Special relativity implies that mass can be created from energy — it implies particle production. A relativistic quantum theory is of necessity a theory of many particles. We shall therefore first describe systems of many identical particles but in the simpler context of non-relativistic theory. As we shall soon see, many-particle systems can be described by field operators that create and destroy particles, with these operators acting in a vastly enlarged vector space that spans states with an arbitrary number of particles, including a state of no particles — the vacuum state. Both Bose and Fermi statistics are handled in this way, and the resulting formalism is compact, convenient, and powerful. For the Bose case, the functional integral representation of the quantum many-particle system is direct generalization of the coherent-state functional integral developed in the previous chapter. To describe the analogous formulation for the Fermi case, we must first develop the theory of anticommuting variables. The power and utility of field operator and functional integral methods will be illustrated by applying it to systems in thermal equilibrium, first to free Fermi and Bose gases, and then to a model of liquid helium which exhibits the phenomenon of spontaneous symmetry breaking, a phenomenon of great importance in condensed matter physics as well as in elementary particle theory.