Path Integral Approach to Fluctuations in Dynamic Processes

A Path Integral (PI) formulation of linear elastostatics is first presented. For this, Navier equations are modified by adding a fictitious 'time' derivative of displacements so that equilibrium corresponds to the steady state of the resulting diffusion-like equations. The evolution of displacement is then represented as the propagation, through the fictitious time co-ordinate, of an initial displacement field corresponding to the unloaded state. The resulting procedure somehow mimics the well-known Feynman path integral of quantum mechanics, which is equivalent to the differential formulation embodied in Schrödinger equation. However, the path integral for elastostatics is formulated in terms of infinitesimal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution, by updating displacements until convergence. This may be advantageous for problems involving a very large number of unknowns. On the other hand, by equating the updated displacement field to the actual one a direct method of solution is obtained, which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles this formulation does not require integration over the whole domain, effectively eliminating the need of a background mesh for integration. Also, it only requires continuity of the displacement field on the propagator's support. As a consequence, the formulation lends itself to very flexible meshless implementations. To demonstrate this we describe a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods. Copyright © 2000 John Wiley & Sons, Ltd.

The geometric phase (GP) is a fundamental quantum effect arising from conical intersections (CIs), with profound consequences for vibronic energy levels. Standard imaginary-time path integral molecular dynamics (PIMD) based on the Born-Oppenheimer approximation does not account for the GP, potentially leading to significant errors in low-temperature thermodynamic properties. In this Perspective, we demonstrate that the multi-electronic-state path integral (MES-PI) formulation in imaginary time─developed in J. Chem. Phys. 2018, 148, 102319─naturally captures the GP effect through the electronic trace of the product of statistically weighted overlap matrices between successive imaginary-time slices. This crucial capability was already implicit in the benchmark MES-PIMD simulations in that foundational work. To isolate this topological effect from other nonadiabatic effects, we introduce a geometric signature matrix (for the CI) and a winding-number-induced phase factor, constructing an ad hoc GP-excluded MES-PI method. Comparing this ad hoc baseline against the rigorous MES-PI approach allows us to unambiguously quantify the impact of the GP on thermodynamic properties. While simpler approximations exist when only the ground electronic state is considered, MES-PIMD is the most general and accurate approach applicable to real complex systems, where the location and topology of CI seams are often not known a priori.

The principle of least action is known to apply to many domains of Physics. Yet its justifications and derivations usually come from archaic classical principles like the Fermat or Maupertuis principles, or mathematical analytical frameworks used for classical mechanics like the Lagrange, Hamilton and D’Alembert formalisms. As is well known, it is widely applicable to other domains, without necessarily a clear justification for it, especially when applied to Physics where Lagrangians or Hamiltonians may not be defined or do not exist. In addition, the Feynman Path Integral formulation of Quantum Mechanics gave it a second life, and another way to justify its validity not only in classical, but also in quantum Physics. However, the conventional logic to reach these conclusions is at time circular, and the principle is applied without proof that it is applicable: it just works and everybody does it. It is this applicability that led us in the past to treat the Action path integral as God’s equation.The paper justifies the principle of least action and Feynman’s path integrals through different physical reasoning instead of the traditional mathematical formalism, and on that basis we confirm its validity in most situations, even if Hamiltonians or Lagrangians may be shaky to define, or the system may be dissipative. We also explain when, and why, sometimes, it may not conventionally apply. The derivation uncovers different links between Entropy and Action, which also opens a different way to look at the origin of the probability distribution of even just a single lone particle; especially when complemented with the W-type multi-fold hypothesis, and makes some considerations on Feynman diagrams, instantons, renormalons and resurgence. Also discontinuities in the paths of the path integrals imply that spacetime is discrete, and non-commutative, and supersymmetry in flat spacetime is unphysical. Surfaceology, as optimized computation of the Feynman diagram is shown to trivially predict that all gauge theories double copy dual to gravity share the same forbidden scatterings. Relying on a recently encountered paper, we show how the results of the multi-fold theory, in particular the multi-fold space time matter induction and scattering, built explicitly on the path integral formalism, its underlying principles, and the principle of least action, allows recovering almost everything in Today’s physics. It leaves out essentially only some of the coupling constants and mixing angles/parameters. This goes a long way towards the ambition of a Theory Of Everything (TOE), even if not presented as usually expected. It maybe the best that can be achieved considering Gödel's incompleteness theorems. Our justification of the principle of least action, and path integrals, and our work on multi-fold theory encounters relationships between Entropy and Action. We review these encounters, including how our early prediction that entanglement might be irreversible match recent published results that argue against the existence of a (second) Law of entanglement analogous to the second Law of Thermodynamics. Also, from our approach, we obtain a formulation of the evolution of entropy, including for irreversible systems, which matches Onsager and Prigogine’s famous models and principles.

The Feynman path integral has revolutionized modern approaches to quantum physics. Although the path integral formalism has proven very successful and spawned several approximation schemes, the direct evaluation of real-time path integrals is still extremely expensive and numerically delicate due to its high-dimensional and oscillatory nature. We propose an efficient method for the numerical evaluation of the real-time world-line path integral for theories where the potential is dominated by a quadratic at infinity. This is done by rewriting the high-dimensional oscillatory integral in terms of a series of low-dimensional oscillatory integrals, that we efficiently evaluate with Picard-Lefschetz theory or approximate with the eikonal approximation. Subsequently, these integrals are stitched together with a series of fast Fourier transformations to recover the lattice regularized Feynman path integral. Our method directly applies to problems in quantum mechanics, the word-line quantization of quantum field theory, and quantum gravity. Published by the American Physical Society 2025

In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is developed for determining the evolution of the response probability density function (PDF). Specifically, the PI approach, utilized for evaluating the response PDF in short time steps based on the Chapman–Kolmogorov equation, is here employed in conjunction with the Laplace’s method of integration. In this manner, an approximate analytical solution of the integral involved in this equation is obtained, thus circumventing the repetitive integrations generally required in the conventional numerical implementation of the procedure. Further, the method is extended to nonlinear oscillators, approximately modeling the amplitude of the system response as a one-dimensional Markovian process. Various nonlinear systems are considered in the numerical applications, including Duffing and Van der Pol oscillators. Appropriate comparisons with Monte Carlo simulation data are presented, demonstrating the efficiency and accuracy of the proposed approach.

Feynman’s path integral approach is studied in the framework of the Wigner–Dunkl deformation of quantum mechanics. We start with reviewing some basics from Dunkl theory and investigate the time evolution of a Gaussian wave packet, which exhibits the same dispersion relation as observed in standard quantum mechanics. Feynman’s path integral approach is then extended to Wigner–Dunkl quantum mechanics. The harmonic oscillator problem is solved explicitly. We then look at the Euclidean time evolution and the related Dunkl process. This process, which exhibit jumps, can be represented by two continuous Bessel processes, one with reflection and one with absorption at the origin. The Feynman–Kac path integral for the harmonic oscillator problem is explicitly calculated.

$\delta'$-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual $\delta$-function or a $\delta'$-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral.

The authors examine several topical subjects, commencing with a general introduction to path integrals in quantum mechanics and the group theoretical backgrounds for path integrals. Applications of harmonic analysis, polar coordinate formulation, various techniques and path integrals of SU(2) and SU(1,1) are discussed. Soluble examples presented include particle-flux system, a pulsed oscillator, magnetic monopole, the Coulomb problems in curved space and others. The second part deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent states, formulation of coherent path integrals for spin and unitary spin, and semiclassical quantization are presented. Applications are made to the study of quantum fluctuation, the nonlinear field model and phase holonomy. The final chapters present the theory of the SU(1,1) coherent states and their applications. The radial Coulomb problem, the Morse oscillator, and the large-N approximation are discussed. Applications to problems in quantum optics such as squeezed states, interaction with the squeezed vacuum states, and phase operator formalism are also included. This book is intended as an introduction to the subject as well as a work of reference.

Superfield formulation of the phase space path integral

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.

We compute four-point functions of two heavy and two “perturbatively heavy” operators in the semiclassical limit of Liouville theory on the sphere. We obtain these “Heavy-Heavy-Light-Light” (HHLL) correlators to leading order in the conformal weights of the light insertions in two ways: (a) via a path integral approach, combining different methods to evaluate correlation functions from complex solutions for the Liouville field, and (b) via the conformal block expansion. This latter approach identifies an integral over the continuum of normalizable states and a sum over an infinite tower of lighter discrete states, whose contribution we extract by analytically continuing standard results to our HHLL setting. The sum over this tower reproduces the sum over those complex saddlepoints of the path integral that contribute to the correlator. Our path integral computations reveal that when the two light operators are inserted at equal time in radial quantization, the leading-order HHLL correlator is independent of their separation, and more generally that at this order there is no short-distance singularity as the two light operators approach each other. The conformal block expansion likewise shows that in the discrete sum short-distance singularities are indeed absent for all intermediate states that contribute. In particular, the Virasoro vacuum block, which would have been singular at short distances, is not exchanged. The separation-independence of equal-time correlators is due to cancelations between the discrete contributions. These features lead to a Lorentzian singularity that, in conformal theories with anti-de Sitter (AdS) duals, would be associated to locality below the AdS scale.

Non-Abelian Bosonization as a Nonholonomic Transformation from a Flat to a Curved Field Space

The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.

Theoretical methods are developed for studying the dynamics of chemical reactions. Many chemical processes can be adequately described as a system'' of one or two reaction coordinates coupled to a bath'' of harmonic oscillators. Such a formulation of the problem has obvious advantages as it is often possible to integrate out the harmonic bath, thus reducing the many degree of freedom problem to a one or two dimensional calculation. A simple semiclassical model for incorporating tunneling effects in classical trajectory simulations is constructed and shown to give excellent results over a variety of conditions. Feynman path integration provides an exact way of calculating quantum, dynamical properties and is in principle applicable to problems of involving many degrees of freedom. However, the standard Monte Carlo methodology which must be used to evaluate the multidimensional integrals that occur, is not directly applicable to the path integral expression for the real time propagator, because the integrand of the latter is highly, oscillatory. A technique which helps circumvent this problem by sampling about the stationary phase points of the integrand is developed and illustrated with several numerical applications. A new effective real time propagator is also constructed, which is localized and devoid of rapid oscillations. Finally, accurate approximations to the short time propagator are derived, which drastically improve the convergence characteristics of path integrals. These techniques are expected to extend the applicability of path integral calculations to more complex chemical problems. 258 refs., 63 figs., 3 tabs.

The fundamental solution for the continuous-time filtering problems can be expressed in terms of Feynman path integrals. This enables one to view the solution of filtering problem in terms of an effective action that is a function of the signal and measurement models. The practical utility of the path integral formula is demonstrated via some nontrivial examples. Specifically, it is shown that the simplest approximation of the path integral formula for the fundamental solution of the Fokker-Planck-Kolmogorov forward equation (termed the Dirac-Feynman approximation) can be applied to solve nonlinear continuous-discrete filtering problems quite accurately using sparse grid filtering and Monte-Carlo approaches.