A remark on formal KMS states in deformation quantization. (English) Zbl 0951.53057

Bordemann, Martin; Römer, Hartmann; Waldmann, Stefan

Lett. Math. Phys. 45, No. 1, 49-61 (1998).

The authors define formal KMS states on a deformed algebra of power series of functions with compact support in phase space as \(C[[ \lambda]]\)-linear functionals obeying a formal variant of the usual KMS conditions known in the theory of \(C^*\)-algebras within the framework of deformation quantization. Existence and uniqueness of KMS states for any star product on a connected symplectic manifold for any inverse temperature \(\beta\) with respect to the time development induced by an arbitrary Hamiltonian vector field has been proven. It is shown that no KMS states for \(\beta\neq 0\) exist for symplectic but non-Hamiltonian vector fields.

Reviewer: Samir Musayev (Baku)

MSC:

53D55	Deformation quantization, star products
37N20	Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
37K05	Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)

Keywords:

deformation quantization; KMS states