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.
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Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)
Besse A.L.: Einstein Manifolds. Springer, Berlin (1987)
Chavel I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge (1993)
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., 1986
Golden S.: Lower bounds for the Helmholtz function. Phys. Rev. B 137, 1127–1128 (1965)
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)
Lawson B.H., Michelsohn M.-L.: Spin Geometry. Princeton University Press, Princeton, NJ (1989)
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)
Lichnerowicz A.: Spineurs harmoniques. C. R. Acad. Sci. Paris 257, 7–9 (1963)
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)
Schoen R., Yau S.-T.: Lectures on Differential Geometry. International Press, Cambridge USA (1994)
Simon, B.: Trace Ideals and their Applications. Second edition. Providence, RI: Amer. Math. Soc. 2005
Simon, B.: Functional Integration and Quantum Physics. Second edition. AMS Chelsea Publishing, Providence, RI: Amer. Math. Soc., 2005
Symanzik K.: Proof and refinements of an inequality of Feynman. J. Math. Phys. 6, 1155–1156 (1965)
Thompson C.J.: Inequality with applications in statistical mechanics. J. Math. Phys. 6, 1812–1813 (1965)
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Communicated by A. Connes
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Bär, C., Pfäffle, F. Asymptotic Heat Kernel Expansion in the Semi-Classical Limit. Commun. Math. Phys. 294, 731–744 (2010). https://doi.org/10.1007/s00220-009-0973-3
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DOI: https://doi.org/10.1007/s00220-009-0973-3