Showing papers in "Journal of Mathematical Physics in 1991"

General theory of fractal path integrals with applications to many‐body theories and statistical physics

Abstract: A general scheme of fractal decomposition of exponential operators is presented in any order m. Namely, exp[x(A+B)]=Sm(x)+O(xm+1) for any positive integer m, where Sm(x)=et1A et2B et3A et4B⋅⋅⋅etMA with finite M depending on m. A general recursive scheme of construction of {tj} is given explicitly. It is proven that some of {tj} should be negative for m≥3 and for any finite M (nonexistence theorem of positive decomposition). General systematic decomposition criterions based on a new type of time‐ordering are also formulated. The decomposition exp[x(A+B)]=[Sm(x/n)]n +O(xm+1/nm) yields a new efficient approach to quantum Monte Carlo simulations.

Examples of braided groups and braided matrices

Abstract: Matrix braided groups are developed as an analog of the ‘‘coordinate functions’’ on a group or supergroup. The ±1 in the super case is replaced by braid statistics. There are braided group analogs of all the classical simple Lie groups as well as braided matrix groups and braided matrices B(R) for every regular solution R of the quantum Yang–Baxter equations. A direct verification of B(R) is provided and some of the simplest examples are computed in detail.

Necessary and sufficient conditions for constructing orthonormal wavelet bases

Abstract: This paper proves a previous conjecture of the author characterizing sequences h∈l2(Z) that yield orthonormal wavelet bases of L2(R) in terms of the multiplicity of the eigenvalue 1 of an operator associated to h. The proof utilizes a result of Cohen characterizing these sequences in terms of the real zeros of their Fourier transforms. The mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2(Z) into L2(R). Related conjectures are discussed.

Group theory of the Smorodinsky-Winternitz system

Abstract: The three degrees of freedom Smorodinsky–Winternitz system is a degenerate or super‐integrable Hamiltonian that possesses five functionally independent globally defined and single‐valued integrals of the motion in both classical and quantum mechanics. This is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the Hamiltonian under a group of symmetry transformations isomorphic to the three‐dimensional unitary unimodular group, SU(3). In turn, this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other, the so‐called dynamical group. All the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group. So, in common with the hydrogen atom and the harmonic oscillator, the quantum mechanics of the Smorodinsky–Winternitz system may be completely solved within the framework of group theory alone.

A new N=2 supersymmetric Korteweg–de Vries equation

Abstract: There are two known integrable N=2 space supersymmetric extensions of the KdV equation. Both can be written as Hamiltonian systems with a common Poisson structure which corresponds to the N=2 supersymmetric form of the second KdV structure. By analyzing the conservation laws of the one parameter family of equations that share this Hamiltonian structure, one finds that a third system is singled out by the existence of nontrivial conservation laws. It is conjectured to be integrable.

Graviton fluctuations in de Sitter space

Abstract: The two‐point correlation function of metric fluctuations in de Sitter space is calculated. The results are expressed in a gauge‐invariant and O(4,1)‐invariant form in terms of elementary functions of z(x,x’) (a biscalar variable simply related to the invariant distance between x and x’). The Feynman functions for the transverse, trace‐free, and scalar metric fluctuations grow without bounds, as ln z and z ln z, respectively, for large z. The consequences of this bad infrared behavior are discussed and it is shown that it leads to divergences in physical quantities at both the classical and quantum levels.

Preliminary group classification of equations vtt=f(x,vx)vxx+g(x,vx)

Abstract: A classification is given of equations vtt=f(v,vx)vxx+g(x,vx) admitting an extension by one of the principal Lie algebra of the equation under consideration. The paper is one of few applications of a new algebraic approach to the problem of group classification: the method of preliminary group classification. The result of the work is a wide class of equations summarized in Table II.

Algebraic invariants of the Riemann tensor in a four‐dimensional Lorentzian space

Abstract: A set of 16 scalar invariants is given of the Riemann tensor which is shown to contain complete minimal sets in the Einstein–Maxwell and perfect fluid cases. All previously known sets fail to be complete in the perfect fluid case.

The quantum Heisenberg group H(1)q

Abstract: The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

Method of self‐similar approximations

Abstract: A new method is suggested to approximately find out the limit for a sequence of functions when solely several first terms of a sequence are known. The method seems to be very useful for those complicated problems of mathematical physics, statistical mechanics, and field theory in which, because of technical difficulties, one is able to calculate, using some iteration procedure or perturbation scheme, only several first approximations for quantities of interest. As an illustration, the quartic oscillator is considered, and it is shown that invoking only the terms of a perturbation theory up to the second order, the ground‐state energy is defined with an accuracy of 0.1% for all values of the anharmonicity parameter, from zero up to infinity.

Unimodular theory of gravity and the cosmological constant

Abstract: The unimodular theory of gravity with a constrained determinant gμν is equivalent to general relativity with an arbitrary cosmological constant Λ. Within this framework Λ appears as an integration constant unrelated to any parameters in the Lagrangian. In a quantum theory the state vector of the universe is thus expected to be a superposition of states with different values of Λ. Following Hawking’s argument one concludes that the fully renormalized Λ=0 completely dominates other contributions to the integral over Λ in the vacuum functional. In this scenario of the unimodular theory of gravity the cosmological constant problem is solved. Furthermore, this formulation naturally provides an external (cosmic) time for time ordering of measurements so that the quantum version of the unimodular theory can have a normal ‘‘Schrodinger’’ form of time development, giving a simpler interpretation to the equation of the universe.

Some multivariable orthogonal polynomials of the Askey tableau-discrete families

Abstract: A multivariable generalization is presented for all the discrete families of the Askey tableau. This significantly extends the multivariable Hahn polynomials introduced by Karlin and McGregor. The latter are recovered as a limit case from a family of multivariable Racah polynomials.

An Sp(2)‐covariant quantization of gauge theories with linearly dependent generators

Abstract: The Sp(2)‐symmetric version of covariant quantization is formulated for the general case of any stage‐reducible gauge theories.

p‐adic quantum mechanics with p‐adic valued functions

Abstract: An extension of the formalism of quantum mechanics to the case where the canonical variables and functions are valued in a field of p‐adic numbers is considered. A new p‐adic integral calculus is used for the realization of the Gauss representation in p‐adic quantum mechanics.

Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure

Abstract: Basic invariants, such as conserved quantities, symmetries, mastersymmetries, and recursion operators are explicitly constructed for the following nonlinear lattice systems: The modified Korteweg–de Vries lattice, the Ablowitz–Ladik lattice, the Brusci–Ragnisco lattice, the Ragnisco–Tu lattice and some cases of the class of integrable systems introduced by Bogoyavlensky. The algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the Lax pair analysis is discussed. By explicit presentation of the higher‐order members of the corresponding hierarchies new completely integrable lattice flows are found. For all systems, multi‐Hamiltonian formulations are given.

The three‐dimensional Euclidean quantum group E(3)q and its R‐matrix

Abstract: A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three‐dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R‐matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R‐matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two‐dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

Onsager’s algebra and the Dolan–Grady condition in the non‐self‐dual case

Abstract: In this paper, it is shown that a pair of operators that satisfy the Dolan–Grady conditions generate an Onsager algebra even in the case that they are not self‐dual. Briefly considered will be the implications for the spectra of superintegrable spin chains, in various sectors.

Bianchi type‐I cosmological models in Lyra’s geometry

Abstract: Exact solutions are obtained for the anisotropic Bianchi type‐I model in normal gauge for Lyra’s geometry. The physical behavior of the models is examined in vacuum and in the presence of perfect fluids.

Geometry of superspace with even and odd brackets

Abstract: The geometry of the superspace provided by two arbitrary Poisson brackets of different gradings is studied. It will be shown that the group of transformations that preserve these two brackets is finite‐dimensional.

Redefinition of position variables and the reduction of higher‐order Lagrangians

Abstract: Single‐time Lagrangians are treated in this paper, describing the dynamics of systems of point particles, which are given as formal power series in some ordering parameter and which may contain higher time derivatives in all terms but the leading one. An efficient method for eliminating the higher time derivatives directly on the Lagrangian level is presented. This method clarifies the meaning of using the lower‐order equations of motion in higher‐order terms in a Lagrangian. The method consists of an iterative use of ‘‘contact’’ transformations in the jet prolongation of the extended configurations space and is called ‘‘the method of redefinition of position variables.’’ Several examples from electrodynamics and relativistic gravity are treated explicitly.

On the geometric quantization of Poisson manifolds

Abstract: In a paper by Huebschmann [J Reine Angew Math 408, 57 (1990)], the geometric quantization of Poisson manifolds appears as a particular case of the quantization of Poisson algebras Here, this quantization is presented straightforwardly The results include a geometric prequantization integrality condition and its discussion in particular cases such as Dirac brackets, an adaptation of the notion of a polarization and a construction of a quantum Hilbert space, and a computational example In the last section of the paper the general prequantization representations in the sense of Urwin [Adv Math 50, 126 (1983)] are described for the Poisson and Jacobi manifolds

Remarks on the Sp(2)-covariant Lagrangian quantization of gauge theories.

Abstract: The existence of a solution of the exact generating equations (with ℏ≠0) for the Sp(2)‐covariant Lagrangian quantization scheme is established The characteristic arbitrariness of this solution is also studied The equivalence between the Sp(2)‐covariant quantization and the standard one is proven

Some exact results for N-point massive Feynman integrals

Abstract: By using the Mellin–Barnes representation for massive denominators, some exact results for classes of one‐loop N‐point massive Feynman integrals are obtained for arbitrary values of the line indices (the powers of denominators) and of the space‐time dimension. A representation for corresponding massless integral is also derived.

Dyson’s correlation functions and graded symmetry

Abstract: A new derivation of Dyson’s k‐level correlation functions of the Gaussian unitary ensemble (GUE) is given. The method uses matrices with graded symmetry. The number of integrations needed for the ensemble average becomes independent of the level number N. For arbitrary level number N, the k‐level correlation function is expressed as an integral involving the eigenvalues of a 2k×2k graded matrix. The limit of infinitely many levels N→∞ is calculated by a simple saddle‐point approximation of this integral, avoiding the introduction of Hermite polynomials and oscillator wave functions.

Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations

Abstract: Given a partial differential equation, its Painleve analysis will first be performed with a built‐in invariance under the homographic group acting on the singular manifold function. Then, assuming an order for the underlying Lax pair, a multicomponent pseudopotential of projective Riccati type, the components of which are homographically invariant, is introduced. If the equation admits a classical Darboux transformation, a very small set of determining equations whose solution yields the Lax pair will be generated in the basis of the pseudopotential. This new method will be applied to find the yet unpublished Lax pair of the scalar Hirota–Satsuma equation.

Continuous quantum jumps and infinite-dimensional stochastic equations

Abstract: From a mathematical point of view, a class of infinite‐dimensional stochastic differential equations describing continuous spontaneous localization in quantum dynamics will be studied. Existence and uniqueness of weak and strong solutions of respective equations are proven via Cameron–Martin–Girsanov transformation. The case of Gaussian initial states is explicitly solved.

A thermodynamical approach to Eddington factors

Abstract: Eddington factors are a common ingredient in many techniques for solving radiation hydrodynamics problems. Usually they are introduced in a phenomenological or ad hoc manner. In this paper a fundamental approach is devised for justifying Eddington factors on the basis of mathematical requirements arising from nonequilibrium thermodynamics.

Negative powers of Olver recursion operators

Abstract: Olver’s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator. Some negative order KdV equations are derived in this way. In particular, the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated Schrodinger equation used in the inverse scattering solution of the KdV equation and this formulation is used to show that this new equation is the Miura transform of a sinh‐Gordon equation.

Nonlinear resonant scattering and plasma instability : an integrable model

Abstract: A detailed study of a system of coupled waves is given for which an initial‐boundary value problem is solved by means of the spectral transform theory. This system represents the nonlinear interaction of an electrostatic high‐frequency wave with the ion acoustic wave in a two component homogeneous plasma. As a result it is understood the plasma instability as (i) a continuous secular transfer of energy from the laser beam to the acoustic wave, (ii) the evolution toward the formation of local singularities of the electrostatic wave (collapsing), (iii) a mutual trapping of the acoustic wave and the scattered Langmuir wave.

Real representations of finite Clifford algebras. I, Classification

Abstract: A classification of real matrix irreducible representations of finite‐dimensional real Clifford algebras has been made. In contrast to the case of complex representation, three distinct types of representations can be obtained which we call normal, almost complex, and quaternionic. The dimension of the latter two cases is twice as large as that of the normal representation. A criteria for a given Clifford algebra to possess a particular type of the representations is also given with some applications.