nLab KMS state



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT

Measure and probability theory




A KMS state (named after Kubo 57, Martin-Schwinger 59) is a vacuum state for a relativistic quantum field theory describing thermal equilibrium at some positive temperature TT.

The final formulation of the KMS condition, now widely accepted, is due to Haag-Hugenholtz-Winnink 67.

Under Wick rotation (if applicable) a relativistic field theory on Minkowski spacetime d,1\mathbb{R}^{d,1} in a KMS state at temperature TT is identified with a Euclidean field theory on the product space d×S β 1\mathbb{R}^d \times S^1_{\beta} with a “compact/periodic Euclidean time” circle S β 1S^1_\beta of length β=1/T\beta = 1/T, this is essentially the KMS condition (see Fulling-Ruijsenaars 87). This relation is the basis of thermal quantum field theory, see there for more.

quantum probability theoryobservables and states


The formulation of the KMS condition is due to

  • R. Kubo Statistical-Mechanical Theory of Irreversible Processes I. General Theory and Simple Applications to Magnetic and Conduction Problems, Journal of the Physical Society of Japan 12, 570-586 1957

  • Paul C. Martin, Julian Schwinger, Theory of Many-Particle Systems. I, Physical Review 115, 1342-1373 (1959)

and found its final, now generally accepted, form in

  • Rudolf Haag, N. M. Hugenholtz, M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. Volume 5, Number 3 (1967), 215-236 (euclid:1103840050)

A good quick survey, putting the KMS states into context with Wick rotation and thermal quantum field theory, is

  • S.A. Fulling, S.N.M. Ruijsenaars, Temperature, periodicity and horizons, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 (pdf, doi:10.1016/0370-1573(87)90136-0)

See also

Last revised on February 8, 2020 at 10:49:30. See the history of this page for a list of all contributions to it.