Causal Field Equations and Real Eigenvalues from a Non-Local Lagrangian

Abstract

Recently, we proposed a non-local relativistic formulation of MOND (Modified Newtonian Dynamics). The equations of motion were not derived, rather they were inferred from the result one would obtain by using the Schwinger-Keldysh formalism. The formalism simultaneously ensures causality of the field equations and the reality of in-out operator amplitudes. This point was avoided in our previous paper as its discussion was too far afield. Here we first demonstrate the features non-local actions generally possess: namely acausal equations of motion and non-real in-out operator amplitudes; and secondly how the Schwinger-Keldysh formalism works to provide the characteristics we usually desire from effective theories.