High temperature heat kernel expansion and~its~applications

V. V. Mikheyev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2013), pp. 369-378.

Résumé

The algorithm constructed to build the high-temperature heat kernel expansion and the statistic sum on the noncompact Lie groups manifolds is discussed in the article. The method is based on the formalism of non-commutative integration which originated from the coadjoint orbits' approach to the problems of integration and quantization. Applications of presented method to the problems of quantum statistic mechanics and quantum field theory are also discussed.

Détail

Keywords: heat kernel, statistic sum, partition function, non-commutative integration, high-temperature asymptotics, effective lagrangian.

Bibliographie

[1] N. E. Hurt, Geometric quantization in action, Mathematics and Its Applications (East European Series), 8, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1983, xiv+336 pp. ; N. Khart, Geometricheskoe kvantovanie v deistvii, Mir, M., 1985, 343 pp. | MR | Zbl | MR

[2] V. N. Shapovalov, “Symmetry and separation of variables in a linear differential equation of second order. I”, Sov. Phys. J., 21:5 (1978), 645–650 | DOI | MR

[3] V. N. Shapovalov, “Symmetry and separation of variables in second-order differential equations. II”, Sov. Phys. J., 21:6 (1978), 693–695 | DOI | MR

[4] V. N. Shapovalov, “Separation of variables in a second-order linear differential equation”, Differ. Equ, 16:10 (1981), 1212–1220 | MR | Zbl | Zbl

[5] D. V. Vassilevich, “Heat kernel expansion: user's manual”, Phys. Rep., 338:5–6 (2003), 279–360 | DOI | MR

[6] A. V. Shapovalov, I. V. Shirokov, “Noncommutative integration of linear differential equations”, Theoret. and Math. Phys., 104:2 (1995), 921–934 | DOI | MR | Zbl

[7] I. V. Shirokov, “Darboux coordinates on $K$-orbits and the spectra of Casimir operators on Lie groups”, Theoret. and Math. Phys., 123:3 (2000), 754–767 | DOI | DOI | MR | Zbl

[8] A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, 220, Springer Verlag, Berlin, New York, Heidelberg, 1976, xi+315 pp. | DOI | MR | MR | Zbl | Zbl

[9] A. O. Barut, R. Razcka, Theory of group representations and applications. 2nd rev. ed., World Scientific, Singapore, 1986, xix+717 pp. | MR | Zbl

[10] S. P. Baranovsky, V. V. Mikheyev, I. V. Shirokov, “Quantum Hamiltonian systems on K-orbits: Semiclassical spectrum of the asymmetric top”, Theoret. and Math. Phys., 129:1 (2001), 1311–1319 | DOI | DOI | MR | MR | Zbl

[11] V. Mikheyev., I. Shirokov, “Building of heat kernel on non-compact homogeneous spaces”, EJTP, Electron. J. Theor. Phys., 3:13 (2006), 99–108 | Zbl

[12] V. V. Mikheyev., I. V. Shirokov, “Method of orbits of coadjoint representation in thermodynamics of noncompact Lie groups”, Russ. Phys. J., 50:3 (2007), 290–295 | DOI | MR | Zbl

[13] V. V. Mikheyev, I. V. Shirokov, “Application of coadjoint orbits in the thermodynamics of non-compact manifolds”, EJTP, Electron. J. Theor. Phys., 2:7 (2005), 1–10 | Zbl

[14] L. D. Landau; E. M. Lifshits, Theoretical physics. In 10 vols, v. 5, Statistical physics. Part 1, Nauka, Moscow, 1995, 606 pp. | Zbl

[15] A. A. Grib, S. G. Mamayev, V. M. Mostepanenko, Vacuum quantum effects in strong fields, Friedmann Lab. Publ., St. Petrsburg, 1994, 361 pp.

[16] N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space. Corrected reprint of the 1982 original, Cambridge University Press, Cambridge, 1984, ix+340 pp. | MR | Zbl

[17] O. A. Chalykh, A. P. Veselov, “Integrability and Huygens' principle on symmetric spaces”, Comm. Math. Phys., 178:2 (1996), 311–338 | DOI | MR | Zbl