Efficient evaluation of real-time path integrals

Testing Monte Carlo methods for path integrals in some quantum mechanical problems

We formulate the quasielastic response of a non-relativistic many-body system at zero temperature in terms of ground state density matrix elements and real time path integrals that embody the final state interactions. While the former provide the weight for a conventional Monte Carlo calculation, the latter require a more sophisticated treatment. We argue that the recently developed Stationary Phase Monte Carlo technique can be used to study the approach to "Y-scaling." We perform calculations for a particle in a potential well in one and three dimensions and compare them to the exact results available for these models. We then derive an eikonal approximation to the Path Integrals. This method is suitably generalized to treat strongly repulsive interactions, and allows comparison to Silver's theory of final state interactions in a straightforward way. We also give an exact prescription to calculate the scaling limit for potentials comprising a hard core. Finally, we study the approach to scaling in a model 4He nucleus, and find good agreement with experimental data.

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.

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.

The continuum calculus, proposed earlier [J. Math. Phys. 17, 1988 (1976)], is here applied to the development of a functional version of Laplace transform and a method for the evaluation of path integrals containing a Gaussian-like measure. Two methods in functional integration are proposed. First, the Gaussian integral for polynomial functionals and consequently functionals that can be expanded in Taylor series are examined. Formula (2.12) is derived. Next, we define the Laplace transform in the function space through the weak distribution formulation of Skorohod. Comparison of both approaches enables us to determine the expression of the functional integeral through a series of Laplace transformations. The second formula is given in Eq. (4.5). The latter formula is applicable to all Laplace transformable functionals. As an illustration of the utility of the formulas derived, we evaluate the integral of a cosine functional by methods 1 and 2, and obtain consistent results. Further applications to quantum mechanics are also presented. We examine the cases of a free particle,the quantum harmonic oscillator, the forced oscillator, and charged particle in a magnetic field. In all cases, we obtain correct results in comparison with known expressions. A numerical procedure is employed in the calculation of infinite products. The usefulness of the p-integral method is stressed.

A polymer density functional theory (P-DFT) has been extended to the case of quantum statistics within the framework of Feynman path integrals. We start with the exact P-DFT formalism for an ideal open chain and adapt its efficient numerical solution to the case of a ring. We show that, similarly, the path integral problem can, in principle, be solved exactly by making use of the two-particle pair correlation function (2p-PCF) for the ends of an open polymer, half of the original. This way the exact data for one-dimensional quantum harmonic oscillator are reproduced in a wide range of temperatures. The exact solution is not, though, reachable in three dimensions (3D) because of a vast amount of storage required for 2p-PCF. In order to treat closed paths in 3D, we introduce a so-called "open ring" approximation which proves to be rather accurate in the limit of long chains. We also employ a simple self-consistent iteration so as to correctly account for the interparticle interactions. The algorithm is speeded up by taking convolutions with the aid of fast Fourier transforms. We apply this approximate path integral DFT (PI-DFT) method to systems within spherical symmetry: 3D harmonic oscillator, atoms of hydrogen and helium, and ions of He and Li. Our results compare rather well to the known data, while the computational effort (some seconds or minutes) is about 100 times less than that with Monte Carlo simulations. Moreover, the well-known "sign problem" is expected to be considerably reduced within the reported PI-DFT, since it allows for a direct estimate of the corresponding partition functions.

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.

Wavelets offer significant advantages for the analysis of problems in quantum mechanics. Because wavelets are localized in both time and frequency they avoid certain subtle but potentially fatal conceptual errors that can result from the use of plane wave or δ function decomposition. Morlet wavelets in particular are well-suited for this work: as Gaussians, they have a simple analytic form and they work well with Feynman path integrals. But to take full advantage of Morlet wavelets we need to supply an explicit form for the inverse Morlet transform and a manifestly covariant form for the four-dimensional Morlet wavelet. We construct both here. Quanta 2012; 1: 58–70.

A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so-called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based.

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.

The method of Path Integrals (PI’s) was developed by Richard Feynman in the 1940’s. It offers an alternate way to look at quantum mechanics (QM), which is equivalent to the Schrodinger formulation. As will be seen in this project work, many elementary problems are much more difficult to solve using path integrals than ordinary quantum mechanics. The benefits of path integrals tend to appear more clearly while using quantum field theory (QFT) and perturbation theory. However, one big advantage of Feynman’s formulation is a more intuitive way to interpret the basic equations than in ordinary quantum mechanics. Here we give a basic introduction to the path integral formulation, starting from the well known quantum mechanics as formulated by Schrodinger. We show that the two formulations are equivalent and discuss the quantum mechanical interpretations of the theory, as well as the classical limit. We also perform some explicit calculations by solving the free particle and the harmonic oscillator problems using path integrals. The energy eigenvalues of the harmonic oscillator is found by exploiting the connection between path integrals, statistical mechanics and imaginary time.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

The spacetime approach to quantum mechanics

In this chapter we will introduce the Feynman path integral. We will derive the simplest path integral from conventional quantum mechanics. In doing so we may perhaps give the impression that path integration is merely a useful trick for solving a certain class of quantum mechanics problems. As familiarity with the technique grows, however, the gentle reader may come to follow Feynman in his belief that the path integral captures the fundamental physics, and that hamiltonians and Hubert space are merely useful mathematical methods for evaluating path integrals.KeywordsPath IntegralPerturbation SeriesGaussian IntegralEuclidean TimeClassical Statistical MechanicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.