Generalised conservation laws in non-local field theories

Abstract

We propose a geometrical treatment of symmetries in non-local field theories, where the nonlocality is due to a lack of identification of field arguments in the action. We show that the existence of a symmetry of the action leads to a generalised conservation law, in which the usual conserved current acquires an additional non-local correction term, obtaining a generalisation of the standard Noether theorem. We illustrate the general formalism by discussing the specific physical example of complex scalar field theory of the type describing the hydrodynamic approximation of Bose-Einstein condensates. We expect our analysis and results to be of particular interest for the group field theory formulation of quantum gravity.1 Notice that most such models are non-local in the sense that they involve an infinite number of field derivatives (see for example [6][7][8][9][10]).It is unclear to us if there is a general connection between this type of non-locality and the one we deal with in our analysis. At any rate, we will not deal with this type of field theories.

Null Lagrangians of non-local field theories

“…This paper completes the former work [1,2] regarding divergence symmetries. While we have worked out the simplest non-local case, a generalisation to arbitrary nonlocal theories on possibly N copies of a single jet space is straightforward.…”

“…The nature of non-locality is reflected in the fact that those are integro-differential equations. A generalisation of this derivation can be found in [1].…”

“…where the subscript 12 denotes the order of the jet spaces in the Cartesian product 1 . For an open, connected region 1 Introduce the permutation operator P :…”

“…Were we not interested in the fundamental mediation by photons nor aware of even their existence, we would describe the phenomenon by an effective interaction "at a distance" whose essence is the electrical potential with its well-known inverse distance proportionality. In this regard, a Lagrangian density is referred to as non-local (sometimes also multi-local ) if it is a functional on possibly N copies of a single jet space, assuming that each of the fundamental fields depends on only one point of the base space [1]. We will see in the following that non-locality is closely connected with the fact that the corresponding Euler-Lagrange equations are in integro-differential form.…”

“…A Noether theorem was proposed for theories with Lagrangian densities of the described kind [1,2]. However, it did not cover the analogue of the divergence symmetry present in the well-known local theorem, i.e.…”