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In the [[statistical mechanics]] of [[quantum mechanics|quantum mechanical]] systems and [[quantum field theory]], the properties of a system in thermal equilibrium can be described by a mathematical object called a [[Ryogo Kubo|Kubo]]-Martin-[[Julian Schwinger|Schwinger]] state or, more commonly, a '''KMS state''': a state satisfying the '''KMS condition'''. {{harvtxt|Kubo|1957}} introduced the condition, {{harvtxt|Martin|Schwinger|1959}} used it to define [[thermodynamic]] [[Green's function|Greens function]]s, and
{{harvs|txt | last1=Haag | first1=Rudolf | author1-link=Rudolf Haag | last2=Winnink | first2=M. | last3=Hugenholtz | first3=N. M. | year=1967 }} 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 transition]]s or [[spontaneous symmetry breaking]]. The [[density matrix]] of a '''thermal state''' is given by

:<math>\rho_{\beta,\mu}=\frac{e^{-\beta \left(H-\mu N\right)}}{\mathrm{Tr}\left[ e^{-\beta \left(H-\mu N\right)} \right]}=\frac{e^{-\beta \left(H-\mu N\right)}}{Z(\beta,\mu)}</math>

where ''H'' is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] [[Operator (physics)|operator]] and ''N'' is the [[particle number operator]] (or [[charge (physics)|charge]] operator, if we wish to be more general) and

:<math>Z(\beta,\mu)\ \stackrel{\mathrm{def}}{=}\ \mathrm{Tr}\left[ e^{-\beta \left(H-\mu N\right)} \right]</math>

is the [[partition function (quantum field theory)|partition function]]. We assume that ''N'' commutes with ''H,'' or in other words, that particle number is [[Conservation law|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

:<math>\alpha_\tau(A)\ \stackrel{\mathrm{def}}{=}\ e^{iH\tau}A e^{-iH\tau}</math>.

A combination of time translation with an [[internal symmetry]] "rotation" gives the more general

:<math>\alpha^{\mu}_{\tau}(A)\ \stackrel{\mathrm{def}}{=}\ e^{i\left(H-\mu N\right)\tau} A e^{-i\left(H-\mu N\right)\tau}</math>

A bit of algebraic manipulation shows that the [[expected value]]s

:<math>\langle\alpha^\mu_\tau(A)B\rangle_{\beta,\mu}=\mathrm{Tr}\left[\rho \alpha^\mu_\tau(A)B\right]=\mathrm{Tr}\left[\rho B \alpha^\mu_{\tau+i\beta}(A)\right]=\langle B\alpha^\mu_{\tau+i\beta}(A)\rangle_{\beta,\mu}</math>

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 class|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 function]]s of ''z'', <math>\langle\alpha^\mu_z(A)B\rangle</math> converges in the complex strip <math>-\beta < \Im{z} < 0</math>
whereas <math>\langle B\alpha^\mu_z(A)\rangle</math> converges in the complex strip
<math>0 < \Im{z} < \beta</math>
if we make certain technical assumptions like the [[spectrum of an operator|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 function|analytic]] within the strip they are defined over as their derivatives,

:<math>\frac{d}{dz}\langle\alpha^\mu_z(A)B\rangle=i\langle\alpha^\mu_z\left(\left[H-\mu N,A\right]\right)B\rangle</math>

and

:<math>\frac{d}{dz}\langle B\alpha^\mu_z(A)\rangle = i\langle B\alpha^\mu_z\left(\left[H-\mu N,A\right]\right)\rangle</math>

exist.

However, we can still define a '''KMS state''' as any state satisfying

:<math>\langle \alpha^\mu_\tau(A)B\rangle=\langle B\alpha^\mu_{\tau+i\beta}(A)\rangle</math>

with <math>\langle\alpha^\mu_z(A)B\rangle</math> and <math>\langle B\alpha^\mu_z(A)\rangle</math> being analytic functions of ''z'' within their domain strips.

<math>\langle\alpha^\mu_\tau(A)B\rangle</math> and <math>\langle B\alpha^\mu_{\tau+i\beta}(A)\rangle</math> are the boundary [[Distribution (mathematics)|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 transformation]]s involving time translations (or time translations and an [[internal symmetry]] transformation for nonzero chemical potentials) via the [[Tomita–Takesaki theory]].

==References==

*{{Citation | last1=Haag | first1=Rudolf | author1-link=Rudolf Haag | last2=Winnink | first2=M. | last3=Hugenholtz | first3=N. M. | title=On the equilibrium states in quantum statistical mechanics | doi=10.1007/BF01646342 | id={{MR|0219283}} | year=1967 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=5 | pages=215–236|bibcode = 1967CMaPh...5..215H }}
*{{citation | last = Kubo | first = R. | authorlink = Ryogo Kubo | title = Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems |journal = [[Journal of the Physical Society of Japan]] |volume = 12 | issue = 6 | pages = 570–586 | year = 1957 | doi = 10.1143/JPSJ.12.570}}
*{{citation | last = Martin | first = Paul C. | last2 = Schwinger | first2 = Julian | title = Theory of Many-Particle Systems. I | year = 1959 | journal = [[Physical Review]] | volume = 115 | issue = 6 | pages = 1342–1373 | doi = 10.1103/PhysRev.115.1342|bibcode = 1959PhRv..115.1342M }}

[[Category:Statistical mechanics]]
[[Category:Quantum field theory]]

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