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Communications in Mathematical Physics

, Volume 294, Issue 3, pp 731–744 | Cite as

Asymptotic Heat Kernel Expansion in the Semi-Classical Limit

  • Christian BärEmail author
  • Frank Pfäffle
Article

Abstract

Let \({H_\hbar = \hbar^{2}L +V}\), where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of \({H_\hbar}\) as \({\hbar \searrow 0}\). As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive \({\hbar}\) by the classical partition function.

Keywords

Partition Function Asymptotic Expansion Dirac Operator Heat Kernel Compact Riemannian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)zbMATHGoogle Scholar
  2. 2.
    Besse A.L.: Einstein Manifolds. Springer, Berlin (1987)zbMATHGoogle Scholar
  3. 3.
    Chavel I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  4. 4.
    Elworthy, K.D., Ndumu, M.N., Trumam, A.: An elementary inequality for the heat kernel on a Riemannian manifold and the classical limit of the quantum partition function. Pitman Res. Notes Math. Ser., 150, Harlow: Longman Sci. Tech., 1986Google Scholar
  5. 5.
    Golden S.: Lower bounds for the Helmholtz function. Phys. Rev. B 137, 1127–1128 (1965)CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Hess H., Schrader R., Uhlenbrock D.A.: Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds. J. Diff. Geom. 15, 27–37 (1980)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Lawson B.H., Michelsohn M.-L.: Spin Geometry. Princeton University Press, Princeton, NJ (1989)zbMATHGoogle Scholar
  8. 8.
    Lenard, A.: Generalization of the Golden-Thompson inequality Tr(e A e B) ≥ Tr e A+B. Indiana Univ. Math. J. 21, 457–467 (1971/1972)Google Scholar
  9. 9.
    Lichnerowicz A.: Spineurs harmoniques. C. R. Acad. Sci. Paris 257, 7–9 (1963)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Schrader R., Taylor M.E.: Small \({\hbar}\) asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials. Commun. Math. Phys. 92(4), 555–594 (1994)CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Schoen R., Yau S.-T.: Lectures on Differential Geometry. International Press, Cambridge USA (1994)zbMATHGoogle Scholar
  12. 12.
    Simon, B.: Trace Ideals and their Applications. Second edition. Providence, RI: Amer. Math. Soc. 2005Google Scholar
  13. 13.
    Simon, B.: Functional Integration and Quantum Physics. Second edition. AMS Chelsea Publishing, Providence, RI: Amer. Math. Soc., 2005Google Scholar
  14. 14.
    Symanzik K.: Proof and refinements of an inequality of Feynman. J. Math. Phys. 6, 1155–1156 (1965)CrossRefADSGoogle Scholar
  15. 15.
    Thompson C.J.: Inequality with applications in statistical mechanics. J. Math. Phys. 6, 1812–1813 (1965)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Universität Potsdam, Institut für MathematikPotsdamGermany

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