Quantum Field Theory of Many-Body Systems

condensation Extended objects, such as strings and membranes, have been studied for many years in the context of sta-tistical physics. In these systems, quantum effects are typically negligible, and the extended objects can be treated classically. Yet it is natural to wonder how strings and membranes behave in the quantum regime. In this chapter, we will investigate the properties of one dimensional, string-like, objects with large quan-tum fluctuations. Our motivation is both intellectual curiosity and (as we will see) the connection between quantum strings and topological/quantum orders in condensed matter systems. It is useful to organize our discussion using the analogy to the well understood theory of quantum par-ticles. One of the most remarkable phenomena in quantum many-particle systems is particle condensation. We can think of particle condensed states as special ground states where all the particles are described by the same quantum wave function. In some sense, all the symmetry breaking phases examples of particle con-densation: we can view the order parameter that characterizes a symmetry breaking phase as the condensed wave function of certain “effective particles. ” According to this point of view, Landau’s theory [Landau (1937)] for symmetry breaking phases is really a theory of “particle ” condensation. The theory of particle condensation is based on the physical concepts of long range order, symmetr