Spatial contraction of the Poincare group and Maxwell's equations in the electric limit

TL;DR

This paper explores the dual Galilei group derived from the Poincare group contraction and shows that in the electric limit of Maxwell's equations, only electrostatics is possible due to the trivial time evolution.

Contribution

It introduces the dual Galilei group as a contraction of the Poincare group and analyzes its implications for the non-relativistic electric limit of Maxwell's equations.

Findings

Time evolution is trivial in the dual Galilei group representations.

Only electrostatics is consistent in the electric limit.

The dual Galilei group underpins the non-relativistic electric limit of Maxwell's equations.

Abstract

The contraction of the Poincare group with respect to the space trans- lations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincare group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIR) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non- relativistic limit of Maxwell's equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.