Abstract
To a gauge field on a principalG-bundleP→M is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant ħ→0 and a sequence of representations π n ofG is taken. This paper studies the associated quantum partition functions, trace exp (−tH n ), and produces a complete asymptotic expansion, as ħ→0, ħ=1/n, of which the principal term, proportional to the classical partition function, is the familiar classical limit.
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Cahn, R., Taylor, M.: Asymptotic behavior of multiplicities of representations of compact groups. Pac. J. Math.83, 17–28 (1979)
Combes, J., Schrader, R., Seiler, R.: Classical bounds and limits of energy distributions of Hamiltonian operators in electromagnetic fields. Ann. Phys.111, 1–18 (1978)
Dresden, M.: Private communication
Ehrenfest, P.: Bemerkung über die genäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z. Physik45, 453–457 (1927)
Fuller, W., Lenard, A.: Generalized quantum spins, coherent states, and Lieb inequalities. Commun. Math. Phys.67, 69–84 (1979)
Gilmore, R.: The classical limit of quantum nonspin systems. J. Math. Phys.20, 891–893 (1979)
Guillemin, V., Sternberg, S.: On the equations of motion of a classical particle in a Yang-Mills field and the principle of general covariance. Hadronic J.1, 1–32 (1978)
Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys.35, 265–277 (1974)
Hepp, K., Lieb, E.: Equilibrium statistical mechanics of matter interacting with the quantized radiation field. Phys. Rev.8A, 2517–2525 (1973)
Hogreve, H.: Classical limits for quantum particles in external metric fields. Thesis, Freie Universität Berlin, 1983
Hogreve, H., Potthoff, J., Schrader, R.: Classical limits for quantum particles in external Yang-Mills potentials. Commun. Math. Phys.91, 573–598 (1983)
Klauder, J.: The action option and a Feynman quantization of spinor fields in terms of ordinaryC-numbers. Ann. Phys.11, 123–168 (1960)
Landau, L., Lifshitz, E.: Statistical physics. Reading, MA; Addison-Wesley, 1969
Lieb, E.: The classical limit of quantum spin systems. Commun. Math. Phys.31, 327–340 (1973)
Mather, J.: Differentiable invariants. Topology16, 145–155 (1977)
Peremelov, S.: Coherent states for arbitrary Lie groups. J. Math. Phys.26, 222–236 (1972)
Schwartz, A.: Smooth functions invariant under the action of a compact group. Topology14, 16–68 (1975)
Simon, B.: The classical limit of quantum partition functions. Commun. Math. Phys.71, 247–276 (1980)
Simon, B.: Functional integration and quantum physics, New York: Academic Press, 1979
Sternberg, S.: Minimal coupling and the symplectic structure of a classical particle in the presence of a Yang-Mills field. Proc. NAS, USA74, 5253–5254 (1977)
Taylor, M.: Pseudodifferential operators. Princeton, NJ: Princeton University Press 1981
Taylor, M.: First order hyperbolic systems with a small viscosity term. Commun. Pure Appl. Math.31, 707–786 (1978)
Taylor, M.: Fourier integral operators and harmonic analysis on compact manifolds. In: Proc. Symp. Pure Math. Vol. 35, Part 2, pp. 115–136, Providence, RI: AMS, 1979
Taylor, M.: Grazing rays and reflection of singularities of solutions to wave equations. Commun. Pure Appl. Math.29, 1–38 (1976)
Taylor, M.: Functions of several self-adjoint operators. Proc. AMS19, 91–98 (1968)
Thirring, W.: A course in mathematical physics, Vol. 3, Berlin, Heidelberg, New York: Springer, 1979
Uhlenbeck, G., Gropper, L.: The equation of state of a non-ideal Einstein-Bose or Fermi-Dirac gas. Phys. Rev.41, 79–90 (1932)
Wallach, N.: Harmonic analysis on homogeneous spaces. New York: Dekker 1973
Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev.40, 749–759 (1932)
Wong, S.: Field and particle equation for the classical Yang-Mills field and particles with isotopic spin. Nuovo Cimento65A, 689–694 (1970)
Zelobenko, D.: Compact Lie groups and their representations. Transl. Math. Monogr., Providence, RI: AMS, 1973
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Communicated by A. Jaffe
Research supported in part by Deutsche Forschungsgemeinschaft and NSF grant NoPhy 81-09011A-01
On leave of absence from Freie Universität, Berlin
Research supported by NSF grant MCS 820176A01
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Schrader, R., Taylor, M.E. Small ħ asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials. Commun.Math. Phys. 92, 555–594 (1984). https://doi.org/10.1007/BF01215284
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DOI: https://doi.org/10.1007/BF01215284