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We analyze the optimal unambiguous discrimination of two arbitrary mixed quantum states. We show that the optimal measurement is unique and we present this optimal measurement for the case where the rank of the density operator of one of the states is at most 2 (“solution in four dimensions”). The solution is illustrated by some examples. The optimality conditions proven by Eldar et al. [Phys. Rev. A 69, 062318 (2004)] are simplified to an operational form. As an application we present optimality conditions for the measurement, when only one of the two states is detected. The current status of optimal unambiguous state discrimination is summarized via a general strategy. [ABSTRACT FROM AUTHOR]

We present an exact solution of the three-dimensional Duffin–Kemmer–Petiau oscillator for spins 1 and 0 in the momentum space with the presence of minimal length uncertainty by the technique of vector spherical harmonics. The eigenfunctions are determined for both cases and the energy eigenvalues equation are obtained. The limiting case is then deduced for a small parameter of deformation. [ABSTRACT FROM AUTHOR]

Recently, the first author [Adachi, “Asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field,” Commun. Math. Phys. 275, 443 (2007)] proved the asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is nonzero by obtaining propagation estimates for the physical propagator. However, in his work, it is needed that potentials under consideration are sufficiently smooth. In this paper, when N=2, we prove the asymptotic completeness of (modified) wave operators under the assumption that the potential has the local singularity of type |x|-1+ε when d≥3 and 0<ε<1, where d is the space dimension. We also discuss the modifiers in the position representation, which are used in the definition of the modified wave operators in the long-range case. [ABSTRACT FROM AUTHOR]

In this paper we study the well posedness of solution to a certain system of nonlinear Klein–Gordon–Schrödinger equations in three space dimensions. Basing on the Strichartz estimates, we obtain the global existence, uniqueness of the solutions, and continuous dependence with respect to initial data in the Sobolev spaces of low regularities by setting appropriate contraction and taking difference estimates. [ABSTRACT FROM AUTHOR]

Polar coordinates are used for the complex scalar free field in D=4 dimensions. The resulting nonrenormalizable theory is healed by using a recently proposed symmetric subtraction procedure. The existence of the coordinate transformation is proved by construction. [ABSTRACT FROM AUTHOR]

We show that appropriate superpositions of motional states are a reference frame resource that enables breaking of time-reversal superselection so that two parties lacking knowledge about the other’s direction of time can still communicate. We identify the time-reversal reference frame resource states and determine the corresponding frameness monotone, which connects time-reversal frameness to entanglement. In contradistinction to other studies of reference frame quantum resources, this is the first analysis that involves an antiunitary rather than unitary representation. [ABSTRACT FROM AUTHOR]

This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term “daseinisation” by which a projection operator is mapped to a subobject of the spectral presheaf—the topos quantum analog of a classical state space. In the second part of the paper, we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study “truth objects” in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence, there are no micro-states in the sense of classical physics. Truth objects therefore play a crucial role in our formalism. [ABSTRACT FROM AUTHOR]

This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we discuss two different types of language that can be attached to a system S. The first is a propositional language PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in Paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our program utilizes the more powerful language L(S) and its representation in an appropriate topos. [ABSTRACT FROM AUTHOR]

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. In Paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow δo(A):Σ_→R≽_, where R≽_ is the quantity-value object for this theory. The construction of δo(A⁁) is an extension of the daseinisation of projection operators that was discussed in Paper II. The object R≽_ is a monoid object only in the topos, τ[lowercase_phi_synonym]=SetsV(H)op, of the theory, and to enhance the applicability of the formalism, we apply to R≽_ a topos analog of the Grothendieck extension of a monoid to a group. The resulting object, k(R≽_), is an abelian group object in τ[lowercase_phi_synonym]. We also discuss another candidate, R↔_, for the quantity-value object. In this presheaf, both inner and outer daseinisations are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme. [ABSTRACT FROM AUTHOR]

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. In Paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow δo(A):Σ_→R≽_, where R≽_ is the quantity-value object for this theory. The construction of δo(A⁁) is an extension of the daseinisation of projection operators that was discussed in Paper II. The object R≽_ is a monoid object only in the topos, τ[lowercase_phi_synonym]=SetsV(H)op, of the theory, and to enhance the applicability of the formalism, we apply to R≽_ a topos analog of the Grothendieck extension of a monoid to a group. The resulting object, k(R≽_), is an abelian group object in τ[lowercase_phi_synonym]. We also discuss another candidate, R↔_, for the quantity-value object. In this presheaf, both inner and outer daseinisations are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme. [ABSTRACT FROM AUTHOR]

Geometry can be an unsuspected source of equations with physical relevance, as everybody is aware since Einstein formulated the general theory of relativity. However, efforts to extend a similar type of reasoning to other areas of physics, namely, electrodynamics, quantum mechanics, and particle physics, usually had very limited success; particularly in quantum mechanics the standard formalism is such that any possible relation to geometry is impossible to detect; other authors have previously trod the geometric path to quantum mechanics, some of that work being referred to in the text. In this presentation we will follow an alternate route to show that quantum mechanics has indeed a strong geometric character. The paper makes use of geometric algebra, also known as Clifford algebra, in five-dimensional space-time. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in a hyperbolic space this fact leads inevitably to a wave equation with planelike solutions. This is also true for five-dimensional space-time and we will explore those solutions, establishing a parallel with the solutions of the free particle Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4×4 matrices, also known as Dirac’s matrices. There is one problem with this isomorphism, because the solutions to Dirac’s equation are usually known as spinors (column matrices) that do not belong to the 4×4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate fourfold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research. [ABSTRACT FROM AUTHOR]

In this paper we prove that the two-dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes. [ABSTRACT FROM AUTHOR]

We prove the existence of two-cluster threshold resonances in N-body problems and study their perturbation by intercluster interactions. As application, we construct concrete examples based on Yukawa potentials for which the N-body Efimov effect happens with N>=4. [ABSTRACT FROM AUTHOR]

We reexamine here the issue of consistency of minimal action formulation with the minimal coupling procedure (MCP) in spaces with torsion. In Riemann­Cartan spaces, it is known that a proper use of the MCP requires that the trace of the torsion tensor be a gradient, Τµ= аµθ, and that the modified volume element τθ=еθ√gdχ¹^⋯^χn be used in the action formulation of a physical model. We rederive this result here under considerably weaker assumptions, reinforcing some recent results about the inadequacy of propagating torsion theories of gravity to explain the available observational data. The results presented here also open the door to possible applications of the modified volume element in the geometric theory of crystalline defects. [ABSTRACT FROM AUTHOR]

We prove that the mathematical framework for the de Sitter top system is the de Sitter fiber bundle. In this context, the concept of soldering associated with a fiber bundle plays a central role. We comment on the possibility that our formalism may be of particular interest in different contexts including MacDowell-Mansouri theory, two time physics, and oriented matroid theory. [ABSTRACT FROM AUTHOR]

We present an extended version of max-plus algebra which includes the inverse operator of “max.” This algebra enables us to ultradiscretize the system including subtractions and obtain new ultradiscrete equations. The known ultradiscrete equations can also be recovered by this construction. [ABSTRACT FROM AUTHOR]

A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R⁁(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R⁁(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2-1) nonzero elements of R⁁(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 1/2(N+3)(N-1) free parameters mα. The diagonalizer of R⁁(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R⁁(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R⁁(θ) is developed in a power series in θ. The basic difference arising for even dimensions is made explicit. Some special features of our R⁁(θ) are discussed in a concluding section. [ABSTRACT FROM AUTHOR]

We complete the classification of almost commutative geometries from a particle physics point of view given by Iochum, Schücker, and Stephan, J. Math. Phys. (to be published). Four missing Krajewski diagrams will be presented after a short introduction into irreducible, nondegenerate spectral triples. [ABSTRACT FROM AUTHOR]

A quantum mechanical version of the Lax–Phillips scattering theory was recently developed. This theory is a natural framework for the description of quantum unstable systems. However, since the spectrum of the generator of evolution in this theory is unbounded from below, the existing framework does not apply to a large class of quantum mechanical scattering problems. It is shown in this work that the fundamental mathematical structure underlying the Lax–Phillips theory, i.e., the Sz.-Nagy–Foias theory of contraction operators on Hilbert space, can be used for the construction of a formalism in which models associated with a semibounded spectrum may be accommodated. [ABSTRACT FROM AUTHOR]

The integrability of the deformed quantum elliptic Calogero–Moser problem introduced by Chalykh, Feigin, and Veselov is proven. Explicit recursive formulas for the integrals are found. For integer values of the parameter this implies the algebraic integrability of the systems. [ABSTRACT FROM AUTHOR]

The Hughston–Jozsa–Wootters theorem shows that any finite ensemble of quantum states can be prepared “at a distance,” and it has been used to demonstrate the insecurity of all bit commitment protocols based on finite quantum systems without superselection rules. In this paper, we prove a generalized HJW theorem for arbitrary ensembles of states on a C*-algebra. We then use this result to demonstrate the insecurity of bit commitment protocols based on infinite quantum systems, and quantum systems with Abelian superselection rules. [ABSTRACT FROM AUTHOR]

If a state is passive for uniformly accelerated observers in n-dimensional (n>=2) anti–de Sitter (Ads) space–time (i.e., cannot be used by them to operate a perpetuum mobile), they will (a) register a universal value of the Unruh temperature, (b) discover a PCT symmetry, and (c) find that observables in complementary wedge-shaped regions necessarily commute with each other in this state. The stability properties of such a passive state induce a “geodesic causal structure” on AdS and concommitant locality relations. It is shown that observables in these complementary wedge-shaped regions fulfill strong additional independence conditions. In two-dimensional AdS these even suffice to enable the derivation of a nontrivial, local, covariant net indexed by bounded space–time regions. All these results are model-independent and hold in any theory which is compatible with a weak notion of space–time localization. Examples are provided of models satisfying the hypotheses of these theorems. [ABSTRACT FROM AUTHOR]

We classify all irreducible, almost commutative geometries whose spectral action is dynamically nondegenerate. Heavy use is made of Krajewski’s diagrammatic language. The motivation for our definition of dynamical nondegeneracy stems from particle physics where the fermion masses are nondegenerate. [ABSTRACT FROM AUTHOR]

One of the simplest pseudo-Hermitian models with real spectrum (viz., square-well on a real interval I of coordinates) is re-examined. A PT-symmetric complex deformation C of I is introduced and shown tractable via an innovated approach to matching conditions. The result is surprising: An arbitrarily small deformation I→C implies a sudden collapse (i.e., the spontaneous PT-symmetry breaking) of virtually all the spectrum (i.e., up to its low-energy part). [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence and uniqueness of the local strong solutions to the initial boundary problem for a one-dimensional shallow-water equation (Camassa–Holm equation) on the half-space {x>0} with initial data uo∈H2(R+)∩H01(R+). The solution is obtained as a limit of the solutions for a class of approximation problems. We also establish the global result of the corresponding solution, provided that the initial data u0 satisfies certain positivity condition. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in 2 variables. Chaundy’s equation, with 4 parameters, is the most general self-adjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a two-dimensional vector space of second-order symmetry operators. Hence a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Historically Lie algebras of first-order symmetry operators have proven to be a useful method for finding equivalence classes of Riemann functions. Here this idea is extended to higher order symmetries. The approach is to seek self-adjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group E(1,1). The equations derived admit no nontrivial first-order operators and can only be obtained from second-order symmetry operators. Using this symmetry structure, a new equivalence class of Riemann functions can then be found. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The recently introduced SLE growth processes are based on conformal maps from an open and simply connected subset of the upper half-plane to the half-plane itself. We generalize this by considering a hierarchy of stochastic evolutions mapping open and simply connected subsets of smaller and smaller fractions of the upper half-plane to these fractions themselves. The evolutions are all driven by one-dimensional Brownian motion. Ordinary chordal SLE appears at grade one in the hierarchy. At grade two we find a direct correspondence to conformal field theory through the explicit construction of a level-four null vector in a highest-weight module of the Virasoro algebra. This conformal field theory has central charge c=-22/5 and is associated with the Yang–Lee singularity. Our construction may thus offer a novel description of this statistical model. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In a previous article, an “invariant method” to calculate monomial integrals over the U(n) group was introduced. In this paper, we study the more traditional group-theoretical method, and compare its strengths and weaknesses with those of the invariant method. As a result, we are able to introduce a “hybrid method” which combines the respective strengths of the other two methods. There are many examples in the paper illustrating how each of these methods works. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We perform the complete group classification in the class of nonlinear Schrödinger equations of the form iψt+ψxx+|ψ|γψ+V(t,x)ψ=0, where V is an arbitrary complex-valued potential depending on t and x, γ is a real nonzero constant. We construct all the possible inequivalent potentials for which these equations have nontrivial Lie symmetries using a combination of algebraic and compatibility methods. The proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(rij) = g(rij²) in thee spatial dimensions. In natural units &hstroke; = c = 1 the system has the semirelativistic ‘spinless-Salpeter’ Hamiltonian H = ∑I=1N √m² + pi² + ∑j>i=jNg(¦ri-rj¦²), where g is monotone increasing and has convexity gn≥0. We use ‘envelope theory’ to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N≥2. In particular, we determine the energy of the N-body oscillator g(r²) = cr² with error less than 0.15% for all m≥0, N≥2, and c>0. [ABSTRACT FROM AUTHOR]

To treat the front-form Hamiltonian approach to quantum field theory, called light cone quantum field theory, in a mathematically rigorous way, the existence of a well-defined restriction of the corresponding free fields to the hypersurface {x0+x3=0} in Minkowski space is of an essential necessity. However, even in the situation of a real scalar free field such a restriction does canonically not exist; this is called the restriction problem. Furthermore, since the beginning of light cone quantum field theory there is the problem of nonexistence of a well-defined Fock space expansion of a free quantum field in terms of light cone momenta which is called the zero-mode problem. In this paper we present solutions to these long outstanding problems where the study of the zero-mode problem (of the corresponding classical field) will lead us to a solution of the restriction problem. We introduce a new function space of “squeezed” smooth functions which can canonically be embedded into the Schwartz space S(R3). The restriction of the free field to {x0+x3=0} is canonically definable on this function space and we show that the covariant field is uniquely determined by this “tame” restriction. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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 Schrödinger 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. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We construct a realization of the L-operator satisfying the RLL-relation of the face-type elliptic quantum group Bq,λ(A2(2)). The construction is based on the elliptic analog of the Drinfeld currents of Uq(A2(2)), which forms the elliptic algebra Uq,p(A2(2)). We give a realization of the elliptic currents E(z), F(z), and K(z) as a tensor product of the Drinfeld currents of Uq(A2(2)) and a Heisenberg algebra. In the level-one representation, we also give a free field realization of the elliptic currents. Applying these results, we derive a free field realization of the Uq,p(A2(2))-analog of the Bq,λ(A2(2))-intertwining operators. The resultant operators coincide with those of the vertex operators in the dilute AL model, which is known to be a RSOS restriction of the A2(2) face model. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We raise the possibility of developing a theory of constructing quantum dynamical observables independent from quantization and deriving classical dynamical observables from pure quantum mechanical consideration. We do so by giving a detailed quantum mechanical derivation of the classical time of arrival at arbitrary arrival points for a particle in one dimension. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We minimize the functional f->∫ afdμ under the entropy condition E(f)=-∫ f log fdμ>=E, ∫ fdμ=1 and f>=0, where E∈R is fixed. We prove that the minimum is attained for f=e-sa/∫ e-sadμ, where s∈R is chosen such that E(f)=E. We apply the result on minimizing problems in pseudodifferential calculus, where we minimize the harmonic oscillator. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An analysis is performed of instanton configurations in pure Euclidean Yang–Mills theory containing small Lorentz-violating perturbations that maintain gauge invariance. Conventional topological arguments are used to show that the general classification of instanton solutions involving the topological charge is the same as in the standard case. Explicit solutions are constructed for general gauge invariant corrections to the action that are quadratic in the curvature. The value of the action is found to be unperturbed to lowest order in the Lorentz-violating parameters. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The concept of partially invariant solutions is discussed in the framework of the group analysis of models derived from the Nambu–Goto action. In particular, we consider the nonrelativistic Chaplygin gas and the relativistic Born–Infeld theory for a scalar field. Using a general systematic approach based on subgroup classification methods, nontrivial partially invariant solutions with defect structure δ=1 are constructed. For this purpose, a classification of the subgroups of the Lie point symmetry group, which have generic orbits of dimension 2, has been performed. These subgroups allow us to introduce the corresponding symmetry variables and next to reduce the initial equations to different nonequivalent classes of partial differential equations and ordinary differential equations. The latter can be transformed to standard form and, in some cases, solved in terms of elementary and Jacobi elliptic functions. This results in a large number of new partially invariant solutions, which are determined to be either reducible or irreducible with respect to the symmetry group. Some physical interpretation of the results in the area of fluid dynamics and field theory are discussed. The solutions represent traveling and centered waves, algebraic solitons, kinks, bumps, cnoidal and snoidal waves. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products is constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf algebra is built. Examples of quantum doubles from a Clifford monoid as well as a noncommutative and noncocommutative weak Hopf algebra are given, generalizing quantum doubles from a group and a noncommutative and noncocommutative Hopf algebra, respectively. Moreover, some characterizations of quantum doubles of finite dimensional biperfect weak Hopf algebras are obtained. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We consider the “thermodynamic limit” of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual “cluster expansion” techniques. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study a (quenched) random-field quantum model of an anharmonic crystal for displacive structural phase transitions in spherical approximation: the random-field quantum spherical (ferroelectric) model. For stationary ergodic random fields its behavior depends on the quantum parameter of the model and on the expectation and covariance of the field. If quantum fluctuations are small enough not to destroy the phase transition, then it can be suppressed when the field fluctuations are large. For the field of independent identically distributed random variables and the short-range interaction we obtain that the lower critical dimensionality dl=4 (dl=2 for the zero-field) and that it decreases for long-range interactions. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

By means of complex representation and companion vector, this paper studies the problems of eigenvalues and eigenvectors of quaternion matrices, and gives a technique of computing the eigenvalues and eigenvectors of the quaternion matrices in quaternionic quantum mechanics. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The Schrödinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrödinger 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 Schrödinger 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. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The unified description of the main thermodynamical functions of the Bose and Fermi ideal gases, obtained by Lee [J. Math. Phys. 36, 1217 (1995)] in terms of the polylogarithmic functions, can also be obtained by analytic continuation in the chemical potential owing to the analytic properties of the Lerch function that is simply related to the polylogarithmic ones. By this procedure we also show that the Fourier coefficients of the thermal Green function of the ideal Bose gas convert into those of the Fermi one. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this work, the Balian–Low theorem is extended to Gabor (also called Weyl–Heisenberg) frames for subspaces and, more particularly, its relationship with the unique Gabor dual property for subspace Gabor frames is pointed out. To achieve this goal, the subspace Gabor frames which have a unique Gabor dual of type I (resp. type II) are defined and characterized in terms of the Zak transform for the rational parameter case. This characterization is then used to prove the Balian–Low theorem for subspace Gabor frames. Along the same line, the same characterization is used to prove a duality theorem for the unique Gabor dual property which is an analogue of the Ron and Shen duality theorem. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

For relativistic stars rotating slowly and differentially with a positive angular velocity, some properties in relation to the positiveness of the rate of rotational dragging and of the angular momentum density are derived. Moreover, the proof for the bounds on the rotational mass-energy, which we have generalized (outside the slow rotation limit) in a previous paper, is briefly exposed. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]