KMS Condition

The Kubo-Martin-Schwinger (KMS) condition is a kind of boundary-value condition which naturally emerges in quantum statistical mechanics and related areas.

Given a quantum system  with finite dimensional Hilbert space , define the function  as

(1)

where  is the imaginary unit and where  is the Hamiltonian, i.e., the sum of the kinetic energies of all the particles in  plus the potential energy of the particles associated with . Next, for any real number , define the thermal equilibrium  as

(2)

where  denotes the matrix trace. From  and , one can define the so-called equilibrium correlation function  where

(3)

whereby the KMS boundary condition says that

(4)

In particular, this identity relates to the state  the values of the analytic function  on the boundary of the strip

(5)

where here,  denotes the imaginary part of  and  denotes the signum function applied to .

In various literature, the KMS boundary condition is stated in sometimes-different contexts. For example, the identity () is sometimes written with respect to integration, yielding

(6)

where here,  is used as shorthand for . In other literature (e.g., Araki and Miyata 1968), the condition looks different still.

REFERENCES:

Araki, H. and Miyata, H. "On KMS Boundary Condition." Publ. RIMS, Kyoto Univ. Ser. A 4, 373-385, 1968.

Cohen, J. S.; Daniëls, H. A. M.; and Winnink, M. "On Generalizations of the KMS-Boundary Condition." Commun. Math. Phys. 84, 449-458, 1982.

Derezński, J. and Pillet, C. "KMS States." https://pillet.univ-tln.fr/data/pdf/KMS-states.pdf.

Nave, C. R. "The Hamiltonian in Quantum Mechanics." HyperPhysics. 2012. https://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hamil.html.