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.
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