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Showing papers in "Journal of Mathematical Physics in 2004"


Journal ArticleDOI
TL;DR: It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.
Abstract: We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

933 citations


Journal ArticleDOI
TL;DR: A new entanglement monotone for bipartite quantum states is presented, inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions: it is convex, additive on tensor products, and superadditive in general.
Abstract: In this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call “squashed entanglement”: it is a lower bound on entanglement of formation and an upper bound on distillable entanglement. Furthermore, it is convex, additive on tensor products, and superadditive in general. Continuity in the state is the only property of our entanglement measure which we cannot provide a proof for. We present some evidence, however, that our quantity has this property, the strongest indication being a conjectured Fannes-type inequality for the conditional von Neumann entropy. This inequality is proved in the classical case.

472 citations


Journal ArticleDOI
TL;DR: In this article, the Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodings equation, and the resulting Hamiltonian is found to be non-Hermitian and non-local in time.
Abstract: The Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodinger equation. The resulting Hamiltonian is found to be non-Hermitian and nonlocal in time. The resulting wave functions are thus not invariant under time reversal. The time fractional Schrodinger equation is solved for a free particle and for a potential well. Probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative. New identities for the Mittag–Leffler function are also found and presented in an Appendix.

402 citations


Journal ArticleDOI
TL;DR: The non-Hermitian quadratic Hamiltonian H =ωa†a+αa2+βa†2 is analyzed in this article, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants.
Abstract: The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.

210 citations


Journal ArticleDOI
TL;DR: In this article, the generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity, and the generalized Peierls-Bogoliubov inequality is also proven.
Abstract: Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity Moreover, the generalized Peierls–Bogoliubov inequality is also proven

176 citations


Journal ArticleDOI
TL;DR: In this article, a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles is presented.
Abstract: As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland–Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established.

168 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present all Hamiltonian systems in E(2) that admit a third-order integral, both in quantum and in classical mechanics, and show that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-de Vries equation and the Painleve transcendents.
Abstract: We present in this article all Hamiltonian systems in E(2) that are separable in Cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg–de Vries equation and the Painleve transcendents.

155 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider the expansion of transcendental functions in a small parameter around rational numbers and present algorithms which are suitable for an implementation within a symbolic computer algebra system The method is an extension of the technique of nested sums The algorithms allow in addition the evaluation of binomial sums, inverse binomial sum and generalizations thereof.
Abstract: I consider the expansion of transcendental functions in a small parameter around rational numbers This includes in particular the expansion around half-integer values I present algorithms which are suitable for an implementation within a symbolic computer algebra system The method is an extension of the technique of nested sums The algorithms allow in addition the evaluation of binomial sums, inverse binomial sums and generalizations thereof

147 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the properties of the associated Yang-Baxter structures and proved a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, the semigroup S(X,r), the group G(X/r), and the k-algebra A(k,X, r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r.
Abstract: A bijective map r: X2→X2, where X={x1,…,xn} is a finite set, is called a set-theoretic solution of the Yang–Baxter equation (YBE) if the braid relation r12r23r12=r23r12r23 holds in X3. A nondegenerate involutive solution (X,r) satisfying r(xx)=xx, for all x∈X, is called square-free solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X,r) and the associated Yang–Baxter algebraic structures—the semigroup S(X,r), the group G(X,r) and the k-algebra A(k,X,r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang–Baxter structures, and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, ...

141 citations


Journal ArticleDOI
TL;DR: In this paper, a gauge covariant quantization of the Weyl pseudodifferential calculus is presented, based on the magnetic canonical commutation relations, defined in terms of the magnetic flux through triangles of the classical Moyal product.
Abstract: In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle” at the level of the classical symbols, does not lead to gauge invariant formulas if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.

128 citations


Journal ArticleDOI
TL;DR: In this paper, the exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-type equation to the complex plane and then performing a method of monodromy matching at several poles in the plane.
Abstract: The exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-type equation to the complex plane and then performing a method of monodromy matching at several poles in the plane. While this method was successfully used in asymptotically flat space–time, as applied to both the Schwarzschild and Reissner–Nordstro/m solutions, its extension to nonasymptotically flat space–times has not been achieved yet. In this work it is shown how to extend the method to this case, with the explicit analysis of Schwarzschild–de Sitter and large Schwarzschild–anti–de Sitter black holes, both in four dimensions. We obtain, for the first time, analytic expressions for the asymptotic quasinormal frequencies of these black hole space–times, and our results match previous numerical calculations with great accuracy. We also list some results concerning the general classification of asymptotic quasinormal frequencies in d-dimensional space–times.

Journal ArticleDOI
TL;DR: In this article, six equivalent definitions of Frobenius algebra in a monoidal category are provided, and what it means for a morphism of a bicategory to be a projective equivalence is defined.
Abstract: Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What it means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to “strongly separable” Frobenius algebras and “weak monoidal Morita equivalence.” Wreath products of Frobenius algebras are discussed.

Journal ArticleDOI
TL;DR: In this article, the modified Dirac equation in the Lorentz-violating standard-model extension (SME) is considered, and the construction of a Hermitian Hamiltonian to all orders in the LDA parameters is investigated, discrete symmetries and the first-order roots of the dispersion relation are determined.
Abstract: The modified Dirac equation in the Lorentz-violating Standard-Model Extension (SME) is considered. Within this framework, the construction of a Hermitian Hamiltonian to all orders in the Lorentz-breaking parameters is investigated, discrete symmetries and the first-order roots of the dispersion relation are determined, and various properties of the eigenspinors are discussed.

Journal ArticleDOI
TL;DR: In this paper, it was shown that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.
Abstract: We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.

Journal ArticleDOI
TL;DR: In this paper, the eigenvalue distribution of the adjacency matrix A(N,p) of weighted random graphs is studied and the moments of the normalized eigen value counting function σp =limN→∞σ n,p are derived.
Abstract: We study eigenvalue distribution of the adjacency matrix A(N,p) of weighted random graphs Γ=ΓN,p. We assume that the graphs have N vertices and the average number of edges attached to one vertex is p. To each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite. In the first part of the paper, we consider the moments of normalized eigenvalue counting function σN,p of A(N,p). Assuming all moments of a finite, we obtain recurrent relations that determine the moments of the limiting measure σp=limN→∞σN,p. The method developed is applied to the Laplace operator ΔΓ closely related with A(N,p). Using the recurrent relations, we analyze the form of σp for the both of random matrix families. In the second part of the paper we consider the resolvents G(A,Δ)(z) of A(N,p) and ΔΓ of ΓN,p and study the functions fN(A,Δ)(z,u)=(1/N)∑k=1N exp{−uGkk(A,Δ)(z)} in the limit N→∞. We derive closed equations that uniquely determine the limiting functio...

Journal ArticleDOI
TL;DR: In this article, the authors investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors, where the nodes carry generalized charge that may be randomly transferred from one node to another.
Abstract: We investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors. The nodes carry generalized charge that may be randomly transferred from one node to another. Our goal is to find the time evolution of the probability distribution of charges in the network. The building blocks of our theoretical approach are (1) known probability distributions for the connector currents, (2) physical constraints such as local charge conservation, and (3) a time scale separation between the slow charge dynamics of the nodes and the fast current fluctuations of the connectors. We integrate out fast current fluctuations and derive a stochastic path integral representation of the evolution operator for the slow charges. The statistics of charge fluctuations may be found from the saddle-point approximation of the action. Once the probability distributions on the discrete network have been studied, the continuum limit is taken to obtain a statistical field theory. We find a correspondence between the diffusive field theory and a Langevin equation with Gaussian noise sources, leading nevertheless to nontrivial fluctuation statistics. To complete our theory, we demonstrate that the cascade diagrammatics, recently introduced by Nagaev, naturally follows from the stochastic path integral. By generalizing the principle of minimal correlations, we extend the diagrammatics to calculate current correlation functions for an arbitrary network. One primary application of this formalism is that of full counting statistics (FCS), the motivation for why it was developed in the first place. We stress however, that the formalism is suitable for general classical stochastic problems as an alternative approach to the traditional master equation or Doi–Peliti technique. The formalism is illustrated with several examples: Both instantaneous and time averaged charge fluctuation statistics in a mesoscopic chaotic cavity, as well as the FCS and new results for a generalized diffusive wire.

Journal ArticleDOI
TL;DR: In this paper, a (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable.
Abstract: A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable. The most important examples include the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics and its six-dimensional generalization arising in the context of sdiff(Σ2) self-dual Yang–Mills equations. Given a multidimensional PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions. As an example of this phenomenon we discuss the second commuting flow of the dispersionless KP hierarchy. Considered separately, this is a four-dimensional PDE which does not pass the integrability test. However, the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full (2+1)-dimensional dis...

Journal ArticleDOI
TL;DR: In this article, the Dirichlet Laplacian in the layer of a complete noncompact surface embedded in R3 is considered and the spectral results of the original paper are generalized to the situation when the surface does not possess poles.
Abstract: Given a complete noncompact surface Σ embedded in R3, we consider the Dirichlet Laplacian in the layer Ω that is defined as a tubular neighborhood of constant width about Σ. Using an intrinsic approach to the geometry of Ω, we generalize the spectral results of the original paper by Duclos et al. [Commun. Math. Phys. 223, 13 (2001)] to the situation when Σ does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain Ω.

Journal ArticleDOI
TL;DR: In this paper, a new application of the original Fisher-Hartwig formula was given to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads to generalizations of the Fisher-Harmwig formula to random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles.
Abstract: Fisher–Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin–spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher–Hartwig formula to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads us to generalize the Fisher–Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher–Hartwig asymptotic form known for Toeplitz determinants.

Journal ArticleDOI
TL;DR: In this article, it was shown that some aspects of the no-go theorems are still present in the differential (but not in the integral) picture of higher gauge theory.
Abstract: The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports. We recall how to circumvent the classic no-go theorems in order to define non-Abelian surface ordered products in the integral picture. We then derive the differential picture from the integral formulation under the assumption that the curve and surface labels depend smoothly on the position of the curves and surfaces. We show that some aspects of the no-go theorems are still present in the differential (but not in the integral) picture. This implies a substantial structural ...

Journal ArticleDOI
TL;DR: In this article, an invariant Lagrangian formalism for scalar single-variable difference schemes is presented, which is used to obtain first integrals and explicit exact solutions of the schemes.
Abstract: One of the difficulties encountered when studying physical theories in discrete space–time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two- and three-dimensional groups of Lagrangian symmetries are considered

Journal ArticleDOI
TL;DR: In this article, a characterization of block-diagonalizable pseudounitary operators with finite-dimensional diagonal blocks is presented, and a proof of the spectral theorem for symplectic transformations of classical mechanics is given.
Abstract: We consider pseudounitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudounitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudounitary matrix is the exponential of i=−1 times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudounitary matrices. In particular, we present a thorough treatment of 2×2 pseudounitary matrices and discuss an example of a quantum system with a 2×2 pseudounitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group Sp(2n) with the real subgroup of a matrix group that is isomorphic to the pseudounitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups.

Journal ArticleDOI
TL;DR: In this article, the authors extend a classification of irreducible almost-commutative geometries, whose spectral action is dynamically non-degenerate, to internal algebras that have six simple summands.
Abstract: We extend a classification of irreducible almost-commutative geometries, whose spectral action is dynamically nondegenerate, to internal algebras that have six simple summands. We find essentially four particle models: an extension of the standard model by a new species of fermions with vectorlike coupling to the gauge group and gauge invariant masses, two versions of the electrostrong model, and a variety of the electrostrong model with Higgs mechanism.

Journal ArticleDOI
TL;DR: In this article, the authors consider the Lie-algebra of the group of diffeomorphisms of a d-dimensional torus, which is also known as the algebra of derivations on a Laurent polynomial ring A in d commuting variables denoted by Der-A.
Abstract: We consider the Lie-algebra of the group of diffeomorphisms of a d-dimensional torus which is also known to be the algebra of derivations on a Laurent polynomial ring A in d commuting variables denoted by Der A. The universal central extension of Der A for d=1 is the so-called Virasoro algebra. The connection between Virasoro algebra and physics is well known. See, for example, the book on Conformal Field Theory by Di Francesco, Mathieu, and Senechal. In this paper we classify (A, Der A) modules which are irreducible and have finite dimensional weight spaces. Earlier Larsson constructed a large class of modules, the so-called tensor fields, based on gld modules which are also A modules. We prove that they exhaust all (A, Der A) irreducible modules.

Journal ArticleDOI
TL;DR: In this paper, the authors derived the multi-instanton expansion for the eigenvalues of the symmetric double well using a Langer-Cherry uniform asymptotic expansion of the solution of the corresponding Schrodinger equation.
Abstract: The multi-instanton expansion for the eigenvalues of the symmetric double well is derived using a Langer–Cherry uniform asymptotic expansion of the solution of the corresponding Schrodinger equation. The Langer–Cherry expansion is anchored to either one of the minima of the potential, and by construction has the correct asymptotic behavior at large distance, while the quantization condition amounts to imposing the even or odd parity of the wave function. This method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi-instanton expansion, and with trivial modifications can also be used for nonsymmetric double wells.

Journal ArticleDOI
TL;DR: In this article, a group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed, and it is shown that there are two equations admitting simple Lie algebras of dimension three.
Abstract: Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation.

Journal ArticleDOI
TL;DR: The present paper clarifies some misunderstandings appearing on the literature of Dirac spinor fields and exhibits a truly useful collection of results concerning the theory of Clifford algebras (including many tricks of the trade) necessary for the intelligibility of the text.
Abstract: Almost all presentations of Dirac theory in first or second quantization in physics (and mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of nonhomogeneous even multivectors fields) is used. However, a careful analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF) and Dirac–Hestenes spinor fields (DHSF) on Minkowski space–time as some equivalence classes of pairs (Ⅺu,ψⅪu), where Ⅺu is a spinorial frame field and ψⅪu is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a careful analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the bilinear covariants (on Minkowski space–time) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary Riemann–Cartan space–times. The present paper contains also Appendixes A–E which exhibits a truly useful collection of results concerning the theory of Clifford algebras (including many tricks of the trade) necessary for the intelligibility of the text.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of finding formulas for preferred deformations of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.
Abstract: Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

Journal ArticleDOI
TL;DR: In this article, an easily programmable algorithm is given for the computation of SO(5) spherical harmonics needed to complement the radial (beta) wave functions to form an orthonormal basis of wave functions for the five-dimensional harmonic oscillator.
Abstract: An easily programmable algorithm is given for the computation of SO(5) spherical harmonics needed to complement the radial (beta) wave functions to form an orthonormal basis of wave functions for the five-dimensional harmonic oscillator. It is shown how these functions can be used to compute the (Clebsch–Gordan a.k.a. Wigner) coupling coefficients for combining pairs of irreps in this space to other irreps. This is of particular value for the construction of the matrices of Hamiltonians and transition operators that arise in applications of nuclear collective models. Tables of the most useful coupling coefficients are given in the Appendix.

Journal ArticleDOI
TL;DR: In this paper, the integral of motion is assumed to be a first or second order Hermitian operator, and quadratic integrability does not imply the separation of variables in the Schrodinger equation.
Abstract: Integrable quantum mechanical systems with magnetic fields are constructed in two-dimensional Euclidean space. The integral of motion is assumed to be a first or second order Hermitian operator. Contrary to the case of purely scalar potentials, quadratic integrability does not imply the separation of variables in the Schrodinger equation. Moreover, quantum and classical integrable systems do not necessarily coincide: the Hamiltonian can depend on the Planck constant ℏ in a nontrivial manner.