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
![S_beta=<span style=]() {z in C:0<I(zsgn(beta))<|beta|}, " src="https://mathworld.wolfram.com/images/equations/KMSCondition/NumberedEquation5.gif" style="height:18px; width:198px" /> |
(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.
