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:

1. Expanding the action about the classical solution of the Lagrangian;

2. 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:

1. • Correlators of exponential functions of the degree of freedom are discussed in Section 12.1.

2. • The generating functional for periodic paths is evaluated in Section 12.2.

3. • 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.

4. • Section 12.4 discusses the evolution kernel for a particle starting at an initial position xi and, after time τ, having a final position that is indeterminate.