Quasiclassical path-integral approach to supersymmetric quantum mechanics

In discussing time slicing approximations of Feynman's path integrals, one can use stationary phase method for some oscillatory integrals over a space of large dimension. Previously estimates of remainder term of such stationary phase method were given in [2] and [6]. In the present note an improved remainder estimate is given for such stationary phase method. The estimate is sharp enough to give us the second term of semiclassical asymptotic expansion if it is applied to Feynman path integrals of some kind discussed in [6] and [4]. See our forthcoming paper [3].

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.

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrodinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) “practical calculations” — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrodinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in “practical” calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.

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.

Atoms and molecules submitted to a strong laser field can emit electrons of high energies in the above-threshold ionization (ATI) process. This process finds a highly intuitive and also quantitative explanation in terms of Feynman's path integral and the concept of quantum orbits [P. Salières et al., Science 292, 902 (2001)]10.1126/science.108836. However, the connection with the Feynman path-integral formalism is explained only by intuition and analogy and within the so-called strong-field approximation (SFA). Using the phase space path-integral formalism we have obtained an exact result for the momentum-space matrix element of the total time-evolution operator. Applying this result to the ATI we show that the SFA and the so-called improved SFA are, respectively, the zeroth- and the first-order terms of the expansion in powers of the laser-free effective interaction of the electron with the rest of the atom (molecule). We have also presented the second-order term of this expansion which is responsible for the ATI with double scattering of the ionized electron.

Resonant tunneling is studied by use of Feynman's path integrals. The semiclassical Green's function is obtained for a one-dimensional double-barrier structure. This approach leads to a clearer and deeper understanding of the physical process of resonant tunneling. It is shown that sequential tunneling never gives rise to negative differential resistance.

Stochastic processes If a classical particle moves in an external force field F(x ) its motion is deterministic and it could be called a ‘deterministic process’. If the particle interacts with randomly distributed obstacles its motion is still deterministic, but the trajectory depends on the random characteristics of the obstacles, and maybe also on the random choice of the initial conditions. Such a motion is an example of a ‘stochastic process’, or rather a ‘random process’. In a more abstract generalized definition, a stochastic process is a random variable X ξ that depends on an additional (deterministic) independent variable ξ, which can be discrete or continuous. In most cases it stands for an index ξ ∈ N or time t , space x or energy E . An almost trivial ubiquitous stochastic process is given by additive noise η( t ) on a time-dependent signal s(t ), i.e. X t = s(t ) + η( t ). As a consequence, X t is no longer continuous. The most apparent applications of stochastic processes are time series of any kind that depend on some random impact. A broad field of applications are time series occurring, for example, in business, finance, engineering, medical applications and of course in physics. Beyond time series analysis, stochastic processes are at the heart of diffusion Langevin dynamics, Feynman's path integrals [43], as well as Klauder's stochastic quantization [62], which represents an unconventional approach to quantum mechanics. Here we will give a concise introduction and present a few pedagogical examples.

Kinetic isotope effect values on the decarboxylation of 3-carboxybenzisoxazole have been computed using the second-order Kleinert's variational perturbation theory in the framework of Feynman's path integrals along with the potential energy surface obtained at the MP2/6-31+G(d) level. Good agreement with the experimental data was obtained, demonstrating that this novel computational approach for computing KIE values of organic reaction is a viable alternative to the traditional method employing the Bigeleisen equation and harmonic vibrational frequencies. Compared with the experimental measurements, consideration of anharmonicity and tunneling effects can significantly improve the calculated KIE values, reducing the root-mean-square deviation from 1.19 % for traditional method to 0.20 % for path-integral method.

Quantisation in multiply connected spaces

Cognitive decisions are best described by quantum mathematics. Do quantum information devices operate in the brain? What would they look like? Fuss and Navarro () describe quantum lattice registers in which quantum superpositioned pathways interact (compute/integrate) as 'quantum walks' akin to Feynman's path integral in a lattice (e.g. the 'Feynman quantum chessboard'). Simultaneous alternate pathways eventually reduce (collapse), selecting one particular pathway in a cognitive decision, or choice. This paper describes how quantum walks in a Feynman chessboard are conceptually identical to 'topological qubits' in brain neuronal microtubules, as described in the Penrose-Hameroff 'Orch OR' theory of consciousness.

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.

A method is developed for calculating the equation of state of a system of quantum particles at a finite temperature, based on the Feynman formulation of quantum statistics. A general analytical expression is found for the virial estimator for the kinetic energy of a system with rigid boundaries at a finite pressure. An effective method is developed for eliminating the unphysical singularity in the electrostatic potential between a discretized Feynman path of an electron and a proton. It is shown that the “refinement” of an expansion of a quantum-mechanical propagator by addition of high powers of time exacerbates, rather than eliminates, the divergence of a Feynman path integral. A brief summary of the current status of the problem is presented. The proposed new approaches are presented in relation to progress made in this field. Path integral Monte Carlo simulations are performed for nonideal hydrogen plasmas in which both indistinguishability and spin of electrons are taken into account under conditions preceding the formation of the electron shells of atoms. The electron permutation symmetry is represented in terms of Young operators. It is shown that, owing to the singularity of the Coulomb potential, quantum effects on the behavior of the electron component cannot be reduced to small corrections even if the system must be treated as a classical system according to the formal de Broglie criterion. Quantum-mechanical delocalization of electrons substantially weakens the repulsion between electrons as compared to protons. In relatively cold plasmas, many-body correlations lead to complex behavior of the potential of the average force between particles and give rise to repulsive forces acting between protons and electrons at distances of about 5 angstroms. Plasma pressure drops with decreasing plasma temperature as the electron shells of atoms begin to form, and the electron kinetic energy reaches a minimum at a temperature of about 31000 K. The minimum point weakly depends on plasma density. Owing to quantum effects, the electron component is “heated” well before electrons are completely bound in the field of protons.

The Monte Carlo Fourier path integral approach has proved to be quite useful in calculating equilibrium thermodynamic properties. One of its advantages is that it can be formulated in such a way as to include higher order terms using the partial averaging technique, which includes the contribution from higher terms usually neglected by the discretized path integral approach. In the original approach, the Feynman path integral is evaluated via a free-particle reference state. Here, using a new expression for the Feynman paths expanded around a harmonic reference path, we derive an alternative formulation for the density matrix element and its corresponding partial averaging expression.

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.

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