Product of Boundary Distributions

TL;DR

This paper explores new parameter structures for boundary Poisson brackets in higher dimensions, linking distribution products with fundamental algebraic identities, and defining higher derivatives independently of integral choices.

Contribution

It introduces new parameter branches for boundary Poisson brackets, relates distribution products to algebraic identities, and defines higher derivatives independently of integral kernels.

Findings

Identified multiple new parameter branches for boundary Poisson brackets.

Established the connection between distribution product consistency and Jacobi identities.

Defined higher functional derivatives independently of integral kernel choices.

Abstract

1) We identify new parameter branches for the ultra-local boundary Poisson bracket in d spatial dimension with a (d-1)-dimensional spatial boundary. There exist 2^{r(r-1)/2} r-dimensional parameter branches for each d-box, r-row Young tableau. The already known branch (hep-th/9912017) corresponds to a vertical 1-column, d-box Young tableau. 2) We consider a local distribution product among the so-called boundary distributions. The product is required to respect the associativity and the Leibnitz rule. We show that the consistency requirements on this product correspond to the Jacobi identity conditions for the boundary Poisson bracket. In other words, the restrictions on forming a boundary Poisson bracket can be related to the more fundamental distribution product construction. 3) The definition of the higher functional derivatives is made independent of the choice of integral kernel representative for a functional.