Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

The Feynman Path Integral

N. Cabibbo, Luciano Maiani, Omar Benhar A

Published Oct 22, 2025• 0 citations
Publisher

Abstract

This Chapter introduces the Feynman Path Integral description of quantum transition amplitudes, considering the simplest case of a quantum system described by one quantum variable q and its conjugate momentum p . The transition amplitude from time https://www.w3.org/1998/Math/MathML" display="inline"> t 1 to time https://www.w3.org/1998/Math/MathML" display="inline"> t 2 = t 1 + T is obtained by discretising the time axis in N time steps of amplitude https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , implying T=N https://www.w3.org/1998/Math/MathML" display="inline"> ϵ . For https://www.w3.org/1998/Math/MathML" display="inline"> ϵ → 0 , the Feynman Path Integral formula is derived: https://www.w3.org/1998/Math/MathML" display="inline"> < q N | e - i T H / ℏ | q 0 > = ∫ d [ q ( t ) ] e i S q ( t ) , q ' ( t ) / ℏ , where integration is over all trajectories https://www.w3.org/1998/Math/MathML" display="inline"> q ( t ) which satisfy the boundary conditions at https://www.w3.org/1998/Math/MathML" display="inline"> t 1 and t 2 . https://www.w3.org/1998/Math/MathML" display="inline"> S ( q , q ' ) is the classical Action, a function of https://www.w3.org/1998/Math/MathML" display="inline"> q and of its time derivative. The classical limit, corresponding to https://www.w3.org/1998/Math/MathML" display="inline"> ℏ → 0 , is considered, whereby the solution of the Feynman Path Integral reduces to the classical trajectory, https://www.w3.org/1998/Math/MathML" display="inline"> q c ( t ) , which minimises the classical Action. The lattice approximation , i.e. small, non vanishing https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , with evaluation of the Feynman integral by numerical integration, provides a useful approximation in the case of the fundamental theory of the strong interactions (Quantum Chromodynamics, or QCD), to be considered later; Chapters 17 to19. This chapter is completed by consideration of the extension of gthe formalism to imaginary time, where the Feynman Path Integral becomes the Partition Function of Statistical Mechanics, and by the definition of the Green's functions, ground-state expectation values of products of quantum operators q(t) taken at different times, https://www.w3.org/1998/Math/MathML" display="inline"> t 1 , t 2 … t N .

DOI
10.1201/9781003560708-2

Related papers

The Feynman Path Integral

N. Cabibbo, Luciano Maiani, Omar Benhar A

In this chapter we will derive the expression for the sum over paths from the usual formulation of non-relativistic quantum mechanics, considering the particularly simple case of a quantum system with only one degree of freedom. After deriving the expression for the transition amplitudes by means of the sum over paths, we will show how this method also allows calculations of the Green’s functions in quantum mechanics. We will conclude the chapter by showing how these results can be extended to the case of systems with more discrete degrees of freedom and to field theory.

CRC Press eBooks•Jul 28, 2017•0 citations

Transition Amplitude

Lukong Cornelius Fai

We examine under this heading the transition amplitude in the Feynman path integral, which we will discuss shortly and will be reduced to an integral over the continuum degrees of freedom, such as a set of p i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_1.jpg"/> and q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_2.jpg"/> . This will involve harmonic (non-interacting) systems that are reduced to the problem of multiple Gaussian integrals over potentially coupled degrees of freedom, q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_3.jpg"/> that we develop appropriate Feynman path integral tools for their solution. Considering that the harmonic oscillator is one of the central objects of theoretical physics then Gaussian integrals are likely to be the key tools of theoretical physics. This will be demonstrated utilizing Feynman path-integral formulation of quantum mechanics where Gaussian integrals will be pivotal for computation in quantum mechanics, quantum statistical mechanics and generally in quantum field theory. It will be interesting to introduce the Maslov correction [ 11 , 12 ] to the wave function which is a jump of π 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_4.jpg"/> in the phase when a system passes through a caustic point, a phenomenon related to the second variation as well as to the geometry of paths. This phenomenon is applicable to any system using the quasi-classical approximation. The harmonic oscillator will be one of the objects amongst others to illustrate this.

CRC Press eBooks•Feb 19, 2021•0 citations

- Path Integral Formulation

S. Rajasekar, R. Velusamy

The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.

CRC Press eBooks•Dec 10, 2014•1 citation

Path Integral Formulation Of Quantum Mechanics

Nelson Enrique Bolivar, Gabriel Abellán

To introduce the formalism of the functional integral in quantum mechanics, this chapter starts from the definition of the temporal evolution operator. It is important to emphasize that in the classical limit is not only the classic trajectory that have the grater contribution, but all the paths close to this also contribute. The chapter shows how integral formulation is perfectly consistent with Schrödinger's time-dependent differential equation. It retrieves this equation from definition of transition amplitude as a path integral. The chapter discusses functional derivative generalized to the case of the ordinary functional derivative. A very useful tool to solve problems related to functional integrals is the Wick rotation or Euclidean rotation. In quantum mechanics there are few problems we can solve exactly, and very often, when possible, we use perturbation theory to find an approximate solution. The chapter follows this approach with formalism of functional integration and construct a perturbation theory from which can be obtained a physical interpretation.

Apple Academic Press eBooks•May 23, 2018•0 citations

The Feynman path integral

Mark E. Tuckerman

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

•Aug 1, 2023•0 citations

Path Integrals in Non-Relativistic Quantum Mechanics

David Bailin, A. Love

Feynman showed that quantum mechanics could be formulated in terms of path integrals. This chapter discusses this approach to quantum mechanics in some detail since it provides the key to the path integral formulation of quantum field theory. It considers a system described by a single generalised coordinate with a conjugate momentum. The chapter shows that analogous objects in non-relativistic quantum mechanics are ground-state expectation values of products of operators.

Routledge eBooks•Jan 22, 2019•0 citations
This Chapter introduces the Feynman Path Integral description of quantum transition amplitudes, considering the simplest case of a quantum system described by one quantum variable q and its conjugate momentum p . The transition amplitude from time https://www.w3.org/1998/Math/MathML" display="inline"> t 1 to time https://www.w3.org/1998/Math/MathML" display="inline"> t 2 = t 1 + T is obtained by discretising the time axis in N time steps of amplitude https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , implying T=N https://www.w3.org/1998/Math/MathML" display="inline"> ϵ . For https://www.w3.org/1998/Math/MathML" display="inline"> ϵ → 0 , the Feynman Path Integral formula is derived: https://www.w3.org/1998/Math/MathML" display="inline"> < q N | e - i T H / ℏ | q 0 > = ∫ d [ q ( t ) ] e i S q ( t ) , q ' ( t ) / ℏ , where integration is over all trajectories https://www.w3.org/1998/Math/MathML" display="inline"> q ( t ) which satisfy the boundary conditions at https://www.w3.org/1998/Math/MathML" display="inline"> t 1 and t 2 . https://www.w3.org/1998/Math/MathML" display="inline"> S ( q , q ' ) is the classical Action, a function of https://www.w3.org/1998/Math/MathML" display="inline"> q and of its time derivative. The classical limit, corresponding to https://www.w3.org/1998/Math/MathML" display="inline"> ℏ → 0 , is considered, whereby the solution of the Feynman Path Integral reduces to the classical trajectory, https://www.w3.org/1998/Math/MathML" display="inline"> q c ( t ) , which minimises the classical Action. The lattice approximation , i.e. small, non vanishing https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , with evaluation of the Feynman integral by numerical integration, provides a useful approximation in the case of the fundamental theory of the strong interactions (Quantum Chromodynamics, or QCD), to be considered later; Chapters 17 to19. This chapter is completed by consideration of the extension of gthe formalism to imaginary time, where the Feynman Path Integral becomes the Partition Function of Statistical Mechanics, and by the definition of the Green's functions, ground-state expectation values of products of quantum operators q(t) taken at different times, https://www.w3.org/1998/Math/MathML" display="inline"> t 1 , t 2 … t N .
In this chapter we will derive the expression for the sum over paths from the usual formulation of non-relativistic quantum mechanics, considering the particularly simple case of a quantum system with only one degree of freedom. After deriving the expression for the transition amplitudes by means of the sum over paths, we will show how this method also allows calculations of the Green’s functions in quantum mechanics. We will conclude the chapter by showing how these results can be extended to the case of systems with more discrete degrees of freedom and to field theory.
We examine under this heading the transition amplitude in the Feynman path integral, which we will discuss shortly and will be reduced to an integral over the continuum degrees of freedom, such as a set of p i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_1.jpg"/> and q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_2.jpg"/> . This will involve harmonic (non-interacting) systems that are reduced to the problem of multiple Gaussian integrals over potentially coupled degrees of freedom, q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_3.jpg"/> that we develop appropriate Feynman path integral tools for their solution. Considering that the harmonic oscillator is one of the central objects of theoretical physics then Gaussian integrals are likely to be the key tools of theoretical physics. This will be demonstrated utilizing Feynman path-integral formulation of quantum mechanics where Gaussian integrals will be pivotal for computation in quantum mechanics, quantum statistical mechanics and generally in quantum field theory. It will be interesting to introduce the Maslov correction [ 11 , 12 ] to the wave function which is a jump of π 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_4.jpg"/> in the phase when a system passes through a caustic point, a phenomenon related to the second variation as well as to the geometry of paths. This phenomenon is applicable to any system using the quasi-classical approximation. The harmonic oscillator will be one of the objects amongst others to illustrate this.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
To introduce the formalism of the functional integral in quantum mechanics, this chapter starts from the definition of the temporal evolution operator. It is important to emphasize that in the classical limit is not only the classic trajectory that have the grater contribution, but all the paths close to this also contribute. The chapter shows how integral formulation is perfectly consistent with Schrödinger's time-dependent differential equation. It retrieves this equation from definition of transition amplitude as a path integral. The chapter discusses functional derivative generalized to the case of the ordinary functional derivative. A very useful tool to solve problems related to functional integrals is the Wick rotation or Euclidean rotation. In quantum mechanics there are few problems we can solve exactly, and very often, when possible, we use perturbation theory to find an approximate solution. The chapter follows this approach with formalism of functional integration and construct a perturbation theory from which can be obtained a physical interpretation.
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