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 :\rho_{\beta,\mu}=\frac{\mathrm{e}^{-\beta \left(H - \mu N\right)}}{\mathrm{Tr}\left[ \mathrm{e}^{-\beta \left(H - \mu N\right)} \right]}=\frac{\mathrm{e}^{-\beta \left(H - \mu N\right)}}{Z(\beta,\mu)} 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(\beta,\mu)\ \stackrel{\mathrm{def}}{=}\ \mathrm{Tr}\left[ \mathrm{e}^{-\beta \left(H-\mu N\right)} \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 :\alpha_\tau(A)\ \stackrel{\mathrm{def}}{=}\ \mathrm{e}^{iH\tau}A \mathrm{e}^{-iH\tau}. A combination of
time translation with an
internal symmetry "rotation" gives the more general :\alpha^{\mu}_{\tau}(A)\ \stackrel{\mathrm{def}}{=}\ \mathrm{e}^{i\left(H-\mu N\right)\tau} A \mathrm{e}^{-i\left(H - \mu N\right)\tau} A bit of algebraic manipulation shows that the
expected values :\left\langle\alpha^\mu_\tau(A)B\right\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] = \left\langle B\alpha^\mu_{\tau+i\beta}(A)\right\rangle_{\beta,\mu} 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, \left\langle\alpha^\mu_z(A)B\right\rangle converges in the complex strip -\beta whereas \left\langle B\alpha^\mu_z(A)\right\rangle converges in the complex strip 0 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, :\frac{\mathrm{d}}{\mathrm{d}z}\left\langle\alpha^\mu_z(A)B\right\rangle = i\left\langle\alpha^\mu_z\left(\left[H - \mu N, A\right]\right)B\right\rangle and :\frac{\mathrm{d}}{\mathrm{d}z}\left\langle B\alpha^\mu_z(A)\right\rangle = i\left\langle B\alpha^\mu_z\left(\left[H - \mu N, A\right]\right)\right\rangle exist. However, we can still define a
KMS state as any state satisfying :\left\langle \alpha^\mu_\tau(A)B\right\rangle = \left\langle B\alpha^\mu_{\tau+i\beta}(A)\right\rangle with \left\langle\alpha^\mu_z(A)B\right\rangle and \left\langle B\alpha^\mu_z(A)\right\rangle being analytic functions of
z within their domain strips. \left\langle\alpha^\mu_\tau(A)B\right\rangle and \left\langle B\alpha^\mu_{\tau+i\beta}(A)\right\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. ==See also==