# nLab KMS state

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

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 $T$.

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

## References

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)