Regularized Path Integral Research Articles

Configuration space for quantum gravity in a locally regularized path integral

We discuss some aspects of the metric configuration space in quantum gravity in the background field formalism. We give a necessary and sufficient condition for the parametrization of Euclidean metric fluctuations such that (i) the signature of the metric is preserved in all configurations that enter the gravitational path integral, and (ii) the parametrization provides a bijective map between full Euclidean metrics and metric fluctuations about a fixed background. For the case of foliatable manifolds, we show how to parametrize fluctuations in order to preserve foliatability of all configurations. Moreover, we show explicitly that preserving the signature on the configuration space for the Lorentzian quantum gravitational path integral is most conveniently achieved by inequality constraints. We discuss the implementation of these inequality constraints in a nonperturbative renormalization group setup.

On Entropy Regularized Path Integral Control for Trajectory Optimization.

In this article, we present a generalized view on Path Integral Control (PIC) methods. PIC refers to a particular class of policy search methods that are closely tied to the setting of Linearly Solvable Optimal Control (LSOC), a restricted subclass of nonlinear Stochastic Optimal Control (SOC) problems. This class is unique in the sense that it can be solved explicitly yielding a formal optimal state trajectory distribution. In this contribution, we first review the PIC theory and discuss related algorithms tailored to policy search in general. We are able to identify a generic design strategy that relies on the existence of an optimal state trajectory distribution and finds a parametric policy by minimizing the cross-entropy between the optimal and a state trajectory distribution parametrized by a parametric stochastic policy. Inspired by this observation, we then aim to formulate a SOC problem that shares traits with the LSOC setting yet that covers a less restrictive class of problem formulations. We refer to this SOC problem as Entropy Regularized Trajectory Optimization. The problem is closely related to the Entropy Regularized Stochastic Optimal Control setting which is often addressed lately by the Reinforcement Learning (RL) community. We analyze the theoretical convergence behavior of the theoretical state trajectory distribution sequence and draw connections with stochastic search methods tailored to classic optimization problems. Finally we derive explicit updates and compare the implied Entropy Regularized PIC with earlier work in the context of both PIC and RL for derivative-free trajectory optimization.

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

A fully regulated definition of Feynman’s path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowski space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier integral operators. Physically, we are replacing the time-evolution semi-group by a holomorphic family of operators such that the corresponding path integrals are well defined in some half space of C. The regularized path integral is, thus, defined through analytic continuation. This regularization can be performed by means of stationary phase approximation or computed analytically depending only on the Hamiltonian and the observable (i.e., known a priori). In either case, the computational effort to evaluate path integrals or expectations of observables reduces to the evaluation of integrals over spheres. Furthermore, computations can be performed directly in the continuum and applications (analytic computations and their implementations) to a number of models including the non-trivial cases of the massive Schwinger model and a φ4 theory.

Regularized path integrals and anomalies: U(1) chiral gauge theory

We analyze the origin of the Adler–Bell–Jackiw anomaly of chiral U(1) gauge theory within the framework of regularized path integrals. Momentum or position space regulators allow for mathematically well-defined path integrals but violate local gauge symmetry. It is known how (nonanomalous) gauge symmetry can be recovered in the renormalized theory in this case [Kopper, C. and Müller, V. F., “Renormalization of spontaneously broken SU(2) Yang-Mills theory with flow equations,” Rev. Math. Phys. 21, 781 (2009)]10.1142/S0129055X0900375X. Here we analyze U(1) chiral gauge theory to show how the appearance of anomalies manifests itself in such a context. We show that the three-photon amplitude leads to a violation of the Slavnov-Taylor identities which cannot be restored on taking the UV limit in the renormalized theory. We point out that this fact is related to the nonanalyticity of this amplitude in the infrared region.

Quantum free-energy differences from nonequilibrium path integrals. II. Convergence properties for the harmonic oscillator

Nonequilibrium path-integral methods for computing quantum free-energy differences are applied to a quantum particle trapped in a harmonic well of uniformly changing strength with the purpose of establishing the convergence properties of the work distribution and free energy as the number of degrees of freedom M in the regularized path integrals goes to infinity. The work distribution is found to converge when M tends to infinity regardless of the switching speed, leading to finite results for the free-energy difference when the Jarzynski nonequilibrium work relation or the Crooks fluctuation relation are used. The nature of the convergence depends on the regularization method. For the Fourier method, the convergence of the free-energy difference and work distribution go as 1/M , while both quantities converge as 1/M(2) when the bead regularization procedure is used. The implications of these results to more general systems are discussed.

Nonperturbative lorentzian path integral for gravity

We construct a well-defined regularized path integral for Lorentzian quantum gravity in terms of dynamically triangulated causal space-times. Each Lorentzian geometry and its action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time and, for finite lattice volume, the associated transfer matrix is self-adjoint, bounded, and strictly positive. The degenerate geometric phases found in dynamically triangulated Euclidean gravity are not present.

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.