Contractions of the Maxwell algebra

Abstract

We construct all the possible non-relativistic, non-trivial, Galilei and Carroll kcontractions also known as k-1 p-brane contractions of the Maxwell algebra in D + 1 space-time dimensions. k has to do with the number of space-time dimensions one is contracting. For non-trivial solutions, we mean the ones with a non-abelian algebra of the momenta. We find in both cases, Galilei and Carroll, eight non trivial solutions. We also study the electromagnetic properties of the solutions, defined according to the scaling performed on the generators present in the Maxwell algebra. We find that besides the electric and magnetic contractions studied in the literature for k = 1, that are related to the magnetic and electric limit of the free Maxwell equations, there also exist contractions where the two types of fields are scaled in the same way.

Non-relativistic limits and three-dimensional coadjoint Poincare gravity

“…This scaling of the generators is different from the scaling used in[37] and resembles more the ones of[38]. Other scalings are also possible[19,20,22].…”

“…Other scalings are also possible[19,20,22]. This is related to the fact that the commutation relations of the Poincaré algebra are invariant under the rescaling H = λH and Pa = λ Pa (see for example[38]). At the field theory level it corresponds to the fact that two different scalings can differ by an overall scaling of the Lagrangian that can be absorbed by a scaling of Newton's constant[9].…”

Infinite-dimensional algebras as extensions of kinematic algebras

“…We note that one can also consider non-relativistic limits of the relativistic Maxwell algebra and there are different limits that arise depending on the scaling of the electric and magnetic fields. The corresponding algebras can be called electric, magnetic and pulse Maxwell algebras [60,16].…”

“…In section 4.6, we consider a particle model on the associated space-time and how it relates to the motion of charged particle in an electro-magnetic background field. We also not that one can consider various non-relativistic limits of Maxwell algebras and space-times [69,60,16].…”

Infinite-Dimensional Algebras as Extensions of Kinematic Algebras

“…We note that one can also consider non-relativistic limits of the relativistic Maxwell algebra and there are different limits that arise depending on the scaling of the electric and magnetic fields. The corresponding algebras can be called electric, magnetic and pulse Maxwell algebras [16,59].…”