Delta distributions supported on quadratic O(p,q)-invariant surfaces

Abstract

An important subset of distributions encountered in physics is the set of multiplet delta distributions δcy(k), with support a quadratic O(p,q)-invariant surface cy≜{x∊Rn:P(x)=y}, n=p+q. The evaluation of these distributions for a general test function is not always sufficiently detailed in the classical literature, especially for those distributions that are defined as a regularization and/or when one needs their causal or anticausal version (for p=1 or q=1). This work intends to improve this situation by deriving explicit expressions for ⟨δcy(k),φ⟩, ∀p,q∊Z+, ∀k∊N, ∀y∊R, and ∀φ∊D(Rn), in a form suitable for practical applications. In addition, we also apply to these distributions a new approach to regularization. Distributions that need to be regularized are (equivalently) defined as extensions of a partial distribution. This extension process reveals in a natural way that regularized multiplet delta distributions are in general, uncountably multivalued. In the work of Gel’fand and Shilov [Generalized Functions (Academic, New York, 1964), Vol. 1] four types of multiplet delta distributions with support the null space c0 were introduced: δ1(k)(P), δ2(k)(P) and δ(k)(P+), δ(k)(P−). Our regularization method explains why this particular nonuniqueness was observed and further discloses the full extent of this nonuniqueness.