Advertisement

Communications in Mathematical Physics

, Volume 129, Issue 3, pp 481–509 | Cite as

Integration on thenth power of a hyperbolic space in terms of invariants under diagonal action of isometries (Lorentz transformations)

  • Bent Fuglede
Article

Abstract

The integral of a function over then'th power of hyperbolicd-dimensional spaceH is decomposed into integration along each orbit under diagonal action onHn of the isometry groupG onH, followed by integration over the orbit space, parametrized in terms of a complete set of invariants. The Jacobian entering in this last integral is expressed explicitly in terms of certain determinants. When viewingH as a half-hyperboloid in ℝ d+1 ,G is induced by the homogeneous Lorentz groupO(1,d) acting on ℝ d+1 .

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bogoliubov, N. N., Logunov, A. A., Todorov, I. T.: Introduction to axiomatic quantum field theory. Reading. Massachusetts: Benjamin 1975Google Scholar
  2. 2.
    Fenchel, W.: Elementary geometry in hyperbolic space. Berlin, New York: De Gruyter 1989Google Scholar
  3. 3.
    Hall, D., Wightman, A. S.: A theorem on invariant analytic functions with applications to relativistic quantum field theory. Mat.-Fys. Medd. Dan. Vid. Selsk.31, 41 (1957)Google Scholar
  4. 4.
    Menger, K.: Untersuchungen über allgemeine. Metrik. Math. Ann.100, 75–164 (1928)CrossRefGoogle Scholar
  5. 5.
    Schoenberg, I. J.: Remarks to Maurice Fréchet's article: Sur la définition axiomatique d'une classe d'espaces distancés vectoriellement applicables sur l'espace de Hilbert. Ann. Math.36, 724–732 (1935)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Bent Fuglede
    • 1
  1. 1.Matematisk InstitutKøbenhavns UniversitetKøbenhavn øDenmark

Personalised recommendations