Abstract Time-ordered products and connection with the path integral. Applications in quantum mechanical problems. T-products and the scattering matrix. The perturbation expansion. The scalar field.
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.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].