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Multiplication of the distributions (x±i0)z

  • Ghislain R. Franssens EMAIL logo
Veröffentlicht/Copyright: 24. April 2014

Abstract.

In previous work of the author, a convolution and multiplication product for the set of Associated Homogeneous Distributions (AHDs) with support in ℝ was defined and fully investigated. Here this definition is used to calculate the multiplication product of homogeneous distributions of the form (x±i0)z, for all z. Multiplication products of AHDs generally contain an arbitrary constant if the resulting degree of homogeneity is a negative integer, i.e., if it is a critical product. However, critical products of the forms (x+i0)a.(x+i0)b and (x-i0)a.(x-i0)b, with a+b-, are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like δ(k).δ(l), η(k).δ(l), δ(k).η(l) and η(k).η(l) for all k,l (η1πx-1). This in turn gives a unique distribution for products like δ(k).η(l)+η(k).δ(l) and δ(k).δ(l)-η(k).η(l). The latter two products are of interest in quantum field theory and appear for instance in products of the partial derivatives of the zero-mass two-point Wightman distribution.

Received: 2012-1-26
Revised: 2013-1-8
Accepted: 2013-4-3
Published Online: 2014-4-24
Published in Print: 2014-6-1

© 2014 by Walter de Gruyter Berlin/Boston

Heruntergeladen am 21.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jaa-2014-0003/html
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