Lattice gauge symmetry in neural networks

The concept of symmetry or equivariance under symmetry transformations is at the theoretical foundation of modern physics, and it is hard to overstate its importance. Noether's first theorem establishes a clear relationship between invariance of Lagrangians under continuous global symmetries and the existence of conserved quantities and conserved currents [1]. Global symmetries, as the name implies, are transformations that are applied the same way at every point in space time. In mechanical systems and field theories, energy and momentum conservation laws follow from invariance under space-time translations, whereas rotational invariance implies the conservation of angular momentum. More generally, global symmetry under the Poincaré group, which includes translations, rotations and boosts, is the foundation of special relativity.