On the efficient Monte Carlo implementation of path integrals

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.

Models for local ohmic quantum dissipation

We construct model master equations for local quantum dissipation. The master equations are in the form of Lindblad generators, with imposed constraints that the dissipations be strictly linear (i.e. ohmic), isotropic and translationally invariant. A particular form for is chosen to satisfy the constraints. The resulting master equations are given in both the Schr\"odinger and Heisenberg forms. We obtain fluctuation-dissipation relations, and discuss the relaxation of average kinetic energy to effective thermal equilibrium values. We compare our results to the Dekker and the Caldeira-Leggett master equations. These master equations allow a more general approach to quantum dissipation and the dynamics of quantum coherence to account for the nontrivial system-environment coupling in a local environment.Comment: 19 pages, REVTEX, PSU/TH/12

Quantum Logic with a Single Trapped Electron

We propose the use of a trapped electron to implement quantum logic operations. The fundamental controlled-NOT gate is shown to be feasible. The two quantum bits are stored in the internal and external (motional) degrees of freedom.Comment: 7 Pages, REVTeX, No Figures, To appear in Phys. Rev.

Proper time and Minkowski structure on causal graphs

For causal graphs we propose a definition of proper time which for small scales is based on the concept of volume, while for large scales the usual definition of length is applied. The scale where the change from "volume" to "length" occurs is related to the size of a dynamical clock and defines a natural cut-off for this type of clock. By changing the cut-off volume we may probe the geometry of the causal graph on different scales and therey define a continuum limit. This provides an alternative to the standard coarse graining procedures. For regular causal lattice (like e.g. the 2-dim. light-cone lattice) this concept can be proven to lead to a Minkowski structure. An illustrative example of this approach is provided by the breather solutions of the Sine-Gordon model on a 2-dimensional light-cone lattice.Comment: 15 pages, 4 figure

Upon the existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods

In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is ``experimental'', I only provide a ``physicist's'' proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in the Appendix, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present article is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.Comment: 19 pages, 4 figures; the discrete short-time approximations are now treated as independent from their continuous version; new examples of discrete short-time approximations of order three and four are given; a new appendix containing a short review on Brownian motion has been added; also, some additional explanations are provided here and there; this is the last version; to appear in Phys. Rev.

Brownian Motion and Polymer Statistics on Certain Curved Manifolds

We have calculated the probability distribution function G(R,L|R',0) of the end-to-end vector R-R' and the mean-square end-to-end distance (R-R')^2 of a Gaussian polymer chain embedded on a sphere S^(D-1) in D dimensions and on a cylinder, a cone and a curved torus in 3-D. We showed that: surface curvature induces a geometrical localization area; at short length the polymer is locally "flat" and (R-R')^2 = L l in all cases; at large scales, (R-R')^2 is constant for the sphere, it is linear in L for the cylinder and reaches different constant values for the torus. The cone vertex induces (function of opening angle and R') contraction of the chain for all lengths. Explicit crossover formulas are derived.Comment: 9 pages, 4 figures, RevTex, uses amssymb.sty and multicol.sty, to appear in Phys. Rev

Polaron Variational Methods In The Particle Representation Of Field Theory : I. General Formalism

We apply nonperturbative variational techniques to a relativistic scalar field theory in which heavy bosons (``nucleons'') interact with light scalar mesons via a Yukawa coupling. Integrating out the meson field and neglecting the nucleon vacuum polarization one obtains an effective action in terms of the heavy particle coordinates which is nonlocal in the proper time. As in Feynman's polaron approach we approximate this action by a retarded quadratic action whose parameters are to be determined variationally on the pole of the two-point function. Several ans\"atze for the retardation function are studied and for the most general case we derive a system of coupled variational equations. An approximate analytic solution displays the instability of the system for coupling constants beyond a critical value.Comment: 33 pages standard LaTeX, 3 uuencoded gzipped postscript figures embedded with psfig.st

Dynamical Vortices in Superfluid Films

The coupling of vortices to phonons in a superfluid is a gauge coupling dictated by topology. The density and current response to a moving vortex are computed and contrasted with the standard backflow picture. Exploiting the analogy to (2+1)-dimensional electrodynamics, we compute the effective vortex mass $M(\omega)$ and find it to be logarithmically divergent in the low frequency limit, leading to a super-Ohmic dissipation in response to an oscillating superflow. Numerical integration of the nonlinear Schroedinger equation supports these conclusions. Interaction of vortices and impurities is also discussed.Comment: 13 pages, 6 figure

Metric Fluctuation Corrections to Hawking Radiation

We study how fluctuations of the black hole geometry affect the properties of Hawking radiation. Even though we treat the fluctuations classically, we believe that the results so obtained indicate what might be the effects induced by quantum fluctuations in a self consistent treatment. To characterize the fluctuations, we use the model introduced by York in which they are described by an advanced Vaidya metric with a fluctuating mass. Under the assumption of spherical symmetry, we solve the equation of null outgoing rays. Then, by neglecting the greybody factor, we calculate the late time corrections to the s-wave contributions of the energy flux and the asymptotic spectrum. We find three kind of modifications. Firstly, the energy flux fluctuates around its average value with amplitudes and frequencies determined by those of the metric fluctuations. Secondly, this average value receives two positive contributions one of which can be reinterpreted as due to the `renormalisation' of the surface gravity induced by the metric fluctuations. Finally, the asymptotic spectrum is modified by the addition of terms containing thermal factors in which the frequency of the metric fluctuations acts as a chemical potential.Comment: 27 pages, 2 figures, LaTeX. Revised versio