NOETHER’S INVISIBLE HAND
A way of directly relating both nonlocal and local dynamical laws to associated conservation laws
Laurence I. Gould
Address: Physics Department, University of Hartford, 200 Bloomfield Ave., West Hartford, CT 06117 U.S.A.
Abstract: The object of this paper is to exhibit a formalism which makes it possible to directly interrelate (i.e., without explicitly employing the Noether theorems and their extensions) “laws (or equations) of motion” to “conservation laws”, “laws (or equations) of continuity”, and “balance laws” for “local” systems (those described by linear second-order differential equations) as well as for “nonlocal” systems (those described by linear second-order integrodifferential equations). The formalism can be applied to physical systems containing particles and fields described by classical or quantum theory and whose relevant equations of motion may be relativistic (Gould, 1989). The formalism can, in particular, facilitate the generation of global conservation laws for such quantities as energy and momentum. Some of the examples will include the usual Schrödinger equation as well as its nonlocal counterpart; both important for a variety of quantum systems. Analogies to more common experience should enable the non-technical reader to gain some helpful understanding of the issues discussed.
Keywords: local, nonlocal, dynamical laws, balance laws, laws of continuity, conserved quantities, variational, Noether.