Path Integrals: in Quantum and Statistical Mechanics

A basic exposition of path integrals, an elegant and powerful technique in relation to non-relativistic quantum mechanics and statistical mechanics, is presented. Some exact propagators are calculated. Several statistical mechanics problems are discussed via a path integral formulation of the statistical density matrix and an application to second quantized Hamiltonians using coherent states is presented. The variational technique is considered with illustrative examples. Some selected recent achievements of the technique are reviewed. The effectiveness of this technique enhances with the evaluation of each new propagator. Central to this presentation have been Gaussian path integrals. The technique of general coordinate transformation and time rescaling is proving to be very valuable as it is even able to deal with originally non integrable problems.