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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 tothe Galilei group, that is, the contraction limit of the Poincare group withrespect to the time translations subgroup. In view of this duality, we call theformer the dual Galilei group. A rather remarkable feature of the dual Galileigroup is that the time translations constitute a central subgroup. Therewith,in unitary irreducible representations (UIR) of the group, the Hamiltonianappears as a Casimir operator proportional to the identity H = EI, with E (anda spin value s) uniquely characterizing the representation. Hence, a physicalsystem characterized by a UIR of the dual Galilei group displays no non-trivialtime evolution. Moreover, the combined U(1) gauge group and the dual Galileigroup underlie a non- relativistic limit of Maxwell's equations known as theelectric limit. The analysis presented here shows that only electrostatics ispossible for the electric limit, wholly in harmony with the trivial nature oftime evolution governed by the dual Galilei group.

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