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## KMS stateThe statistical mechanics of quantum field theory (see thermal quantum field theory) at the inverse temperature β and chemical potential μ can be characterized by a condition called the ## Additional recommended knowledge## PreliminariesThe simplest case to study is that of a finite-dimensional Hilbert space, in which one does not encounter complications like phase transitions or spontaneous symmetry breaking. The density matrix of a thermal state is given by where is the partition function. We assume that In the Heisenberg picture, the density matrix does not change with time, but the operators are time-dependent. In particular, translating an operator - .
A combination of time translation with an internal symmetry "rotation" gives the more general A bit of algebraic manipulation shows that the expected values for any two operators As hinted at earlier, with infinite dimensional Hilbert spaces, we run into a lot of problems like phase transitions, spontaneous symmetry breaking, operators which aren't trace class, divergent partition functions, etc.. The complex functions of and exist. However, we can still define a with and being analytic functions of and are the boundary distribution values of the analytic functions in question. This gives the right large volume, large particle number thermodynamic limit. If there is a phase transition or spontaneous symmetry breaking, the KMS state isn't unique. The density matrix of a KMS state is related to unitary transformations involving time translations (or time translations and an internal symmetry transformation for nonzero chemical potentials) via the Tomita-Takesaki theory. |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "KMS_state". A list of authors is available in Wikipedia. |