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 nonzero 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 (4) is sometimes written with respect to integration, yielding
 |
(6)
|
where 
is an entire test function
that decays rapidly and uniformly on every horizontal strip. For 
, the Gibbs state is the normalized
matrix trace, and the KMS condition is the trace
identity 
rather than a condition on an empty strip. In other literature (e.g., Araki and Miyata
1968), the condition looks different still.
