This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Notes and References
R. SCHRADER and R. SEILER, A Uniform Lower Bound on the Renormalized Euclidean Functional Determinant, to appear in Commun.math.Phys.
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.
T. KATO, Schrödinger Operators with Singular Potentials, Israel J. Math. 13, 135–148 (1972).
B. SIMON, An abstract Kato's Inequality for Generators of Positivity Preserving Semigroups, Indiana University Math. Journal 26, 1067 (1977)
Kato's Inequality and the Comparison of Semigroups, submitted to Journ. Funct. Anal.
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).
B. SIMON, Universal Diamagnetism of Spinless Bose Systems, Phys.Rev.Letters 36, 1083 (1976).
R.T. SEELEY, Complex Powers of an Elliptic Operator, Amer. Math. Soc. Proc. Sympos. Pure Math. 10, 288–307 (1967).
R.T. SEELEY, Analytic Extension of the Trace associated with Elliptic Boundary Problems, Amer. J. Math. 91, 963–983 (1969).
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.
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.
W. HEISENBERG, H. EULER; Folgerungen aus der Diracschen Theorie des Positrons, Z. Physik 98, 714–732 (1936).
J. SCHWINGER; On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664–679 (1951).
V. WEISSKOPF, über die Electrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons; Kongelige Danske Videnskabernes Selskas, Mathematisk-fysike Meddelelser XIV, No 6 (1936).
V. FOCK, Physik.Z.Sowjetunion 12, 404 (1937).
See e.g. H. ARAKI and E. LIEB, Entropy Inequalities, Commun.math.Phys. 18, 160–170 (1970).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1979 Springer-Verlag
About this paper
Cite this paper
Hogreve, H., Schrader, R., Seiler, R. (1979). Bounds on the Euclidean functional determinant. In: Albeverio, S., et al. Feynman Path Integrals. Lecture Notes in Physics, vol 106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09532-2_82
Download citation
DOI: https://doi.org/10.1007/3-540-09532-2_82
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09532-3
Online ISBN: 978-3-540-35039-2
eBook Packages: Springer Book Archive