F. Langouche

Functional integration and semiclassical expansions

I: Functional integrals defined as limits of discretized expressions.- II: Correspondence rules and functional integral representations.- III: Functional integral representations of expectation values. Time-ordered products.- IV: Perturbation expansions.- V: Short time propagators and the relations between them.- VI: Covariant definitions of functional integrals.- VII: Functional integral methods in Fokker-Planck dynamics.- VIII: Product integrals.- IX: The semiclassical expansion in phase space.- X: The semiclassical expansion in configuration space.- XI: Other approaches.- XII: Computation of the propagator on the sphere S3.- References.

Functional integral methods for stochastic fields

We study a method proposed recently that gives a unified view of the properties of stochastic fields determined by Langevin-type equations. The functional integrals involved are shown to need an additional prescription to be defined, this prescription being related to the relative order of noncommuting operators in the corresponding operator formalism which we characterize in detail. It is proved that the prescription is responsible for an additional term which has given rise to difficulties of interpretation in the Onsager-Machlup path probability density. We prove that all results are independent of this term. In particular in perturbation theory the cancellation mechanism of the prescription dependence is studied in detail.

SHORT DERIVATION OF FEYNMAN LAGRANGIAN FOR GENERAL DIFFUSION PROCESSES

A short derivation of the Feynman Lagrangian for general diffusion processes is given by a technique relying on the use of different discretisations which are related by equivalence relations under the n-dimensional integral whose limit is the path integral. In this way calculation of the differential equation satisfied by the path integral is avoided.

Steepest descent approximation for the Fokker-Planck equation

The steepest descent approximation is used in the functional integral representation of the solution of the Fokker-Planck equation. The calculation includes the Gaussian integrals of fluctuations around the classical solution. The problem of the prescription dependence of the functional integral is solved previously by transforming the representation to one involving a prescription independent functional integral.

WKB-type expansion for Langevin equations

A systematic WKB-type expansion in a parameter measuring the strength of the fluctuations is performed for the transition probability density of the Markovian processes generated by Langevin equations. We use functional integral techniques and treat the general case in which the diffusion matrix determines a Riemannian space.

WKB expansions with higher order corrections: The general case

Abstract Using the properties of the discretization techniques for functional integrals that we have developed before, we compute explicitly the higher-order corrections to the WKB approximation of the quantum system correspondingto the curved-space classical lagrangian L c 1 = 1 2 g μν q μ q ν +A μ q μ−V .

On the Most Probable Path for Diffusion Processes

The Lagrangian which determines the most probable path for diffusion processes is obtained by a simple and intuitive technique based entirely on the use of path integrals.

Master equation approach to the dynamical critical behaviour of the Schlögl model

Abstract Starting from the multivariate master equation we construct a reduced model that gives the critical theory. The relation to the static problem is studied using the minimal subtraction technique and the critical exponents are calculated up to O(ϵ 2 ).

Semi-classical approximation for a class of quantum dynamical semigroups

We determine the complex semi-classical approximation to the propagator for a class of quantum dynamical semigroups and study a simple model. The method uses functional integral techniques developed earlier.

Wkb-type Expansion for Langevin-equations .2. Covariant Expansion

Abstract We show how to compute in a covariant way the WKB-type expansion for the transition probability density of the Markovian processes generated by general multiplicative noise Langevin equations. The method uses phase space functional integrals and normal coordinates around the classical path. We compare with our earlier non-covariant methods and compute explicitly the first order correction to the WKB-approximation.