Showing papers in "Journal of Mathematical Physics in 1995"

The world as a hologram

Abstract: According to ’t Hooft the combination of quantum mechanics and gravity requires the three‐dimensional world to be an image of data that can be stored on a two‐dimensional projection much like a holographic image. The two‐dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three‐dimensional phenomena. After outlining ’t Hooft’s proposal we give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high‐energy particle collisions are described. The phenomenon of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound fro...

Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

Abstract: Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain–Kuchař model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

Noncommutative geometry and reality

Abstract: We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR‐theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the ‘‘Connes–Lott’’ description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition.

Higher dimensional algebra and topological quantum field theory

Abstract: The study of topological quantum field theories increasingly relies upon concepts from higher‐dimensional algebra such as n‐categories and n‐vector spaces. We review progress towards a definition of n‐category suited for this purpose, and outline a program in which n‐dimensional topological quantum field theories (TQFTs) are to be described as n‐category representations. First we describe a ‘‘suspension’’ operation on n‐categories, and hypothesize that the k‐fold suspension of a weak n‐category stabilizes for k≥n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n‐dimensional unitary extended TQFTs as weak n‐functors from the ‘‘free stable weak n‐category with duals on one object’’ to the n‐category of ‘‘n‐Hilbert spaces.’’ We conclude by describing n‐categorical generalizations of deformation quantization and the quantum double construction.

Projective techniques and functional integration for gauge theories

Abstract: A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non‐linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular, prior knowledge of projective techniques is not assumed.

Numerical path integral techniques for long time dynamics of quantum dissipative systems

Abstract: Recent progress in numerical methods for evaluating the real‐time path integral in dissipative harmonic environments is reviewed. Quasi‐adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi‐adiabatic propagator path integral is evaluated by efficient system‐specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunnelingdynamics in the condensed phase are described.

Linking topological quantum field theory and nonperturbative quantum gravity

Abstract: Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory in which the pullback of the curvature to the boundary is self‐dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all SU(2) Chern–Simon theories defined by all choices of punctures and representations on the spatial boundary S. The integer level k of Chern–Simons theory is found to be given by k=6π/G2Λ+α, where Λ is the cosmological constant and α is a CP breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. Given these results, it is natural to make the conjecture that the quantum states of the system are completely determined by measurements mad...

Wavelet spectra compared to Fourier spectra

Abstract: The relation between Fourier spectra and spectra obtained from wavelet analysis is established. Small scale asymptotic analysis shows that the wavelet spectrum is meaningful only when the analyzing wavelet has enough vanishing moments. These results are related to regularity theorems in Besov spaces. For the analysis of infinitely regular signals, a new wavelet, with an infinite number of cancellations is proposed.

Clock and category: Is quantum gravity algebraic?

Abstract: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods The algebraic tools are similar to ones used in constructing topological quantum field theories The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context

Hamiltonian structures for the n‐dimensional Lotka–Volterra equations

Abstract: Under what algebraic conditions n‐dimensional Lotka–Volterra equations are Hamiltonian for a suitable Poisson structure is investigated herein.

Localization and Diagonalization: A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories

Abstract: Localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low‐dimensional gauge theories are reviewed. These are the functional integral counterparts of the Mathai–Quillen formalism, the Duistermaat–Heckman theorem, and the Weyl integral formula, respectively. In each case, the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups) is introduced, and the finite dimensional integration formulae described. Then some applications to path integrals are discussed and an overview of the relevant literature is given. The applications include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two‐dimensional Yang–Mills theory.

Continuous histories and the history group in generalized quantum theory

Abstract: We treat continuous histories within the histories approach to generalized quantum mechanics. The essential tool is the ‘‘history group:’’ the analog, within the generalized history scheme, of the canonical group of single‐time quantum mechanics.

A class of exact, periodic solutions of nonlinear envelope equations

Abstract: A class of periodic solutions of nonlinear envelope equations, e.g., the nonlinear Schrodinger equation (NLS), is expressed in terms of rational functions of elliptic functions. The Hirota bilinear transformation and theta functions are used to extend and generalize this class of solutions first reported for NLS earlier in the literature. In particular a higher order NLS and the Davey–Stewartson (DS) equations are treated. Doubly periodic standing waves solutions are obtained for both the DSI and DSII equations. A symbolic manipulation software is used to confirm the validity of the solutions independently.

Topological BF theories in 3 and 4 dimensions

Abstract: In this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2‐knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3 dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals.

An algebraic method to construct the Darboux matrix

Abstract: We present an effective procedure to construct the 1‐soliton Darboux matrix. Our approach, based on the Zakharov–Shabat–Mikhailov’s dressing method, is especially useful in the case of non‐canonical normalization and for non‐isospectral linear problems. The construction is divided into two steps. First, we represent a given linear problem as a system of some algebraic constraints on two matrices. In this context we introduce and discuss invariants of the Darboux matrix. Second, we derive the Darboux matrix demanding that it preserves the algebraic constraints. In particular, we consider in details the restrictions imposed by various reduction groups on the form of the Darboux matrix.

Functional integration over geometries

Abstract: The geometric construction of the functional integral over coset spaces M/G is reviewed. The inner product on the cotangent space of infinitesimal deformations of M defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber G, the functional measure on the coset space M/G is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev–Popov determinant of the more traditional gauge fixed approach in non‐abelian gauge theory. If the general construction is applied to the case where G is the group of coordinate reparameterizations of spacetime, the continuum functional integral over geometries, i.e. metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the metric in two and four dimensional spacetime. In two dimensions this approach generates the Polyakov–Liouville action of closed bosonic non‐critical string theory. In four dimensions the corresponding effective action leads to novel conclusions on the importance of quantum effects in gravity in the far infrared, and in particular, a dramatic modification of the classical Einstein theory at cosmological distance scales, signaled first by the quantum instability of classical de Sitter spacetime. Finite volume scaling relations for the functional integral of quantum gravity in two and four dimensions are derived, and comparison with the discretized dynamical triangulation approach to the integration over geometries are discussed. Outstanding unsolved problems in both the continuum definition and the simplicial approach to the functional integral over geometries are highlighted.

The geometrical approach to multidimensional inverse scattering

Abstract: We prove that in multidimensional short‐range potential scattering the high velocity limit of the scattering operator of an N‐body system determines uniquely the potential. For a given long‐range potential the short‐range potential of the N‐body system is uniquely determined by the high velocity limit of the modified Dollard scattering operator. Moreover, we prove that any one of the Dollard scattering operators determines uniquely the total potential. We obtain as well a reconstruction formula with an error term. Our simple proof uses a geometrical time‐dependent method.

Difference Calogero-Moser systems and finite Toda chains

Abstract: Limits of a recently introduced n‐particle difference Calogero–Moser system with elliptic potentials are studied. We obtain hyperbolic and rational difference Calogero–Moser systems with an eight‐parameter external field and (finite) difference Toda chains with four‐parameter potentials acting on the boundary particles. Hamiltonians for a number of known integrable n‐particle systems, such as Ruijsenaars’ relativistic Calogero–Moser and Toda models and their generalizations associated with classical root systems, can be seen as special cases of the Hamiltonians considered in this paper.

Integrable dynamics of a discrete curve and the Ablowitz–Ladik hierarchy

Abstract: It is shown that the following elementary geometric properties of the motion of a discrete (i.e., piecewise linear) curve select the integrable dynamics of the Ablowitz–Ladik hierarchy of evolution equations: (i) the set of points describing the discrete curve lie in the sphere S3; (ii) the distance between any two subsequent points does not vary in time; (iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. These results generalize to a discrete context the previous work on continuous curves [A. Doliwa and P. M. Santini, Phys. Lett. A 185, 373 (1994)].

SU(2) coherent‐state path integral

Abstract: The SU(2) coherent‐state path integral is used to represent the matrix element of a propagator in the SU(2) coherent‐state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary‐phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.

Nonsymmetric Gravitational Theory

Abstract: A nonsymmetric gravitational theory is presented which is free of ghost poles, tachyons, and higher‐order poles and there are no problems with asymptotic boundary conditions. A static spherically symmetric solution in the long‐range approximation is everywhere regular and does not contain a black hole event horizon. The information loss problem is resolved at the classical level.

GR via characteristic surfaces

Abstract: We reformulate the Einstein equations as equations for families of surfaces on a four‐manifold. These surfaces eventually become characteristic surfaces for an Einstein metric (with or without sources). In particular they are formulated in terms of two functions on R4×S2, i.e., the sphere bundle over space–time, one of the functions playing the role of a conformal factor for a family of associated conformal metrics, the other function describing an S2’s worth of surfaces at each space–time point. It is from these families of surfaces themselves that the conformal metric, conformal to an Einstein metric, is constructed; the conformal factor turns them into Einstein metrics. The surfaces are null surfaces with respect to this metric.

Properties of q‐entropies

Abstract: Basic properties of the q‐entropy Sq[ρ]=(q−1)−1(1−tr(ρq)) (0

Mixed states with positive Wigner functions

Abstract: The Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent, according to a result of Hudson. The characterization of mixed states with a positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e., functions with given ‘‘Wigner spectrum’’ in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. It is shown by explicit examples that the converses of both implications are false. The examples are taken from a low‐dimensional section of the state space, in which all Wigner spectra can be computed. In this set counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products are also found, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.

Lie symmetries of finite‐difference equations

Abstract: Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations are determined and it is found that they retain the same Lie symmetry algebras as their continuum limits. Solutions with definite transformation properties are obtained; identities and formulas for these functions are then derived using the symmetry algebra.

The gravitational sector in the Connes–Lott formulation of the standard model

Abstract: We studied the Riemannian aspect and the Hilbert–Einstein gravitational action of the noncommutative geometry underlying the Connes–Lott construction of the action functional of the standard model. This geometry involves a two‐sheeted, Euclidian space–time. We show that if we require the space of forms to be locally isotropic and the Higgs scalar to be dynamical, then the Riemannian metrics on the two sheets of Euclidian space–time must be identical. We also show that the distance function between the two sheets is determined by a single, real scalar field whose vacuum expectation value (VEV) sets the weak scale.

Relativistic theories of dissipative fluids

Abstract: The simple Navier–Stokes equations for dissipative fluids, translated into relativity, form a parabolic—and hence mathematically nonviable—system. There have been formulated numerous alternative theories, consisting instead of hyperbolic equations—theories that necessarily involve more dynamical variables and more free functions than does the simple Navier–Stokes theory. It is argued that, these mathematical differences notwithstanding, the physical content of these hyperbolic theories is in most cases precisely the same as that of Navier–Stokes.

Contractions of Lie algebras: Generalized Inönü-Wigner contractions versus graded contractions

Abstract: Recently, a new notion of contraction has been introduced, the purely algebraically defined graded contractions that fall into two disjoint classes: continuous and discrete. Our main result is that all continuous graded contractions can be realized by generalized Inonu–Wigner contractions. Furthermore, we give a complete characterization of the discrete graded contractions via certain higher‐order identities inherent to the grading group.

A q deformation of the Gauss distribution

Abstract: The q deformed commutation relation aa*−qa*a=1 for the harmonic oscillator is considered with q∈[−1,1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a+a* in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian.

Berry phases for Landau Hamiltonians on deformed tori

Abstract: Parametrized families of Landau Hamiltonians are introduced, where the parameter space is the Teichmuller space (topologically the complex upper half plane) corresponding to deformations of tori. The underlying SO(2,1) symmetry of the families enables an explicit calculation of the Berry phases picked up by the eigenstates when the torus is slowly deformed. It is also shown that apart from these phases that are local in origin, there are global non‐Abelian ones too, related to the hidden discrete symmetry group Γϑ (the theta group, which is a subgroup of the modular group) of the families. The induced Riemannian structure on the parameter space is the usual Poincare metric on the upper half plane of constant negative curvature. Due to the discrete symmetry Γϑ the geodesic motion restricted to the fundamental domain of this group is chaotic.