Section
Quantum Statistical Mechanics
Quantum ensembles and KMS states
Let be a finite-dimensional complex Hilbert space. An observable algebra is a unital -subalgebra
where is the algebra of operators on . Thus contains the identity and is closed under linear combinations, products, and adjoints. The physical observables are the self-adjoint elements of .
A quantum Hamiltonian is a densely defined self-adjoint operator on a nonzero complex Hilbert space , representing the system's energy. When is bounded and belongs to an observable algebra, it is an observable; in finite dimension it has a spectral decomposition
where the real numbers are energy levels and the are their spectral projections.
A finite quantum statistical system consists of a triple , where is a finite-dimensional complex Hilbert space, is a unital observable algebra, and is the Hamiltonian. A state is a positive normalized functional on ; on it is represented uniquely by a density operator.
Let be a Hamiltonian on a finite-dimensional Hilbert space and let . The quantum Gibbs state at inverse temperature is
It is a density operator, and its expectation functional is .
Let be a density operator on a finite-dimensional Hilbert space and a self-adjoint element of the observable algebra. The expectation value of in the state is
For a pure state , this is .
Let be a Hamiltonian on a finite-dimensional Hilbert space and let . The quantum partition function is
If the eigenvalues of are , counted with multiplicity, then . It is finite and strictly positive.
Let for a finite-dimensional Hilbert space, let , fix , and use units with . Write
A state satisfies the -KMS condition if, for every , there is a function continuous on , analytic in its interior, and satisfying
for every .