Bounds on the Euclidean functional determinant

  • H. Hogreve
  • R. Schrader
  • R. Seiler
Section VI
Part of the Lecture Notes in Physics book series (LNP, volume 106)


Partition Function Constant Magnetic Field External Electromagnetic Field Functional Determinant Duke Mathematical Journal 
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Notes and References

  1. [1]
    R. SCHRADER and R. SEILER, A Uniform Lower Bound on the Renormalized Euclidean Functional Determinant, to appear in Commun.math.Phys.Google Scholar
  2. [2]
    H. HESS, R. SCHRADER, D.A. UHLENBROCK, Kato's Inequality and Spectral Distribution of Laplace Operators on Compact Riemannian Manifolds, to appear in Journ.Diff.Geom.Google Scholar
  3. [2a]
    T. KATO, Schrödinger Operators with Singular Potentials, Israel J. Math. 13, 135–148 (1972).Google Scholar
  4. [2b]
    B. SIMON, An abstract Kato's Inequality for Generators of Positivity Preserving Semigroups, Indiana University Math. Journal 26, 1067 (1977)Google Scholar
  5. [2c]
    Kato's Inequality and the Comparison of Semigroups, submitted to Journ. Funct. Anal.Google Scholar
  6. [2d]
    H. HESS, R. SCHRADER, D.A. UHLENBROCK, Domination of Semigroups and Generalization of Kato's Inequality, Proceedings of the Conference on Mathematical Physics, Rome 1977, Springer Lecture Notes in Physics, and Duke Mathematical Journal 44, 893 (1977).Google Scholar
  7. [2e]
    B. SIMON, Universal Diamagnetism of Spinless Bose Systems, Phys.Rev.Letters 36, 1083 (1976).Google Scholar
  8. [3]
    R.T. SEELEY, Complex Powers of an Elliptic Operator, Amer. Math. Soc. Proc. Sympos. Pure Math. 10, 288–307 (1967).Google Scholar
  9. [3a]
    R.T. SEELEY, Analytic Extension of the Trace associated with Elliptic Boundary Problems, Amer. J. Math. 91, 963–983 (1969).Google Scholar
  10. [4]
    Using Fermion functional integration and a lattice cutoff, a proof of the paramagnetic inequality (4) was given by D. Bridges, J. Fröhlich and E. Seiler; On the Construction of Quantized Gauge Fields. I. General Results; IHES Preprint June 1978. The proof works as yet up to dimension three. A particular case of our conjecture has been demonstrated by E. Lieb, private communication. He has shown for the case of a constant magnetic field and an arbitrary potential V in three dimensions that the groundstate is lowered by the magnetic field. In fact, this particular case was the subject of an independent conjecture by I. Herbst, J. Avron and B. Simon in their article “Schrödinger Operators with Magnetic Fields. I, General Interactions; to appear in Duke Math. Journ. 1978, Princeton University Preprint, where also the argument by Lieb can be found. Lieb's argument can be extended to the case of nonconstant magnetic fields, see J. Avron and R. Seiler, Paramagnetism for Nonrelativistic Electrons and Euclidean Massless Dirac Particles, Preprint 7B/P.1038, CNRS Marseille.Google Scholar
  11. [5]
    H. HOGREVE, R. SCHRADER and R. SEILER; A Conjecture on the Spinor Functional Determinant, to appear in Nucl. Phys. B, FUB-HEP Preprint, April 1978.Google Scholar
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    W. HEISENBERG, H. EULER; Folgerungen aus der Diracschen Theorie des Positrons, Z. Physik 98, 714–732 (1936).Google Scholar
  13. [6a]
    J. SCHWINGER; On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664–679 (1951).Google Scholar
  14. [6b]
    V. WEISSKOPF, über die Electrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons; Kongelige Danske Videnskabernes Selskas, Mathematisk-fysike Meddelelser XIV, No 6 (1936).Google Scholar
  15. [6c]
    V. FOCK, Physik.Z.Sowjetunion 12, 404 (1937).Google Scholar
  16. [7]
    See e.g. H. ARAKI and E. LIEB, Entropy Inequalities, Commun.math.Phys. 18, 160–170 (1970).Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • H. Hogreve
    • 1
  • R. Schrader
    • 1
  • R. Seiler
    • 1
  1. 1.Institut für Theoretische PhysikFreie Universität BerlinGermany

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