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A Smooth Parametric Path-Family Method for FeynmanPath-Integral Amplitude Estimation

Rajah Iyer

Published Jan 1, 2019• 0 citations
Publisher

Abstract

We present novel path modeling techniques suitable for use in the Path-Integral formulation of Quantum Mechanics. Our proposed platform aims to address existing challenges encountered in Monte Carlo and other similar path modeling methods. By introducing 'smooth' path modeling techniques, we demonstrate how they can be seamlessly integrated with current approaches, facilitating more accessible amplitude estimations in this invaluable formulation of Quantum Mechanics.

DOI
10.13140/rg.2.2.36270.36162

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We present novel path modeling techniques suitable for use in the Path-Integral formulation of Quantum Mechanics. Our proposed platform aims to address existing challenges encountered in Monte Carlo and other similar path modeling methods. By introducing 'smooth' path modeling techniques, we demonstrate how they can be seamlessly integrated with current approaches, facilitating more accessible amplitude estimations in this invaluable formulation of Quantum Mechanics.
The path integral formulation of quantum mechanics, i.e., the idea that the evolution of a quantum system is determined as a sum over all the possible trajectories that would take the system from the initial to its final state of its dynamical evolution, is perhaps the most elegant and universal framework developed in theoretical physics, second only to the standard model of particle physics. In this Tutorial, we retrace the steps that led to the creation of such a remarkable framework, discuss its foundations, and present some of the classical examples of problems that can be solved using the path integral formalism, as a way to introduce the readers to the topic and help them get familiar with the formalism. Then, we focus our attention on the use of path integrals in optics and photonics and discuss in detail how they have been used in the past to approach several problems, ranging from the propagation of light in inhomogeneous media to parametric amplification and quantum nonlinear optics in arbitrary media. To complement this, we also briefly present the path integral Monte Carlo method, as a valuable computational resource for condensed matter physics, and discuss its potential applications and advantages if used in photonics.
The path integral technique is an alternative formulation of quantum mechanics that is based on a Lagrangian approach. In its exact form, it is completely equivalent to the Hamiltonian-based Schrödinger equation approach. We have used the path integral formalism to develop our Path Integral Quantum Trajectory (PIQTr) model for use in the study of charged particle dynamics. One of the challenges of the PIQTr model is the need to integrate a highly oscillatory function. Here we explore the application of several integration techniques for use in the PIQTr model and benchmark their computational requirements.
The path integral formulation in quantum mechanics corresponds to the first quantization since it is just to rewrite the quantum mechanical amplitude into many dimensional integrations over discretized coordinates xnx_nxn​. However, the path integral expression cannot be connected to the dynamics of classical mechanics, even though, superficially, there is some similarity between them. Further, the field theory path integral in terms of many dimensional integrations over fields does not correspond to the field quantization. We clarify the essential difference between Feynman's original formulation of path integral in QED and the modern version of the path integral method prevailing in lattice field theory calculations, and show that the former can make a correct second quantization while the latter cannot quantize fields at all and its physical meaning is unknown.
Abstract The focus of Chapter 12 is the Feynman path integral formulations of quantum mechanics and quantum statistical mechanics. The connection between real-time quantum propagator and the density matrix is established via the Wick rotation, and a Trotter expansion is used to derive discrete path integral expressions for both. The continuous limit of the discrete path integral is derived to yield the functional integral formulation of the path integral. The construction of estimators and their averages for computing quantum observables is discussed. Next, the path integral is extended to incorporate Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, and a useful recursive formula for the partition function is derived that accelerates convergence by building up N-body partition functions from those of smaller systems. Quantum free energies are discussed, and algorithms for the evaluation of path integrals using molecular dynamics or Monte Carlo are presented. The chapter concludes with a discussion of higher-order factorizations and ring-polymer contraction methods to reduce the computational overhead of path-integral calculations
Abstract The last two decades have seen a consistent growth in the significance of path integrals. They are useful in quantum as well as statistical mechanics, providing more intuitive and easier methods of handling equations than more commonly-known treatments. However, the method is still greeted with skepticism. Where has the operator ordering problem gone? Do polar coordinates work in PI, despite the fact that they cannot exist in the canonical operator formalism? Is it possible to obtain a quantum mechanics on the sphere by inserting a delta function into the unconstrained formula? This book addresses and solves these fundamental issues, and covers more recent topics, such as the lattice fermion problem in quantum theory and the Monte Carlo method in statistical mechanics.