that it is not always necessary to construct the metric for the whole set of observables under consideration, but that it is sufficient for some calculational purposes to construct it for a subset only, even though this metric is, in general, not unique. The restricted metric turns out to be particularly useful in the implementation of a variational principle, which we also formulate.
We use the formulation of the quantum mechanics of first-quantized Klein–Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein–Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits.
In infinite-dimensional Hilbert spaces, the application of the concept of quasi-Hermiticity to the description of non-Hermitian Hamiltonians with real spectra may lead to problems related to the definition of the metric operator. We discuss these problems by examining some examples taken from the recent literature and propose a formulation that is free of these difficulties.
We derive an explicit manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein–Gordon equation. This expression involves a one-parameter family of conserved current densities , with a ∈ (−1, 1), that are analogous to the chiral current density for spin half fields. The conservation of is related to a global gauge symmetry of the Klein–Gordon fields whose gauge group is U (1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein–Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.
Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
M. Znojil. Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation. J Phys A: Math Gen 37:9557-9571, 2004.
Solvable relativistic quantum dots with vibrational spectra
M. Znojil. Solvable relativistic quantum dots with vibrational spectra.
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Three-hilbert-space formulation of quantum mechanics. Symmetry
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M. Znojil. N-site-lattice analogues of V (x) = ix 3. Ann Phys 327(3):893-913, 2012.