文档介绍：Quantum Field Theory of Many-body Systems
– from the Origin of Sound
to an Origin of Light and Fermions
Xiao-Gang Wen
Department of Physics, MIT
May 28, 2004
Chapter 11
Tensor category theory of
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 anize 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, symmetry
breaking, and order parameters, and the mathematical theory of groups. These tools allow us to solve two
important problems in the study of quantum many-particle systems. First, they lead to a classification of all
symmetry-breaking/particle-condensed states. For example, we know that there are only 230 different crys-
tal phases in t