Representation of non-semibounded quadratic forms and orthogonal additivity

TL;DR

This paper proves a representation theorem for non-semibounded Hermitian quadratic forms using orthogonal additivity, with applications to quantum mechanics and group-invariant forms.

Contribution

It introduces a new representation theorem for non-semibounded quadratic forms based on orthogonal additivity and applies it to quantum and group-invariant cases.

Findings

Representation theorem for non-semibounded forms established

Applications to quantum position operator demonstrated

Quadratic forms invariant under group actions analyzed

Abstract

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.