Automorphism Induced Nonlocal Conservation Laws

TL;DR

This paper extends electromagnetic conservation laws to nonlocal forms linked with symmetries of correlation functions, providing a broad method to generate nonlocal conservation laws for various PDEs.

Contribution

It introduces a novel framework connecting nonlocal conservation laws with symmetries of correlation functions, applicable to a wide class of nonlinear PDEs.

Findings

Nonlocal conservation laws associated with symmetries of correlation functions.

Different nonlocal Noether currents can be derived from various nonlocal Lagrangians.

A general procedure to generate nonlocal conservation laws for nonlinear PDEs.

Abstract

The conservation laws of electromagnetism, and implicitly all theories built from quadratic Lagrangians, are extended to a continuum of nonlocal versions. These are associated with symmetries of a class of equal time field correlation functions and give results for both connected and disconnected branches of the general linear group of the space. It is generally assumed that manifestly covariant Lagrangians are the necessary starting point for physical theories. Here we show that the EOM derived from any of these can also follow from a broad class of nonlocal ones and each generally gives a different nonlocal Noether current. When the equations are put into a linear form and evaluated on a flat spacetime, a simple ansatz exists to give a class of conservation laws corresponding to all affine transformations of the underlying space. A general procedure is given to generate a class of nonlocal conservation laws for solutions to a very large class of nonlinear PDEs.