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

##### 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
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