Essential aspects of Wiener-measure regularization for quantum-mechanical path integrals (original) (raw)

Wiener Measure Regularization for Quantum Mechanical Path Integrals

Recent Developments in Mathematical Physics, 1987

The problems associated with a regularization of quantum mechanical path integrals using continuous-time (as opposed to discrete-time) schemes are examined. All such proposals insert regularizing Wiener measures and consider the limit as the diffusion constant diverges as the final step. Two unsuccessful approaches in the Schrodinger representation are reviewed before a fairly complete treatment of the successful coherent-state representation approach is presented. Not only does the coherent-state approach provide a rigorous continuous-time regularization scheme for quantum mechanical path integrals but it also offers a natural and physically appealing formulation that is covariant under classical canonical transformations.

Path Integrals in Quantum Physics

Lecture Notes in Physics Monographs

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, manybody physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to "Details" are marked by a ⋆ as well.

Path Integrals in Quantum Physics (English Version)

arXiv:1209.1315v4, 2017

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.

The Feynman Path Integral: An Historical Slice

Essays in Honor of Hiroshi Ezawa, 2003

Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.

Path Integrals in Quantum Physics Lectures given at ETH Zurich

2016

Wiener Integration for Quantum Systems: A Unified Approach to the Feynman-Kac Formula

NATO ASI Series, 1997

A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.

PATH INTEGRALS IN QUANTUM MECHANICS-Oxford Scholarship

2019

This chapter constructs the path integral associated with the statistical operator e-βH in the case of Hamiltonians of the simple form p2/2m + V (q). The path integral corresponding to a harmonic oscillator coupled to an external, timedependent force is then calculated. This result allows a perturbative evaluation of path integrals with general analytic potentials. The results are applied to the calculation of the partition function tr e-βH using perturbative and semi-classical methods. The integrand for this class of path integrals defines a positive measure on paths. It is thus natural to introduce the corresponding expectation values, called correlation functions. Moments of such a distribution can be generated by a generating functional, and recovered by functional differentiation. These results can be applied to the determination of the spectrum of a class of Hamiltonians in several approximation schemes.

On the Path Integral Approach to Quantum Mechanics

On the Path Integral Approach to Quantum Mechanics, 2021

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.

Further uses of the quantum mechanical path integral in quantum field theory

Journal of Mathematical Physics, 1995

It has been shown how matrix elements of the form (xlexp( -iHr)ly) which arise when using operator regularization to do perturbative calculations in quantum field theory can be evaluated using the quantum mechanical path integral (QMPI). This technique has the advantage of eliminating loop momentum integrals and algebraically complicated vertices in gauge theories. A similar (but distinct) approach of Polyakov and Strassler has been applied to one loop processes with external vector particles and is related to the string based methods of Bern and Kosower. In this article, several features of the QMPI technique are examined. First, it is demonstrated how the path ordering in the QMPI can be handled by considering a model of three interacting scalar fields, each with a distinct mass. Next, it is shown how the QMPI can be used when the external wave function is not a plane wave field. The particular case of having an exponentially damped wave function is considered. Next, a discussion of the difference between the approach of Polyakov and Strassler and that employed here is given. Finally, it is demonstrated how the QMPI can be used to considerably simplify calculations in quantum gravity. 0 1995 American Institute of Physics.

Coherent State Path Integrals at (Nearly) 40

1998

Coherent states can be used for diverse applications in quantum physics including the construction of coherent state path integrals. Most definitions make use of a lattice regularization; however, recent definitions employ a continuous-time regularization that may involve a Wiener measure concentrated on continuous phase space paths. The introduction of constraints is both natural and economical in coherent state path integrals involving only the dynamical and Lagrange multiplier variables. A preliminary indication of how these procedures may possibly be applied to quantum gravity is briefly discussed.

On the approximation of Feynman-Kac path integrals

A general framework is proposed for the numerical approximation of Feynman-Kac path integrals in the context of quantum statistical mechanics. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional Sobolev space by restricting the integrand to a subspace of all admissible paths. Through this process, a wide class of methods is derived, with each method corresponding to a different choice for the approximating subspace. It is shown that the traditional "short-time" approximation and "Fourier discretization" can be recovered by using linear and spectral basis functions, respectively. As an illustration of the flexibility afforded by the subspace approach, a novel method is formulated using cubic elements and is shown to have improved convergence properties when applied to model problems.

Path integral for quantum operations

Journal of Physics A: Mathematical and General, 2004

In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation).

Introduction to path-integral Quantum Field Theory – A toolbox

Revista Brasileira de Ensino de Física, 2021

Lecture notes on path-integrals, suitable for an undergraduate course with prerequisites such as: Classical Mechanics, Electromagnetism and Quantum Mechanics. The aim is to provide the reader, who is familiar with the major concepts of Solid State Physics, to study these topics couched in the language of path integrals. We endeavor to keep the formalism to the bare minimum.

A riemann integral approach to Feynman's path integral

Foundations of Physics Letters, 1995

ABSTRACT It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrdinger'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 Schrdinger'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.

Path Integrals for (Complex) Classical and Quantum Mechanics

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolant of quantum mechanics; additional degrees of freedom permit 'tunnelling' without recourse to instantons and lead to time/energy uncertainty. In practice, 'classical' particle trajectories with additional degrees of freedom have arisen in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that hbar\hbarhbar has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.

Path integrals: From quantum mechanics to photonics

APL Photonics, 2021

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 nonlin...

Wiener and Poisson process regularization for coherent‐state path integrals

Journal of Mathematical Physics, 1993

By introducing a suitable continuous-time regularization into a formal phasespace path integral it follows that the propagator is given by the limit of welldefined functional integrals involving standard stochastic processes and their associated probability measures. Such regularizations require pinning of both coordinate and momenta variables, and automatically lead to coherent-state representations. It is found that each standard independent increment process, involving a superposition of a Wiener and a Poisson process, is associated with a specific, generally non-Gaussian, fiducial vector with which the coherent states are defined.