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Path Integral Representation

Tarō Kashiwa, Yoshio Ohnuki, Masuo Suzuki

Published Jul 3, 1997• 2 citations
Publisher

Abstract

Abstract In this chapter path integral formulae in various representations will be derived by starting from the Schrodinger equation. In order to obtain a c-number expression for the time evolution operator given in terms of the momentum and the coordinate, first we need to deal with the issue of operator ordering. Then a discussion on the Euclidean technique, with which path integration can prove its real worth, that is where path integration can be regarded as a powerful tool for acquiring quantum information, will be presented. Further, alternative path integral expressions other than those obtained by using the eigen states of the momentum and the coordinate will be presented by utilizing the coherent states. Path integral formulae for the Fermi system will be finally presented, together with guidance on Grass man numbers, the indispensable rudiments for doing this.

DOI
10.1093/oso/9780198517719.003.0001

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Abstract In this chapter path integral formulae in various representations will be derived by starting from the Schrodinger equation. In order to obtain a c-number expression for the time evolution operator given in terms of the momentum and the coordinate, first we need to deal with the issue of operator ordering. Then a discussion on the Euclidean technique, with which path integration can prove its real worth, that is where path integration can be regarded as a powerful tool for acquiring quantum information, will be presented. Further, alternative path integral expressions other than those obtained by using the eigen states of the momentum and the coordinate will be presented by utilizing the coherent states. Path integral formulae for the Fermi system will be finally presented, together with guidance on Grass man numbers, the indispensable rudiments for doing this.
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.
Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach. The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space , defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian. To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.
A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.
This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.
Abstract Chapter 4 has introduced the functional integral representation of the quantum statistical operators and thus, formally, evolution in imaginary or Euclidean time. By contrast, to calculate the evolution operator and the scattering S-matrix elements, quantities relevant to particle physics, it is necessary to make a continuation from imaginary to real time. However, the representation of the S-matrix follows from additional considerations. To illustrate the power of the formalism, we show how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Chapter 4 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas, the evolution operator is then given by a holomorphic field integral. A parallel formalism leads to an analogous representation for the evolution operator of a system of non-relativistic fermions.
Abstract This chapter discusses the Feymann path-integral approach to quantum mechanics. First, it derives a path integral expression for the evolution operator. Next, it shows that the classical equations of motion, that is, those obtained from the principle of least action, are obtained from this path integral formulation in the limit where the variation in the action of the problem at hand is small as compared to the Planck’s constant h. Last, it discusses how the WKB approximation is derived from the path integral formulation.