Evaluation Of Path Integrals Research Articles

Path integral evaluation of the one-loop effective potential in field theory of diffusion-limited reactions

The well-established effective action and effective potential framework from the quantum field theory domain is adapted and successfully applied to classical field theories of the Doi and Peliti type for diffusion controlled reactions. Through a number of benchmark examples, we show that the direct path integral calculation of the effective potential in fixed space dimension d = 2 to one-loop order reduces to a small set of simple elementary functions, irrespective of the microscopic details of the specific model. Thus the technique, which allows one to obtain with little additional effort, the potentials for a wide variety of different models, represents an alternative to the standard model-dependent diagram-based calculations. The renormalized effective potential, effective equations of motion and the associated renormalization group equations are computed in d = 2 spatial dimensions for a number of single species field theories of increasing complexity.

Chern-Simons theory onS1-bundles: abelianisation and q-deformed Yang-Mills theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.

Moments of spectral functions: Monte Carlo evaluation and verification

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval . The algorithmic detail that leads to robust numerical approximations is the fact that the path-integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is nonlinear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.

Polymer density functional approach to efficient evaluation of path integrals

A polymer density functional theory (P-DFT) has been extended to the case of quantum statistics within the framework of Feynman path integrals. We start with the exact P-DFT formalism for an ideal open chain and adapt its efficient numerical solution to the case of a ring. We show that, similarly, the path integral problem can, in principle, be solved exactly by making use of the two-particle pair correlation function (2p-PCF) for the ends of an open polymer, half of the original. This way the exact data for one-dimensional quantum harmonic oscillator are reproduced in a wide range of temperatures. The exact solution is not, though, reachable in three dimensions (3D) because of a vast amount of storage required for 2p-PCF. In order to treat closed paths in 3D, we introduce a so-called "open ring" approximation which proves to be rather accurate in the limit of long chains. We also employ a simple self-consistent iteration so as to correctly account for the interparticle interactions. The algorithm is speeded up by taking convolutions with the aid of fast Fourier transforms. We apply this approximate path integral DFT (PI-DFT) method to systems within spherical symmetry: 3D harmonic oscillator, atoms of hydrogen and helium, and ions of He and Li. Our results compare rather well to the known data, while the computational effort (some seconds or minutes) is about 100 times less than that with Monte Carlo simulations. Moreover, the well-known "sign problem" is expected to be considerably reduced within the reported PI-DFT, since it allows for a direct estimate of the corresponding partition functions.

Closed Superstring Emission from Rolling Tachyon Backgrounds

We compute the lowest components of the Type II Ramond-Ramond boundary state for the tachyon profile $T (X) = \lambda e ^{X ^ 0/\sqrt{2}}$ by direct path integral evaluation. The calculation is made possible by noting that the integrals involved in the requisite disk one-point functions reduce to integrals over the product group manifold $U (n)\times U (m)$. We further note that one-point functions of more general closed string operators in this background can also be related to $U (n)\times U (m)$ group integrals. Using this boundary state, we compute the closed string emission from a decaying unstable D$p$-brane of Type II string theory. We also discuss closed string emission from the tachyon profile $T (X) =\lambda\cosh (X ^ 0/\sqrt{2})$. We find in both cases that the total number of particles produced diverges for $p = 0$, while the energy radiated into closed string modes diverges for $p\leq 2$, in precise analogy to the bosonic case.

Different time slices for different degrees of freedom in Feynman path integration

A general scheme is presented for using different numbers of ‘time slices’ for different degrees of freedom in a path integral evaluation of the Boltzmann operator for a large molecular system. This will be particularly useful, for example, in evaluating the ‘quantum instanton’ rate constant [cf. W.H. Miller, Y. Zhao, M. Ceotto, S. Yang. J. Chem. Phys., 119, 1329 (2003)] for H atom transfer reactions, or any applications involving atoms with largely differing masses.

Path integral evaluation of the quantum instanton rate constant for proton transfer in a polar solvent

The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST's, namely, the centroid-density and Hansen-Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz-Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of approximately 2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.

On the efficient path integral evaluation of thermal rate constants within the quantum instanton approximation.

We present an efficient path integral approach for evaluating thermal rate constants within the quantum instanton (QI) approximation that was recently introduced to overcome the quantitative deficiencies of the earlier semiclassical instanton approach [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)]. Since the QI rate constant is determined solely by properties of the (quantum) Boltzmann operator (specifically, by the zero time properties of the flux-flux and delta-delta correlation functions), it can be evaluated by well-established techniques of imaginary time path integrals even for quite complex chemical reactions. Here we present a series of statistical estimators for relevant quantities which can be evaluated straightforwardly with any nonlinear reaction coordinates and general Hamiltonians in Cartesian space. To facilitate the search for the optimal dividing surfaces required by the QI approximation, we introduce a two-dimensional quantum free energy surface associated with the delta-delta correlation function and describe how an adaptive umbrella sampling can be used effectively to construct such a free energy surface. The overall computational procedure is illustrated by the application to a hydrogen exchange reaction in gas phase, which shows excellent agreement of the QI rates with those obtained from quantum scattering calculations.

Path integrals evaluation in two-dimensional de Sitter space

The propagator in de Sitter space is calculated based on the path integrals. The method of evaluation of path integrals for particles with spin is proposed. The calculations are compared with the quantum mechanical ones.

A PATH INTEGRAL APPROACH TO DERIVATIVE SECURITY PRICING II: NUMERICAL METHODS

We discuss two numerical techniques, based on the path integral approach described in a previous paper, for solving the stochastic equations underlying the financial markets: the path integral Monte Carlo, and the path integral deterministic evaluation. In particular, we apply the latter to some specific financial problems: the pricing of a European option, a zero-coupon bond, a caplet, an American option, and a Bermudan swaption.

Path integral centroid molecular-dynamics evaluation of vibrational energy relaxation in condensed phase

We present a method based on centroid molecular dynamics (CMD) to calculate nonlinear quantum force correlation functions important in the golden rule approach for studying vibrational energy relaxation (VER) in condensed phases. We consider a model of a diatomic molecule in a two-dimensional neon liquid and also a diatomic coupled to a small Helium cluster. The predictions of the theory for the neon bath are compared and found in close agreement with available theories for VER based on the Egelstaff correction factor and Feynman–Kleinert variational theory. For the Helium cluster, the force spectrum obtained from CMD is found to be in slightly better agreement with the exact result than a method based on a cumulant approach. The results support the use of CMD in condensed phase studies of VER when quantum effects are important.

Path integral representation for Wilson loops and the non-Abelian Stokes theorem

We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in $\mathrm{SU}(N)$ gauge theory and 4-dimensional Euclidean space-time by using well-known properties of group characters. A discretized form of the path integral is naturally provided by the properties of group characters and does not need any artificial regularization. We show that the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov [Phys. Lett. B 224 131 (1989)] by using a special regularization is erroneous and predicts zero for the Wilson loop. This property is obtained by direct evaluation of path integrals for Wilson loops defined for pure $\mathrm{SU}(2)$ gauge fields and $Z(2)$ center vortices with spatial azimuthal symmetry. Further we show that both derivations given by Diakonov and Petrov for their regularized path integral, if done correctly, predict also zero for Wilson loops. Therefore, the application of their path integral representation of Wilson loops cannot give a new way to check confinement in lattice as has been declared by Diakonov and Petrov [Phys. Lett. B 242 425 (1990)]. From the path integral representation which we consider we conclude that no new non-Abelian Stokes theorem can exist for Wilson loops except the old-fashioned one derived by means of the path-ordering procedure.

Numerical evaluation of coherent-state path integrals in quantum dynamics

The numerical evaluation of coherent-state path integrals for quantum dynamical problems is discussed for one-dimensional examples. To propagate an initial state, we use the normal and antinormal ordered coherent-state path integrals combined with a split-operator technique dividing the Hamiltonian into harmonic and anharmonic parts. For numerical purposes integrations must be approximated by quadrature formulae. This leads to a matrix multiplication scheme which is systematically tested for the double-well and Morse potentials. The method is accurate for propagation times much longer than the natural time scale of the system, and it allows for short as well as long time steps without loss of stability.

The Electron Localization Problem Via a Local Functional Integral Approach

We employ a path integral representation for the three-dimensional Schrödinger equation describing the motion of a quantum particle in a static random potential. Within a semi-classical approximation (pure phase wave function) and a one-loop order path integral evaluation, we obtain three-dimensional electron localization in the presence of static disorder and zero temperature.

Corrections to the semiclassical approximation by numerical evaluation of real time path integrals in complex space

A numerical method for evaluating real time path integrals by analytical continuation of the coordinate space variables into the complex plane is presented. With this method, a Monte Carlo gaussian sampling function around the classical paths is obtained. The function to be sampled is smooth near the stationary path and good Monte Carlo convergence is expected.

Evaluation of coherent-state path integrals in statistical mechanics by matrix multiplication

The numerical evaluation of coherent-state path-integral representations for partition functions and other quantities in equilibrium quantum statistical mechanics is discussed. Several coherent-state path-integral schemes are introduced, which differ from each other by the order of approximation and by the operator ordering employed in the high-temperature approximation of the density operator. Simple one-dimensional systems are used to test these schemes. For the harmonic oscillator, finite-dimensional approximations to the coherent-state path integral are calculated analytically and compared to each other and to the real-space path integral. For anharmonic systems, integrations must be approximated by quadrature formulas. This leads to a matrix multiplication scheme which is tested for the double-well potential. The results obtained are accurate from zero temperature way up into the high-temperature regime where quantum effects become negligible. This is a significant advantage over traditional real-space path integral schemes which break down at low temperatures.

Path integral of the hydrogen atom, Jacobi's principle of least action and one-dimensional quantum gravity

A path integral evaluation of the Green's function for the hydrogen atom initiated by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville type. The basic dynamical principle involved is identified as Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. The separation of variables in the operator formalism corresponds to a choice of gauge in the path integral, and the Green's function is shown to be gauge independent if the operator ordering is properly taken into account. Unlike the conventional Feynman path integral, which deals with a space-time picture of particle motion, the path integral on the basis of Jacobi's principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates, and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation, the Hamiltonian for the Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from the view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem with an evaluation of the Weyl anomaly is presented.

Method for approximate evaluation of path integrals using perturbation theory with convergent series. II. Euclidean quantum field theory

The proof is given of the method suggested in the previous paper for an approximate evaluation of path integrals in Hilbert space where the Gaussian measure is defined by a kernel operator.

Method of approximate evaluation of path integrals using perturbation theory with convergent series. I

We propose a method of evaluating path integrals such that the integral is approximated with any accuracy by summations of a finite number of terms of an absolutely convergent series.

On localization and regularization

Different regularizations are studied in localization of path integrals. We discuss the effect of the choice of regularization by evaluating the partition functions for the harmonic oscillator and the Weyl character for SU(2). In particular, we solve the Weyl shift problem that arises in path integral evaluation of the Weyl character by using the Atiyah–Patodi–Singer η-invariant and the Borel–Weil theory.