Scalar products of elementary distributions

TL;DR

This paper explores defining a scalar product for distributions with finite support within extended real number fields, leading to a positive definite inner product space with applications to quantum mechanics.

Contribution

It introduces explicit formulas for scalar products of elementary distributions in extended fields, creating a new inner product space framework with practical applications.

Findings

Defined scalar product formulas for distributions with finite support

Constructed a positive definite inner product space structure

Applied the framework to quantum mechanics with point interactions

Abstract

The field of real numbers being extended as a larger commutative field, we investigate the possibility of defining a scalar product for the distributions of finite discrete support. Then we focus on the most simple possible extension (which is an ordered field), we provide explicit formulas for this scalar product, and we exhibit a structure of positive definite inner-product space. In a one-dimensional application to the Schroedinger equation, the distributions supported by the origin are embedded into a bra-ket vector space, where the "singular" potential describing point interaction is defined in a natural way. A contact with the hyperreal numbers that arise in nonstandard analysis is possible but not essential, our extensions of $\bf R$ and $\bf C$ being obtained by a quite elementary method.