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Journal of Mathematical Physics / Volume 49 / Issue 5 / RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)

J. Math. Phys. 49, 052302 (2008); http://dx.doi.org/10.1063/1.2909197 (34 pages)

Abelian gauge theories on compact manifolds and the Gribov ambiguity

Gerald Kelnhofer

Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria

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(Received 4 November 2007; accepted 20 March 2008; published online 8 May 2008)

We study the quantization of Abelian gauge theories of principal torus bundles over compact manifolds with and without boundary. It is shown that these gauge theories suffer from a Gribov ambiguity originating in the nontriviality of the bundle of connections whose geometrical structure will be analyzed in detail. Motivated by the stochastic quantization approach, we propose a modified functional integral measure on the space of connections that takes the Gribov problem into account. This functional integral measure is used to calculate the partition function, Green’s functions, and the field strength correlating functions in any dimension by using the fact that the space of inequivalent connections itself admits the structure of a bundle over a finite dimensional torus. Green’s functions are shown to be affected by the nontrivial topology, giving rise to nonvanishing vacuum expectation values for the gauge fields.

© 2008 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. A MODIFIED FUNCTIONAL INTEGRAL MEASURE FOR GAUGE THEORIES
  3. THE GEOMETRICAL SETTING FOR THE ABELIAN GAUGE THEORY
  4. ABELIAN GAUGE THEORIES ON CLOSED MANIFOLDS
    1. The geometry of the Abelian gauge fields
    2. The partition function and the VEV of gauge invariant observables
    3. Green’s functions for the gauge fields
  5. ABELIAN GAUGE THEORIES ON MANIFOLDS WITH BOUNDARY
    1. The geometry of gauge fields
    2. The partition function, VEV of gauge invariant observables, and Green’s functions
  6. TWO EXAMPLES
    1. The Maxwell theory on the circle math1
    2. Abelian gauge theory on two-dimensional manifolds
  7. CONCLUDING REMARKS

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KEYWORDS and PACS

Keywords

gauge field theory, geometry, Green's function methods, integral equations, quantisation (quantum theory), stochastic processes, topology

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.2909197

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    D. Zwanziger, Phys. Rev. D 69, 016002 (2004).

    D. V. Vassilevich, Phys. Rev. D 52, 999 (1995).



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