Unsharp Degrees of Freedom and the Generating of Symmetries

TL;DR

This paper explores symmetric operators in quantum theory that are not self-adjoint, demonstrating they can generate the full unitary group and thus may underpin symmetries in quantum gravity and space-time at the Planck scale.

Contribution

It shows that symmetric operators, despite not being self-adjoint, can generate unitaries and symmetries, expanding their potential role in quantum physics and quantum gravity.

Findings

Symmetric operators can generate the entire unitary group.

Such operators may describe unsharp or fuzzy degrees of freedom.

Potential relevance to space-time at the Planck scale.

Abstract

In quantum theory, real degrees of freedom are usually described by operators which are self-adjoint. There are, however, exceptions to the rule. This is because, in infinite dimensional Hilbert spaces, an operator is not necessarily self-adjoint even if its expectation values are real. Instead, the operator may be merely symmetric. Such operators are not diagonalizable - and as a consequence they describe real degrees of freedom which display a form of "unsharpness" or "fuzzyness". For example, there are indications that this type of operators could arise with the description of space-time at the string or at the Planck scale, where some form of unsharpness or fuzzyness has long been conjectured. A priori, however, a potential problem with merely symmetric operators is the fact that, unlike self-adjoint operators, they do not generate unitaries - at least not straightforwardly. Here, we show for a large class of these operators that they do generate unitaries in a well defined way, and that these operators even generate the entire unitary group of the Hilbert space. This shows that merely symmetric operators, in addition to describing unsharp physical entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g. within a fundamental theory of quantum gravity.