Path integral in constrained classical mechanics

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

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.

We present a fast, high precision and easily implementable path integral method for numerically solving Fokker–Planck equations. It is based on a generalized Trotter formula, which permits one to attain an adequate description of dynamical and equilibrium properties even though the time increment τ=t/N is rather large. A remarkable property of the symmetric Trotter splitting is used to systematically eliminate the lower-order errors resulting from time discretization. This means a significant reduction of the number of time steps that are required to retain a given accuracy for a given net increment t=Nτ, and, therefore, significantly increasing the feasibility of path integral calculations. Yet another attractive feature of the present technique is that it allows for equations with singular diffusion matrices that are known to present a special problem within the scope of the path integral formalism. The favorable scaling of the fast Fourier transform is used to numerically evaluate the path integral on a grid. High efficiency is achieved due to the Stirling interpolation which dynamically readjusts the distribution function every time step with a mild increase in cost and with no loss of precision. These developments substantially improve the path integral method and extend its applicability to various time-dependent problems which are difficult to treat by other means. One can even afford to extract information on eigenvalues and eigenfunctions from a time-dependent solution thanks to the numerical efficiency of the present technique. This is illustrated by calculating the propagator and the lowest eigenvalues of a one-dimensional Fokker–Planck equation. The method is also applied to a two-dimensional Fokker–Planck equation, whose diffusion matrix does not possess an inverse (a so-called Klein-Kramers equation). The numerical applications show our method to be a dramatic improvement over the standard matrix multiplication techniques available for evaluating path integrals in that it is much more efficient in terms of speed and storage requirements.

The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path-integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The construction is complicated by the fact that the amplitudes (class-operators) calculated using a path integral typically do not satisfy the constraint equation everywhere, but we show how they may be systematically modified in such a way that they do satisfy the constraint. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the ``evolving constants'' method).

Nuclear quantum effects play critical roles in a variety of molecular processes, especially in systems that contain hydrogen and other light nuclei, such as water. For water under ambient conditions, nuclear quantum effects are often interpreted as local effects resulting from a smearing of the hydrogen atom distribution. However, the orientational structure of water at interfaces determines long-range effects, such as electrostatics, through the O–H bond ordering that is impacted by nuclear quantum effects. In this work, I examine nuclear quantum effects on long-range electrostatics of water confined between hydrophobic walls using path integral simulations. To do so, I combine concepts from local molecular field theory with path integral methods at varying levels of approximation to develop efficient and physically intuitive approaches for describing long-range electrostatics in nonuniform quantum systems. Using these approaches, I show that quantum water requires larger electrostatic forces to achieve interfacial screening than the corresponding classical system. This work highlights the subtleties of electrostatics in nonuniform classical and quantum molecular systems, and the methods presented here are expected to be of use to efficiently model nuclear quantum effects in large systems.

Describing the spin degree of freedom in terms of Grassmann variables, we study the quantization problem of nonrelativistic and relativistic spinning particles interacting with crossed electric and magnetic fields by using the path integral method. The problem presents some difficulties due to the odd-dimensional nature of the phase space for the spin variables. We show how to deal with these difficulties by bringing back all the problems to the study of the quantum mechanics of a single Grassmann variable. Then it is straightforward to perform the path integrals in all the cases considered here, simply by solving the classical equations of motion.

The path-integral method seems to be the most suitable for the quantization of gravity. One would expect the dominant contribution to the path integral to come from metrics which are near background metrics that are solutions of classical Einstein equations. The action of these background metrics gives rise to a new phenomenon in field theory, intrinsic quantum entropy. This is shown to be related to the scaling behavior of the gravitational action and to the topology of the gravitational field. The quadratic terms in the Taylor series of the action about the background metrics give the one-loop corrections. In a supersymmetric theory the quartic and quadratic but not the so-called logarithmic divergences cancel to give a one-loop term that is finite without regularization. From the one-loop term one can obtain the effective energy-momentum tensor on the background metric. In the case of an evaporating black hole, the energy-momentum tensor will be regular on the future horizon. The usual perturbation expansion breaks down for quantum gravity because the higher (interaction) terms in the Taylor series are not bounded by the quadratic (free) ones. To overcome this I suggest that one might replace the path integrals over the terms in the Taylor series by a discrete sum of the exponentials of the actions of all complex solutions of the Einstein equations, each solution being weighted by its one-loop term. This approach seems to give a picture of the gravitational vacuum as a sea of virtual Planck-mass black holes.

This paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem via a Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). Through Lagrangian relaxation, we convert the chance-constrained (risk-constrained) SOC problem to a risk-minimizing SOC problem, the cost function of which possesses the time-additive Bellman structure. We show that the risk-minimizing control synthesis is equivalent to solving an HJB PDE whose boundary condition can be tuned appropriately to achieve a desired level of safety. Furthermore, it is shown that the proposed risk-minimizing control problem can be viewed as a generalization of the problem of estimating the risk associated with a given control policy. Two numerical techniques are explored, namely the path integral and the finite difference method (FDM), to solve a class of risk-minimizing SOC problems whose associated HJB equation is linearizable via the Cole-Hopf transformation. Using a 2D robot navigation example, we validate the proposed control synthesis framework and compare the solutions obtained using path integral and FDM.

Dynamic programming gives a new method for computing the wave function in quantum mechanics. The paper compares this method with Feynman's path integral method (1965).

The path integral method is used to construct the quantum propagator for a particle moving under a constant force.

Stochastic ship roll motion via path integral method

The asymptotic forms of the distribution functions for two- and three-dimensional continuous wormlike chains near the rod limit are obtained by the path integral method with the use of the WKB approximation. The derived distributions give correctly the first-order corrections to the rod limiting values of all the moments.

Certain problems in the theory of developed hydrodynamical turbulence

Quantum restrictions on the measurement of the trajectory of a harmonic oscillator are calculated by the method of path integration. A minimum value is found for the force acting on an oscillator or a free particle which can be discovered by this sort of measurement. Some examples are considered.

The method of path integration is used to study the effects of quantum fluctuations in the space-time geometry near the classical singularity of general relativity. It is shown that in certain special cases explicit Feynman propagators can be constructed which enable us to evaluate these fluctuationsquantitatively. The cases discussed are (i) the gravitational collapse of a uniform dust ball, (ii) the Friedmann cosmologies, (iii) the axisymmetric Bianchi type I cosmological model, and (iv) the general anisotropic Bianchi type I cosmological model. In all cases discussed here the quantum uncertainty grows to infinity as the classical space-time singularity is approached. In this wider regime of quantum gravitation nonsingular solutions can occur with finite probabilities.