Volume 45, Issue 8, August 2004
Index of content:
 METHODS OF MATHEMATICAL PHYSICS


Symmetry operators for Riemann’s method
View Description Hide DescriptionRiemann’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 selfadjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a twodimensional vector space of secondorder symmetry operators. Hence a new equivalence class of Riemann functions, admitting no firstorder symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived.

Riemann functions and the group
View Description Hide DescriptionHistorically Lie algebras of firstorder 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 selfadjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group The equations derived admit no nontrivial firstorder operators and can only be obtained from secondorder symmetry operators. Using this symmetry structure, a new equivalence class of Riemann functions can then be found.

Invariant and group theoretical integrations over the group
View Description Hide DescriptionIn a previous article, an “invariant method” to calculate monomial integrals over the group was introduced. In this paper, we study the more traditional grouptheoretical 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.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


SLEtype growth processes and the Yang–Lee singularity
View Description Hide DescriptionThe recently introduced SLE growth processes are based on conformal maps from an open and simply connected subset of the upper halfplane to the halfplane 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 halfplane to these fractions themselves. The evolutions are all driven by onedimensional 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 levelfour null vector in a highestweight module of the Virasoro algebra. This conformal field theory has central charge and is associated with the Yang–Lee singularity. Our construction may thus offer a novel description of this statistical model.

 METHODS OF MATHEMATICAL PHYSICS


Group classification of (1+1)dimensional Schrödinger equations with potentials and power nonlinearities
View Description Hide DescriptionWe perform the complete group classification in the class of nonlinear Schrödinger equations of the form where is an arbitrary complexvalued potential depending on and 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.

 STATISTICAL PHYSICS


Symmetry of matrixvalued stochastic processes and noncolliding diffusion particle systems
View Description Hide DescriptionAs an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian randommatrix ensembles, we report a systematic study of Hermitian matrixvalued processes and their eigenvalue processes associated with the chiral and nonstandard randommatrix ensembles. In addition to the noncolliding Brownian motions, we introduce a oneparameter family of temporally homogeneous noncolliding systems of the Bessel processes and a twoparameter 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 stochasticcalculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Relativistic Nboson systems bound by pair potentials
View Description Hide DescriptionWe study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form in three spatial dimensions. In natural units the system has the semirelativistic “spinlessSalpeter” Hamiltonian where g is monotone increasing and has convexity 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 In particular, we determine the energy of the Nbody oscillator with error less than 0.15% for all and

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Langer–Cherry derivation of the multiinstanton expansion for the symmetric double well
View Description Hide DescriptionThe multiinstanton 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 multiinstanton expansion, and with trivial modifications can also be used for nonsymmetric double wells.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On the restriction of quantum fields to a lightlike surface
View Description Hide DescriptionTo treat the frontform Hamiltonian approach to quantum field theory, called light cone quantum field theory, in a mathematically rigorous way, the existence of a welldefined restriction of the corresponding free fields to the hypersurface 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 welldefined Fock space expansion of a free quantum field in terms of light cone momenta which is called the zeromode problem. In this paper we present solutions to these long outstanding problems where the study of the zeromode 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 The restriction of the free field to is canonically definable on this function space and we show that the covariant field is uniquely determined by this “tame” restriction.

 METHODS OF MATHEMATICAL PHYSICS


The Drinfeld realization of the elliptic quantum group
View Description Hide DescriptionWe construct a realization of the operator satisfying the relation of the facetype elliptic quantum group The construction is based on the elliptic analog of the Drinfeld currents of which forms the elliptic algebra We give a realization of the elliptic currents and as a tensor product of the Drinfeld currents of and a Heisenberg algebra. In the levelone representation, we also give a free field realization of the elliptic currents. Applying these results, we derive a free field realization of the analog of the intertwining operators. The resultant operators coincide with those of the vertex operators in the dilute model, which is known to be a RSOS restriction of the face model.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Shouldn’t there be an antithesis to quantization?
View Description Hide DescriptionWe 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.

 METHODS OF MATHEMATICAL PHYSICS


Minimization under entropy conditions, with applications in lower bound problems
View Description Hide DescriptionWe minimize the functional under the entropy condition and where is fixed. We prove that the minimum is attained for where is chosen such that We apply the result on minimizing problems in pseudodifferential calculus, where we minimize the harmonic oscillator.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Yang–Mills instantons with Lorentz violation
View Description Hide DescriptionAn analysis is performed of instanton configurations in pure Euclidean Yang–Mills theory containing small Lorentzviolating 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 Lorentzviolating parameters.

 METHODS OF MATHEMATICAL PHYSICS


Partially invariant solutions of models obtained from the Nambu–Goto action
View Description Hide DescriptionThe 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 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.

Quantum doubles from a class of noncocommutative weak Hopf algebras
View Description Hide DescriptionThe concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasibicrossed products is constructed by means of weak Hopf skewpairs 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.

 STATISTICAL PHYSICS


Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus
View Description Hide DescriptionWe consider the “thermodynamic limit” of a ddimensional lattice of hyperbolic dynamical systems on the 2torus, 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 dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual “cluster expansion” techniques.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Randomfield quantum spherical ferroelectric model
View Description Hide DescriptionWe study a (quenched) randomfield quantum model of an anharmonic crystal for displacive structuralphase transitions in spherical approximation: the randomfield 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 shortrange interaction we obtain that the lower critical dimensionality for the zerofield) and that it decreases for longrange interactions.

 METHODS OF MATHEMATICAL PHYSICS


Partial classification of modules for Liealgebra of diffeomorphisms of dimensional torus
View Description Hide DescriptionWe consider the Liealgebra of the group of diffeomorphisms of a dimensional torus which is also known to be the algebra of derivations on a Laurent polynomialring in commuting variables denoted by The universal central extension of for is the socalled 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 modules which are irreducible and have finite dimensional weight spaces. Earlier Larsson constructed a large class of modules, the socalled tensor fields, based on modules which are also modules. We prove that they exhaust all irreducible modules.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


An algorithm for eigenvalues and eigenvectors of quaternion matrices in quaternionic quantum mechanics
View Description Hide DescriptionBy 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.

Time fractional Schrödinger equation
View Description Hide DescriptionThe 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 nonHermitian 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.
