The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.