An explicit expression is presented for the path integral of an harmonic oscillator interacting with itself in the past. The time dependence of the memory kernel is assumed to be exponential. The relation with the Feynman trial action for the variational calculation of the polaron ground state energy is discussed.
Using the prodistribution theory proposed by DeWitt-Morette [C. DeWitt-Morette, Commun. Math. Phys. 28, 47 (1972); C. DeWitt-Morette, A. Maheshwari, and B. Nelson, Phys. Rep. 50, 257 (1979)], the path integration of a time-dependent forced harmonic oscillator with a two-time quadratic action has been given in terms of the solutions of some integrodifferential equations. We then evaluate explicitly both the classical path and the propagator for the specific kernel introduced by Feynman in the polaron problem. Our results include the previous known results as special cases.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to nd a complete set of wave functions for the oscillator. Harmonic oscillators with dierent (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) invariance under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) {\em invariance} under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.
Several path integrals are exactly evaluated here using Gaussian path integration. A few general ideas are illustrated using the advantage of being able to exactly evaluate Gaussian path integrals. Path integrals defined over a particular collection of allowed indeterminate paths can sometimes be represented by a Fourier expansion of the paths. This leads to two important techniques for performing path integrations: Expanding the action about the classical solution of the Lagrangian; Expanding the degree of freedom in a Fourier expansion of the allowed paths. Various cases are considered to illustrate the usage of classical solutions and Fourier expansions, and these also provide a set of relatively simple examples to familiarize oneself with the nuts and bolts of the path integral. The Lagrangian of the simple harmonic oscillator is used for all of the following examples; all the computations are carried out explicitly and exactly. The following different cases are considered: • Correlators of exponential functions of the degree of freedom are discussed in Section 12.1. • The generating functional for periodic paths is evaluated in Section 12.2. • The path integral required for evaluating the normalization constant for the oscillator evolution kernel is discussed in Section 12.3. The path integral entails summing over all paths that start from and return to the same fixed position. • Section 12.4 discusses the evolution kernel for a particle starting at an initial position x i and, after time τ, having a final position that is indeterminate.