A Feynman-Kac formula for the quantum Heisenberg ferromagnet. II

This article continues the analysis of the first arcticle under the same title. Using methods of stochastic analysis we prove Feynman-Kac formulas for the relevant heat kernels. We also present classical limit theorems.Dedicated to Res Jost and Arthur Wightman     

A Feynman-Kac formula for the quantum Heisenberg ferromagnet. I

The Hamiltonian of the (anisotropic) quantum Heisenberg (anti-) ferromagnet on an arbitrary finite lattice is lifted to a Hamiltonian acting on sections of the bundle obtained by twisting a certain line bundle over the classical spin configuration space (which is a Kähler manifold) with the Dolbeault complex. This procedure is extended fromSU(2) to arbitrary compact semi-simple Lie groups and arbitrary irreducible representations. The Bott-Borel-Weil theorem gives a heat kernel representation for the original partition function in an external magnetic field. TheU(1)-gauged local Hamiltonian is the sum of the free, supersymmetric, twisted Dolbeault Laplace operator (multiplied by the inverse of an arbitrary small mass parameter) plus the lifted Hamiltonian.The resulting (Euclidean) Lagrangian is nonlocal and describes bosons which do and fermions which do not propagate through the lattice. All fields couple to the external magnetic field. The Lagrangian contains Yukawa and Luttinger type interactions.     

Functional Feynman-Kac Equations for Limit Lognormal Multifractals

A novel technique of functional Feynman-Kac equations is developed for the probability distribution of the limit lognormal multifractal process introduced by Mandelbrot [in Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta, eds., Springer, New York (1972)] and constructed explicitly by Bacry, Delour, and Muzy [Phys. Rev. E 64:026103 (2001)]. The distribution of the process is known to be determined by the complicated stochastic dependence structure of its increments (SDSI). It is shown that the SDSI has two separate layers of complexity that can be captured in a precise way by a pair of functional Feynman-Kac equations for the Laplace transform. Exact solutions are obtained as power series expansions in the intermittency parameter using a novel intermittency differentiation rule. The expansion of the moments gives a new representation of the Selberg integral. The author wishes to express gratitude to the Mathematics Department of Lehigh University for generous support during his stay at Lehigh University, where this article was written.     

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.     

A simple picture for structural glasses

This paper is an essay on mosaic structures in glasses, and their possible role in relaxation phenomena. (a) Near T_g the fluid is assumed to contain clusters, slightly more compact than the matrix: they cannot grow in size because of frustration effects, as noted by Kivelson and others. (Mode/mode coupling theories cannot describe this, because they do not incorporate frustration.) The size of the clusters corresponds to the Boson peak wavelength. (b) We propose that the standard free volume picture may be transposed: (i) clusters move rather than molecules, (ii) the required cavity space (‘vacancy’) for cluster motion is not empty, but filled with the low density matrix. (The old criticism against free volume based on pressure effects is thus removed.) (c) To reach one cluster, a ‘vacancy’ must hop through many ‘traps’: the distribution of hopping times ultimately leads to a stretched exponential for the relaxation, as argued by Bendler and Schlesinger. To cite this article: P.-G. de Gennes, C. R. Physique 3 (2002) 1263–1268.     

Solution of Coulomb path integral in momentum space

The path integral for a point particle in a Coulomb potential is solved in momentum space. The solution permits us to answer for the first time an old question of quantum mechanics in curved spaces raised in 1957 by DeWitt: The Hamiltonian of a particle in a curved space must not contain an additional term proportional to the curvature scalar R, since this would change the level spacings in the hydrogen atom.     

A singularity theorem based on spatial averages

Inspired by Raychaudhuri’s work, and using the equation named after him as a basic ingredient, a new singularity theorem is proved. Open non-rotating Universes, expanding everywhere with a non-vanishing spatial average of the matter variables, show severe geodesic incompletness in the past. Another way of stating the result is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies. In memory of Amal Kumar Raychaudhuri (1924–2005).     

Superfield formulation of the phase space path integral

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.     

Feynman-Kac and feynman formulas for evolution pseudodifferential equations in superspace

In the paper, evolution pseudodifferential equations in the space of superanalytic functions (X) of an infinite-dimensional argument with symbols in the space (Y) of Fourier supertransforms of distributions on the dual superspace are considered. For these equations, the “weak” Cauchy problem is posed and the existence theorem for the solutions of this problem is proved. The main result of the paper is the theorem concerning the representation of solutions of the “weak” Cauchy problem by the Feynman path integral in the phase superspace (the Feynman-Kac formula). The Feynman integral is understood in the sequential sense. Thus, the Feynman formula becomes an immediate consequence of the Feynman-Kac formula.     

The differential equation for the Feynman-Kac formula with a Lebesgue-Stieltjes measure

We investigate the Feynman-Kac formula with a Lebesgue-Stieltjes measure, in which the time integration is performed with respect to a Borel measure and not just an ordinary lebesgue measure as in the classical case. We derive the corresponding differential equation, which reduces to the heat or the Schrödinger equation in the standard cases. Actually, we observe two distinct phenomena: first, the differential equation is governed by the continuous part of the measure: secondly, the solution undergoes a discontinuity at every point in the support of the discrete part of the measure. Alternately, we obtain a Volterra-Stieltjes integral equation. We conclude by finding an explicit expression for the solution and a certain product integral representation. These results and the methods of proof suggest possible physical interpretations and various mathematical developments.Dedicated to the Memory of Mark KacResearch partially supported by NSF Grant 8120790. The author was a Member of the Mathematical Sciences Research Institute of Berkeley while this work was carried out, and a visiting faculty member at the University of Nebraska at Lincoln when this paper was completed.     

Spectral decomposition of path space in solvable lattice model

We give thespectral decomposition of the path space of the vertex model with respect to the local energy functions. The result suggests the hidden Yangian module structure on the levell integrable modules, which is consistent with the earlier work [1] in the level one case. Also we prove the fermionic character formula of the levell integrable representations in consequence.     

Coherent nuclear resonant scattering of X-rays: Time and space picture

Hyperfine Interactions - The problem of coherent resonant scattering of X-rays by an ensemble of nuclei is solved directly in time and space. In a first step the problem with a single coherently...     

Feynman path integral and ordering rules on discrete finite space

The Feynman path integral is constructed for systems whose configuration space is a discrete finite set. The construction is based on the operator formulation of quantum mechanics on a finite discrete space. We derive connections between operators and introduce the analogue of the^*-multiplication for discrete symbols.     

A path integral method for data assimilation

Described here is a path integral, sampling-based approach for data assimilation, of sequential data and evolutionary models. Since it makes no assumptions on linearity in the dynamics, or on Gaussianity in the statistics, it permits consideration of very general estimation problems. The method can be used for such tasks as computing a smoother solution, parameter estimation, and data/model initialization.Speedup in the Monte Carlo sampling process is essential if the path integral method has any chance of being a viable estimator on moderately large problems. Here a variety of strategies are proposed and compared for their relative ability to improve the sampling efficiency of the resulting estimator. Provided as well are details useful for its implementation and testing.The method is applied to a problem in which standard methods are known to fail, an idealized flow/drifter problem, which has been used as a testbed for assimilation strategies involving Lagrangian data. It is in this kind of context that the method may prove to be a useful assimilation tool in oceanic studies.     

Avoiding traps in trajectory space: Metadynamics enhanced transition path sampling

We propose a transition path sampling (TPS) scheme designed to enhance sampling in systems with multiple reaction channels. In this method, based on a combination of the metadynamics algorithm with the TPS shooting move, a history dependent bias drives the simulation towards unexplored reaction channels. The bias, constructed as a superposition of repulsive Gaussian potentials deposited on the trajectories harvested in the course of the simulation, acts only during the initial stage of the trajectory generation, but leaves the dynamics along the trajectories unaffected such that the sampled pathways are true dynamical trajectories. Simulations carried out for two test systems indicate that the new approach effortlessly switches between distinct reaction channels even if they are separated by high barriers in trajectory space.     

Time vs. ensemble averages for nonstationary time series

We analyze whether sliding window time averages applied to stationary increment processes converge to a limit in probability. The question centers on averages, correlations, and densities constructed via time averages of the increment x(t,T)=x(t+T)−x(t), e.g. x(t,T)=ln(p(t+T)/p(t)) in finance and economics, where p(t) is a price, and the assumption is that the increment is distributed independently of t. We apply Tchebychev’s Theorem to the construction of statistical ensembles, and then show that the convergence in probability condition is not satisfied when applied to time averages of functions of stationary increments. We further show that Tchebychev’s Theorem provides the basis for constructing approximate ensemble averages and densities from a single, historic time series where, as in FX markets, the series shows a definite ‘statistical periodicity’. The convergence condition is not satisfied strongly enough for densities and certain averages, but is well-satisfied by specific averages of direct interest. Rates of convergence cannot be established independently of specific models, however. Our analysis shows how to decide which empirical averages to avoid, and which ones to construct.