Path Integral Framework Research Articles

A path integral approach to random motion with nonlinear friction

Using a path integral approach, we derive an analytical solution of a nonlinear and singular Langevin equation, which has been introduced previously by P-G de Gennes as a simple phenomenological model for the stick-slip motion of a solid object on a vibrating horizontal surface. We show that the optimal (or most probable) paths of this model can be divided into two classes of paths, which correspond physically to a sliding or slip motion, where the object moves with a non-zero velocity over the underlying surface, and a stick-slip motion, where the object is stuck to the surface for a finite time. These two kinds of basic motions underlie the behavior of many more complicated systems with solid/solid friction and appear naturally in de Gennes' model in the path-integral framework.

Ice limit of Coulomb gauge Yang-Mills theory

In this paper we describe gauge invariant multiquark states generalizing the path integral framework developed by Parrinello, Jona-Lasinio, and Zwanziger to amend the Faddeev-Popov approach. This allows us to produce states such that, in a limit which we call the ice limit, fermions are dressed with glue exclusively from the fundamental modular region associated with Coulomb gauge. The limit can be taken analytically without difficulties, avoiding the Gribov problem. This is illustrated by an unambiguous construction of gauge invariant mesonic states for which we simulate the static quark-antiquark potential.

Worldline path integrals for a Dirac particle in a weak gravitational plane wave

The problem of a relativistic spinning particle interacting with a weak gravitational plane wave in (3+1) dimensions is formulated in the frame work of covariant supersymmetric path integrals. The relative Green function is expressed through a functional integral over bosonic trajectories that describe the external motion and fermionic variables that describe the spin degrees of freedom. The (3+1) dimensional problem is reduced to the (1+1) dimensional one by using an identity. Next, the relative propagator is exactly calculated and the wave functions are extracted.

Covariant path integral for the Dirac equation with pseudoscalar potentials

The problem of a relativistic spinning particle interacting with pseudoscalar potentials in (3 + 1) dimensions is formulated in the framework of covariant supersymmetric path integrals. The relative Green's function is expressed through a functional integral over bosonic trajectories that describe the external motion and fermionic variables that describe the spin degrees of freedom. As an application, we have considered the case of the plane wave, where the pseudoscalar potential is an arbitrary function of the variable (kμxμ). The (3 + 1)-dimensional problem is reduced to a (1 + 1)-dimensional one by using an identity. For the case of k2 = 0, the relative propagator is exactly calculated and the wavefunctions are extracted.

The multilevel pairing Hamiltonian versus the degenerate case

We study the pairing Hamiltonian in a set of non-degenerate levels. First, we review in the path integral framework the spontaneous breaking of the U(1) symmetry occurring in such a system for the degenerate situation. Then the behaviors with the coupling constant of the ground state energy in the multilevel and in the degenerate case are compared. Next we discuss, in the multilevel case, an exact strong coupling expansion for the ground state energy which introduces the moments of the single particle level distribution. The domain of validity of the expansion, which is known in the macroscopic limit, is explored for finite systems and its implications for the energy of the latter is discussed. Finally the seniority and Gaudin excitations of the pairing Hamiltonian are addressed and shown to display the same gap in leading order.

THE EXPONENT EXPANSION: AN EFFECTIVE APPROXIMATION OF TRANSITION PROBABILITIES OF DIFFUSION PROCESSES AND PRICING KERNELS OF FINANCIAL DERIVATIVES

A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. Several examples are presented, and the application of these results to increase the efficiency of numerical approaches to derivative pricing is discussed.

Quantifying the Kinetic Paths of Flexible Biomolecular Recognition

Biomolecular recognition often involves large conformational changes, sometimes even local unfolding. The identification of kinetic pathways has become a central issue in understanding the nature of binding. A new approach is proposed here to study the dynamics of this binding-folding process through the establishment of a path-integral framework on the underlying energy landscape. The dominant kinetic paths of binding and folding can be determined and quantified. The significant coupling between the binding and folding of biomolecules often exists in many important cellular processes. In this case, the corresponding kinetic paths of binding are shown to be intimately correlated with those of folding and the dynamics becomes quite cooperative. This implies that binding and folding happen concurrently. When the coupling between binding and folding is weak (strong), the kinetic process usually starts with significant folding (binding) first, with the binding (folding) later proceeding to the end. The kinetic rate can be obtained through the contributions from the dominant paths. The rate is shown to have a bell-shaped dependence on temperature in the concentration-saturated regime consistent with experiment. The changes of the kinetics that occur upon changing the parameters of the underlying binding-folding energy landscape are studied.

Dominant Kinetic Paths on Biomolecular Binding-Folding Energy Landscape

The identification of kinetic pathways is a central issue in understanding the nature of flexible binding. A new approach is proposed here to study the dynamics of this binding-folding process through the establishment of a path integral framework on the underlying energy landscape. The dominant kinetic paths of binding and folding can be determined and quantified. In this case, the corresponding kinetic paths of binding are shown to be intimately correlated with those of folding and the dynamics becomes quite cooperative. The kinetic time can be obtained through the contributions from the dominant paths and has a U-shape dependence on temperature.

SOME NON-PERTURBATIVE PATH INTEGRALS STUDIES IN STATISTICAL BURGER TURBULENCE: STRINGS AND STOCHASTIC INSTANTONS CONFIGURATIONS

In this paper, we study Burger turbulence and the appearance of spatial coherent string dynamics structures for the statistical averaged vortex phase factor in a path integral framework. This result gives a strong mathematical method support to the unsolved long-standing Landau 1941's conjecture that in a turbulent field the flow vorticity should be confined to a limited region.Additionally, we present the exact instanton path-integral solubility — from a stochastic point of view — of a fluid flow bounded by a moving string (random) boundary together with a study of another exact path-integral solubility of a weakly stirred complete Navier–Stokes flow.

Dual linearized gravity in arbitrary dimensions

We construct a dual formulation of linearized gravity in a first-order tetrad formalism in arbitrary dimensions within the path integral framework following the standard duality algorithm making use of the global shift symmetry of the tetrad field. The dual partition function is in terms of the (mixed symmetric) tensor field in a frame-like formulation. We obtain in d dimensions the dual Lagrangian in a closed form in terms of the field strength of the dual frame-like field. Next, by coupling a source with the (linear) Riemann tensor in d dimensions, a dual generating functional is obtained. Using this an operator mapping between (linear) Riemann tensor and Riemann tensor corresponding to the dual field is derived and we also discuss the exchange of equations of motion and Bianchi identity.

Dirac particle in the presence of a plane waveand constant magnetic fields: path integral approach

The Green function (GF) related to the problem of a Dirac particle interacting with a plane wave and constant magnetic fields is calculated in the framework of path integral via Alexandrou et al. formalism according to the so-called global projection. As a tool of calculation, we introduce two identities (constraints) into this formalism, their main role is the reduction of integrals dimension and the emergence in a natural way of some classical paths, and due to the existence of constant electromagnetic field, we have used the technique of fluctuations. Hence the calculation of the (GF) is reduced to a known gaussian integral plus a contribution of the effective classical action.

Quantifying Kinetic Paths of Protein Folding

We propose a new approach to activated protein folding dynamics via a diffusive path integral framework. The important issues of kinetic paths in this situation can be directly addressed. This leads to the identification of the kinetic paths of the activated folding process, and provides a direct tool and language for the theoretical and experimental community to understand the problem better. The kinetic paths giving the dominant contributions to the long-time folding activation dynamics can be quantitatively determined. These are shown to be the instanton paths. The contributions of these instanton paths to the kinetics lead to the “bell-like” shape folding rate dependence on temperature, which is in good agreement with folding kinetic experiments and simulations. The connections to other approaches as well as the experiments of the protein folding kinetics are discussed.

Noncommutative quantum electrodynamics in path integral framework

In this paper, the dynamics of a relativistic particle of spin 1/2, interacting with an external electromagnetic field in noncommutative space, is studied in the path integral framework. By adopting the Fradkin–Gitman formulation, the exact Green's function in noncommutative space (NCGF) for the quadratic case of a constant electromagnetic field is computed, and it is shown that its form is similar to its counterpart given in commutative space. In addition, it is deduced that the effect of noncommutativity has the same effect as an additional constant field depending on a noncommutative θ matrix.

Decoupling quantum dissipation interaction via stochastic fields.

Based on the Hubbard-Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.

Particle on a conical surface

The motion of a particle on the conical surface interacting with a scalar and a vector potential is studied in the path integral framework following the technique of constraints. The propagators are evaluated taking into account the problem of the angle periodicity. The result is given in a compact form.

Neural network learning dynamics in a path integral framework

A path-integral formalism is proposed for studying the dynamical evolution in time of patterns in an artificial neural network in the presence of noise. An effective cost function is constructed which determines the unique global minimum of the neural network system. The perturbative method discussed also provides a way for determining the storage capacity of the network.

Nonlinear σ model treatment of quantum antiferromagnets in a magnetic field

The dynamics of relativistic particles of spin 0 and 1/2, interacting with an external electromagnetic field and a quantized plane wave, is studied using the path integral framework. We take advantage of the existing properties of the interaction to introduce a delta functional in order to calculate Green's functions. This simply reduces the problem to two coupled oscillators. The energy spectrum and wave functions are calculated for the spin 0 case and the analogy with Jaynes-Cummings model is made to derive the energy spectrum for the spin 1/2 case.

Path integral treatment for relativistic particles in external fields and quantized plane wave

AbstractThe dynamics of relativistic particles of spin 0 and 1/2, interacting with an external electromagnetic field and a quantized plane wave, is studied using the path integral framework. We take advantage of the existing properties of the interaction to introduce a delta functional in order to calculate Green's functions. This simply reduces the problem to two coupled oscillators. The energy spectrum and wave functions are calculated for the spin 0 case and the analogy with Jaynes‐Cummings model is made to derive the energy spectrum for the spin 1/2 case.

Single-particle treatment of quantum stochastic resonance

We first consider a test particle subjected to the simultaneous influence of a double-well mean field, thermal velocity distribution, external white noise and sinusoidal modulation. Within a path-integral framework the resulting power spectrum along with the signal-to-noise ratio (SNR) are expressed perturbatively in terms of the unperturbed eigenfunctions ψ n and eigenvalues E n , permitting us to demonstrate the existence of the phenomenon of quantum stochastic resonance (SR) in the dissipationless case. Next, we show that with the help of suitable transformations (viz., a complex Fourier frequency or a scaled-up potential) systems with weak friction or strong damping can be approximately mapped onto the dissipationless case. This yields a convenient method for studying frictional SR because the ψ n 's and E n 's can still be generated through a short-time propagator. Our formulation is illustrated numerically by displaying the variation of the SNR with the external noise strength and comparing the results vis-á-vis those obtained from classical stochastic simulation and quantum bath models. The usefulness of the single-particle approach to analyze the quantum SR phenomenon is emphasized.

An application of the self-consistent variational effective potential against the path-integral to compute equilibrium properties of quantum simple fluids

The thermodynamic and structural features obtained from the self-consistent variational effective potential SCP (developed by Feynman and Kleinert and by Giachetti and Tognetti) in the study of quantum fluids are analysed, taking the full path-integral (PI) treatment as a reference and helium-4 as a probe. Within the SCP method the pair radial distribution functions are of two types: one describing the correlations between the centroids of the quantum particles, and the other describing the true correlations between quantum particles. An analytical derivation states that the SCP picture for quantum particles does not distinguish between the instantaneous and the linear response pair radial distribution functions, which otherwise can be defined within a full path-integral framework by considering equal imaginary time or mixed imaginary time averages, respectively. It is also shown how the centroid and true quantum particle pair radial distribution functions are connected through a convolution. Explicit formulae for both the isotropic effective potential (ISCP) and the more general ani-sotropic effective potential (ASCP) pair radial distribution functions are given, and the possibilities of computing SCP static structure factors are discussed. Monte Carlo simulations of dense fluid Lennard-Jones helium-4 have been performed with ISCP and PI (PIMC) to determine the proximity between both methods, and to avoid spurious singularities in ISCP, use of a fourth-order Taylor expansion is made (TISCP). Within the conventional Ornstein-Zernike framework two evaluations of the structure factor are carried out. Wherever possible results are compared with experiment and data reported in the literature, and reliable limits of applicability for TISCP are reported.