# Reduction of quantum systems and the local Gauss law

## TL;DR

This paper provides an operator-algebraic framework for understanding ideals generated by unbounded operators in quantum systems, linking Rieffel induction to local Gauss laws in lattice gauge theories.

## Contribution

It introduces a novel operator-algebraic interpretation of ideals related to Lie algebra elements, connecting Rieffel induction with local Gauss law implementation in lattice gauge theories.

## Key findings

- Established a link between Rieffel induction and local Gauss law implementation.
- Provided an operator-algebraic interpretation of ideals generated by Lie algebra operators.
- Applied the framework to lattice gauge theories.

## Abstract

We give an operator-algebraic interpretation of the notion of an ideal generated by the unbounded operators associated to the elements of the Lie algebra of a Lie group that implements the symmetries of a quantum system. We use this interpretation to establish a link between Rieffel induction and the implementation of a local Gauss law in lattice gauge theories similar to the method discussed by Kijowski and Rudolph in [5, 6].