nLab KMS state

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.

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

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)

See also

• Wikipedia KMS state, Tomita-Takesaki theory

• Martin Bordemann, Hartmann Römer, Stefan Waldmann, KMS states and star product quantization, Reports on Mathematical Physics 44, n. 1-2 (1999) 45-52, doi

• W. Pusz, S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58 (1978), no. 3, 273–290, MR471796, euclid

• S. L. Woronowicz, On the existence of KMS states, Letters on Math. Phys. 10, no 1, 29–31 (1985) MR86m:46066, euclid, doi

• Alain Connes, Matilde Marcolli, Noncommutative geometry, quantum fields and motives, Amer. Math. Soc. Coll. Publ. 55, 2008. xxii+785 pp. gBooks, draft pdf

• Matthew Eric Bassett, math blog: KMS states and symmetries, Class Field Theory and the Bost-Connes System

• Neverendingbooks blog KMS, Gibbs and zeta function