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) · Zbl 0589.53037
[3] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Ann. Phys., 111, 61-151 (1978) · Zbl 0377.53025
[4] Bordemann, M.; Neumaier, N.; Waldmann, S., Commun. Math. Phys., 198, 363-396 (1998) · Zbl 0968.53056
[5] Bordemann, M.; Römer, H.; Waldmann, S., Lett. Math. Phys., 45, 49-61 (1998) · Zbl 0951.53057
[6] Bordemann, M.; Waldmann, S., Formal GNS Construction and WKB Expansion in Deformation Quantization, (Sternheimer, D.; Rawnsley, J.; Gutt, S., Deformation Theory and Symplectic Geometry, Mathematical Physics Studies, 20 (1997), Kluwer: Kluwer Dordrecht), 315-319 · Zbl 1166.53321
[7] Bordemann, M.; Waldmann, S., Commun. Math. Phys., 195, 549-583 (1998) · Zbl 0989.53057
[8] DeWilde, M.; Lecomte, P. B.A., Lett. Math. Phys., 7, 487-496 (1983) · Zbl 0526.58023
[9] Fedosov, B. V., J. Diff. Geom., 40, 213-238 (1994) · Zbl 0812.53034
[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) · Zbl 0887.58050
[12] Nest, R.; Tsygan, B., Adv. Math., 113, 151-205 (1995) · Zbl 0837.58029
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.