Letters in Mathematical Physics

, Volume 28, Issue 2, pp 97–106 | Cite as

Hilbert space cocycles as representations of (3 + 1)-d current algebras

  • Jouko Mickelsson


It is proposed that instead of normal representations, one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g. the Fock space of chiral fermions), when dealing with groups associated to current algebras in gauge theories in 3 + 1 spacetime dimensions. The appropriate cocycle is evaluated in the case of the group of smooth maps from the physical three-space to a compact Lie group.

The cocyclic representation of a componentX of the current is obtained through two regularizations, (1) a conjugation by a background potential dependent unitary operatorhA, (2) by a subtraction-h A -1 xhA, where ℒ x is a derivative along a gauge orbit. It is only the total operatorh A -1 XhA -h A -1 xhA which is quantizable in the Fock space using the usual normal ordering subtraction.

Mathematics Subject Classifications (1991)

81D15 17B65 58B25 


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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Jouko Mickelsson
    • 1
  1. 1.Department for Theoretical PhysicsUniversity of FreiburgGermany

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