KMS-States

Abstract

In the previous section we analyzed two C*-algebras which describe the kinematics of particle systems and also analyzed the simplest examples of equilibrium states. Now we discontinue this specific analysis and describe instead various general characterizations of equilibrium phenomena. Principally, we investigate the Kubo—Martin—Schwinger, or KMS, condition briefly outlined in the Introduction and used in the calculation of the Gibbs states of the ideal Fermi and Bose gases. Our description of this condition was, hitherto, rather sketchy and this will be corrected in the sequel. Recall that if 𝕬 = ℒC(𝕳), H is a selfadjoint operator on , 𝕳, β ∈ ℝ, and exp {-βH} is of trace-class, then the Gibbs equilibrium state

$$\omega (A) = \frac{{T{r_h}({a^{ - \beta H}}A)}}{{T{r_h}({e^{ - \beta H}})}} $$

formally satisfies the condition

$$\omega (A{\tau _t}(B))\left| {_{t = i\beta }} \right. = \omega (BA)$$

with respect to the automorphism group

$${\tau _t}(A) = {e^{itH}}A{e^{ - itH}}$$

.