KMS states and star product quantization. (English) Zbl 0964.81045

Summary: The concept of KMS states used to describe thermodynamics is transferred to deformation quantization by defining formal KMS states on the star product algebra of formal power series in \(\hbar\) with coefficients in the smooth functions with compact support on phase space endowed with a star product. Then we prove the existence and uniqueness of these KMS states in the case of a connected phase space for any inverse temperature \(\beta\), and show that they can be described in terms of the star product trace and a certain star exponential analog to the usual Boltzmann factor resulting a formal analogue of the Gibbs states.


81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
Full Text: DOI


[1] Basart, H.; Flato, M.; Lichnerowicz, A.; Sternheimer, D., Lett. math. phys., 8, 394-483, (1984)
[2] Basart, H.; Lichnerowicz, A., Lett. math. phys., 10, 167-177, (1985)
[3] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Ann. phys., 111, 61-151, (1978)
[4] Bordemann, M.; Neumaier, N.; Waldmann, S., Commun. math. phys., 198, 363-396, (1998)
[5] Bordemann, M.; Römer, H.; Waldmann, S., Lett. math. phys., 45, 49-61, (1998)
[6] Bordemann, M.; Waldmann, S., Formal GNS construction and WKB expansion in deformation quantization, (), 315-319 · Zbl 1166.53321
[7] Bordemann, M.; Waldmann, S., Commun. math. phys., 195, 549-583, (1998)
[8] DeWilde, M.; Lecomte, P.B.A., Lett. math. phys., 7, 487-496, (1983)
[9] Fedosov, B.V., J. diff. geom., 40, 213-238, (1994)
[10] Kontsevich, M., deformation quantization of Poisson manifolds I, Preprint q-alg/9709040, (1997)
[11] Nest, R.; Tsygan, B., Commun. math. phys., 172, 223-262, (1995)
[12] Nest, R.; Tsygan, B., Adv. math., 113, 151-205, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.