No abstract

In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is a C 00 principal fibration. The underlying quotient space, the gauge orbit space, is seen explicitly to be a C 00 manifold modelled on a Hubert space.

The Euclidean (φ 4 ) 3, ǫ model in R 3 corresponds to a perturbation by a φ 4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0 ≤ ε ≤ 1. For ε = 1 one recovers the covariance of a massless scalar field in R 3 . For ε = 0 φ 4 is a marginal interaction. For 0 ≤ ε < 1 the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for ε > 0, sufficiently small, there exists a non-gaussian fixed point ( with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed.

Let ∆ be the finite difference Laplacian associated to the lattice Z d . For dimension d ≥ 3, a ≥ 0 and L a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G a := (a − ∆) −1 can be decomposed as an infinite sum of positive semi-definite functions V n of finite range, V n (x−y) = 0 for |x−y| ≥ O(L) n . Equivalently, the Gaussian process on the lattice with covariance G a admits a decomposition into independent Gaussian processes with finite range covariances. For a = 0, V n has a limiting scaling formx−y L n as n → ∞. As a corollary, such decompositions also exist for fractional powers (The results of this paper give an alternative to the block spin renormalization group on the lattice.