Generalized Yang-Mills actions from Dirac operator determinants

We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present an explicit computation proving this fact, generalized to the chiral case. We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension.Comment: LaTex, 26 page

On anomalies and noncommutative geometry

I discuss examples where basic structures from Connes' noncommutative geometry naturally arise in quantum field theory. The discussion is based on recent work, partly collaboration with J. Mickelsson.Comment: 6 pages, latex, no figures. Proceedings of ``34. Internationale Universit\"atswochen f\"ur Kern- und Teilchenphysik Schladming'', Schladming March 1995, Springer Verlag (to appear

Cocycles for Boson and Fermion Bogoliubov Transformations

Unitarily implementable Bogoliubov transformations for charged, relativistic bos\-ons and fermions are discussed, and explicit formulas for the 2-cocycles appearing in the group product of their implementers are derived. In the fermion case this provides a simple field theoretic derivation of the well-known cocycle of the group of unitary Hilbert space operators modeled on the Hilbert Schmidt class and closely related to the loop groups. In the boson case the cocycle is obtained for a similar group of pseudo-unitary (symplectic) operators. I also derive explcite formulas for the phases of one-parameter groups of implementers and, more generally, families of implementers which are unitary propagators with parameter dependent generators.Comment: 23 pages, UBC preprint (August 92

An explicit solution of the (quantum) elliptic Calogero-Sutherland model

We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders in the nome of the elliptic functions, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials.Comment: 16 pages, Contribution to SPT 2004 in Cala Gonone (Sardinia, Italy) v2 and v3: minor correction

The \beta-function in duality-covariant noncommutative \phi^4-theory

We compute the one-loop \beta-functions describing the renormalisation of the coupling constant \lambda and the frequency parameter \Omega for the real four-dimensional duality-covariant noncommutative \phi^4-model, which is renormalisable to all orders. The contribution from the one-loop four-point function is reduced by the one-loop wavefunction renormalisation, but the \beta_\lambda-function remains non-negative. Both \beta_\lambda and \beta_\Omega vanish at the one-loop level for the duality-invariant model characterised by \Omega=1. Moreover, \beta_\Omega also vanishes in the limit \Omega \to 0, which defines the standard noncommutative \phi^4-quantum field theory. Thus, the limit \Omega \to 0 exists at least at the one-loop level.Comment: 11 pages, LaTe