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

Bordemann, Martin; Römer, Hartmann; Waldmann, Stefan

Rep. Math. Phys. 44, No. 1-2, 45-52 (1999).

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.

MSC:

81S10	Geometry and quantization, symplectic methods
53D55	Deformation quantization, star products

Keywords:

deformation quantization; existence and uniqueness; formal analogue of the Gibbs states

References:

[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)