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

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.

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.

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.