The is a fundamental concept in that describes equilibrium states of quantum systems. It connects physical systems to mathematical structures in theory, providing a rigorous framework for understanding thermal equilibrium in infinite quantum systems.
Introduced in the 1950s, the has become crucial in studying thermal states, , and critical phenomena. It extends the notion of Gibbs states to infinite-dimensional systems, offering powerful tools for analyzing and operator algebras.
Definition of KMS condition
Fundamental concept in quantum statistical mechanics describes equilibrium states of quantum systems
Plays crucial role in von Neumann algebra theory connecting physical systems to mathematical structures
Provides rigorous framework for understanding thermal equilibrium in infinite quantum systems
Origin and history
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Introduced by Kubo, Martin, and Schwinger in 1950s to study quantum statistical mechanics
Emerged from attempts to generalize Gibbs ensembles to infinite systems
Haag, Hugenholtz, and Winnink formalized KMS condition in algebraic quantum field theory in 1967
Named after Kubo, Martin, and Schwinger who independently discovered its importance
Mathematical formulation
Defined for a C*-dynamical system (A, αt) where A is a C*-algebra and αt is a one-parameter group of automorphisms
State ω on A satisfies KMS condition at inverse β if:
ω(Aαt(B))=ω(Bαt−iβ(A))
for all A, B in a dense subset of A and t ∈ ℝ
Involves of correlation functions to complex time
Equivalent to the existence of a periodic boundary condition in imaginary time
Physical interpretation
Describes in quantum systems at finite temperature
Reflects detailed balance between forward and backward processes in time
Encodes information about energy distribution and correlations in the system
Generalizes notion of Gibbs states to infinite-dimensional systems
Provides mathematical framework for understanding thermodynamic limit
Properties of KMS states
Crucial for understanding equilibrium behavior in quantum statistical mechanics
Connect abstract mathematical structures to physical observables in von Neumann algebras
Provide powerful tools for analyzing infinite quantum systems and their phase transitions
Invariance under time evolution
KMS states remain unchanged under time evolution governed by the system's Hamiltonian
Satisfy ω(αt(A)) = ω(A) for all observables A and times t
Reflect time-translation invariance of equilibrium states in quantum systems
Preserve expectation values of observables over time
Crucial for maintaining thermodynamic equilibrium in infinite systems
Uniqueness and existence
KMS state uniqueness depends on system's properties and temperature
Existence guaranteed for finite systems at any positive temperature
Infinite systems may have multiple KMS states (phase transitions)
Uniqueness often holds above critical temperature in many physical systems
Non-uniqueness indicates presence of symmetry breaking or phase coexistence
Relationship to equilibrium states
KMS states provide rigorous definition of thermal equilibrium in quantum systems
Generalize notion of Gibbs states to infinite-dimensional systems