翻訳と辞書 Words near each other ・ KMRI ・ KMRJ ・ KMRK-FM ・ KMRN ・ KMRO ・ KMRQ ・ KMRR ・ KMRS ・ KMRT ・ KMRX ・ KMRY ・ KMRZ-FM ・ KMS ・ KMS (hypertext) ・ KMS Fusion ・ KMS state ・ KMS Tools ・ KMSA ・ KMSB ・ KMSC ・ KMSC (AM) ・ KMSC (FM) ・ Kmscon ・ KMSD ・ KMSE ・ KMSG-LD ・ KMSI ・ KMSJ-LP ・ KMSL ・ KMSM-FM Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク KMS state ： ウィキペディア英語版 KMS state In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo-Martin-Schwinger state or, more commonly, a KMS state: a state satisfying the KMS condition. introduced the condition, used it to define thermodynamic Greens functions, and used the condition to define equilibrium states and called it the KMS condition. == Preliminaries == The 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 :$\backslash rho\_=\backslash frac\backslash left(e^\; \backslash right)\}=\backslash frac$ where ''H'' is the Hamiltonian operator and ''N'' is the particle number operator (or charge operator, if we wish to be more general) and :$Z(\backslash beta,\backslash mu)\backslash \; \backslash stackrel\backslash \; \backslash mathrm\backslash left(e^\; \backslash right)$ is the partition function. We assume that ''N'' commutes with ''H,'' or in other words, that particle number is conserved. In the Heisenberg picture, the density matrix does not change with time, but the operators are time-dependent. In particular, translating an operator ''A'' by τ into the future gives the operator :$\backslash alpha\_\backslash tau(A)\backslash \; \backslash stackrel\backslash \; e^A\; e^$. A combination of time translation with an internal symmetry "rotation" gives the more general :$\backslash alpha^\_(A)\backslash \; \backslash stackrel\backslash \; e^\; A\; e^$ A bit of algebraic manipulation shows that the expected values :$\backslash langle\backslash alpha^\backslash mu\_\backslash tau(A)B\backslash rangle\_=\backslash mathrm\backslash left(\backslash alpha^\backslash mu\_\backslash tau(A)B\backslash right)=\backslash mathrm\backslash left(B\; \backslash alpha^\backslash mu\_(A)\backslash right)=\backslash langle\; B\backslash alpha^\backslash mu\_(A)\backslash rangle\_$ for any two operators ''A'' and ''B'' and any real τ (we are working with finite-dimensional Hilbert spaces after all). We used the fact that the density matrix commutes with any function of (''H''-μ''N'') and that the trace is cyclic. As hinted at earlier, with infinite dimensional Hilbert spaces, we run into a lot of problems like phase transitions, spontaneous symmetry breaking, operators that are not trace class, divergent partition functions, etc.. The complex functions of ''z'', $\backslash langle\backslash alpha^\backslash mu\_z(A)B\backslash rangle$ converges in the complex strip whereas $\backslash langle\; B\backslash alpha^\backslash mu\_z(A)\backslash rangle$ converges in the complex strip if we make certain technical assumptions like the spectrum of ''H''-μ''N'' is bounded from below and its density does not increase exponentially (see Hagedorn temperature). If the functions converge, then they have to be analytic within the strip they are defined over as their derivatives, :$\backslash frac\backslash langle\backslash alpha^\backslash mu\_z(A)B\backslash rangle=i\backslash langle\backslash alpha^\backslash mu\_z\backslash left(\backslash left(N,A\backslash right)\backslash right)B\backslash rangle$ and :$\backslash frac\backslash langle\; B\backslash alpha^\backslash mu\_z(A)\backslash rangle\; =\; i\backslash langle\; B\backslash alpha^\backslash mu\_z\backslash left(\backslash left(N,A\backslash right)\backslash right)\backslash rangle$ exist. However, we can still define a KMS state as any state satisfying :$\backslash langle\; \backslash alpha^\backslash mu\_\backslash tau(A)B\backslash rangle=\backslash langle\; B\backslash alpha^\backslash mu\_(A)\backslash rangle$ with $\backslash langle\backslash alpha^\backslash mu\_z(A)B\backslash rangle$ and $\backslash langle\; B\backslash alpha^\backslash mu\_z(A)\backslash rangle$ being analytic functions of ''z'' within their domain strips. $\backslash langle\backslash alpha^\backslash mu\_\backslash tau(A)B\backslash rangle$ and $\backslash langle\; B\backslash alpha^\backslash mu\_(A)\backslash rangle$ 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 is not 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「KMS state」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.