KMSTheorem

Consider a C-algebra $\mathcal{A}$, which represents the algebra of observables in a quantum system. Let $\omega$ be a state on $\mathcal{A}$. A state is a positive linear functional that satisfies $\omega(\mathbb{I}) = 1$, where $\mathbb{I}$ is the identity in $\mathcal{A}$. Furthermore, consider a one-parameter group of *-automorphisms ${\alpha_t}{t \in \mathbb{R}}$ on $\mathcal{A}$, representing the time evolution of the system, where $\alpha{t+s} = \alpha_t \circ \alpha_s$ for all $t, s \in \mathbb{R}$.

The state $\omega$ is said to satisfy the Kubo-Martin-Schwinger (KMS) condition at an inverse temperature $\beta$ (where $\beta$ is a positive real number) if and only if for every pair of elements $A, B \in \mathcal{A}$, there exists a complex function $F_{A,B}$ defined and analytic on the strip ${z \in \mathbb{C} : 0 < \text{Im}(z) < \beta}$ and continuous on its closure. This function $F_{A,B}$ must satisfy the following boundary conditions: For all $t \in \mathbb{R}$, it holds that

$$F_{A,B}(t) = \omega(A \alpha_t(B))$$

and

$$F_{A,B}(t + i\beta) = \omega(\alpha_t(B) A)$$

This condition effectively links the quantum mechanical description of a system in thermal equilibrium at temperature $T$ (related to $\beta$ via $\beta = \frac{1}{kT}$, with $k$ being the Boltzmann constant) with its statistical mechanical properties. The KMS condition is a cornerstone in the mathematical formulation of quantum statistical mechanics and quantum field theory, providing a rigorous framework to discuss thermal states.