D.C. Khandekar

Path integration of a two-time quadratic action

Path integration of a general two-time quadratic action characterising memory effects is performed within the framework of Feynmans polygonal path approach. Explicit evaluation of the propagator in exact analytical form is further carried out for the specific kernel used by Feynman in the polaron problem.

On a path integral with a topological constraint

We discuss a new method to evaluate a path integral with a topological constraint involving a point singularity in a plane. The path integration is performed explicitly in the universal covering space. Our method is an alternative to an earlier method of Inomata.

Path integration of a quadratic action with a generalized memory

Abstract Path integration of an action representing a harmonic oscillator with a generalized memory is carried out within the framework of Feynmans polygonal approach. The exact propagator obtained is in the form of an exponential integral over a single variable. Closed analytical results are available for special cases of the memory function.

Exact analysis of the critical behaviour in the driven Dicke model of interacting atoms

A system of N identical two-level atoms occupying the same site (the Dicke model) interacting with one another and driven by a coherent field is considered. From the known exact steady-state atomic density operator, expressions for the atomic observables, their fluctuations and correlation functions are constructed. Asymptotic analysis is used to obtain the behaviour of these atomic variables with respect to the parameters of the driving field in the thermodynamic limit N → ∞. It is shown that the interplay between the parameters of the interatomic interactions and the driving field leads to a critical behaviour as N → ∞. The conditions under which the atomic system shows either a first order or a second order phase transition with respect to the field variables are discussed.

Exact propagator for a quadratic lagrangian

Abstract An exact propagator for a damped oscillator with a time-dependent frequency and a perturbative force is obtained. This illustrates a new method of determining the classical action S based on a simple relation between S and the generating function of a canonical transformation involving an explicitly time-dependent invariant.

On the eigenvalue problem associated with Feynman path integration

Abstract When the lagrangian is not explicit function of time, the Nth approximation to the propagator may be viewed as the Nth power of anitary operator — the infinitesimal time propagator. We solve the eigenvalue problem associated with the operator for some special cases. In the limit of large N the eigenfunctions are shown to be identical to those of the finite time propagator. We also present an elementary method to evaluate the propagator corresponding to an action function encountered in the study of electron gas in a random potential. The evaluation of this propagator within Feynmans polygonal approach was not possible until recently.

Exact analysis of the fluorescent Dicke model in the thermodynamic limit

A system of N identical two-level atoms occupying the same site (Dicke model) driven by a coherent field is considered. From the known exact steady state atomic density operator, exact closed form expressions for the atomic observables, their fluctuations and correlation functions are constructed. Asymptotic analysis is used to obtain the behaviour of these atomic variables with respect to the driven field parameters in the thermodynamic limit N → ∞.

Path integration of a generalized quadratic action

Abstract Path integration of a class of generalized quadratic actions first proposed by Feynman is performed within the framework of Feynmans polygonal path approach. The exact propagator has the form of a free particle propagator with an “effective mass” apart from the normalization factor. Relation between the propagator and the usual Van Vleck-Pauli formula is discussed.

Stochastic area distributions: optimal trajectories, Maslov indices and asymptotic results

In this paper we study the semi-classical approximation for the distribution of area associated with (i) planar polymer rings constrained to enclose a fixed algebraic area and (ii) planar rings subject to an external electric field and constrained to enclose a fixed algebraic area. We demonstrate that the results are accurate in the asymptotic regime. Moreover, we also show that in case (i) it is possible to reconstruct the exact expression for the distribution, provided the contributions from all optimal trajectories are taken into account, as well as the proper Maslov indices.

Energy deposition of fast α particles in a fully ionized deuterium-tritium plasma

Abstract An α particle transport equation has been solved analytically in one-dimensional spherical geometry when the plasma properties are uniform in space and time. The results agree well with earlier results.