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In a previous paper, we connected the phenomenological non-commutativeinflation of Alexander, Brandenberger and Magueijo (2003) and Koh S andBrandenberger (2007) with the formal representation theory of groups andalgebras and analyzed minimal conditions that the deformed dispersion relationshould satisfy in order to lead to a successful inflation. In that paper, weshowed that elementary tools of algebra allow a group like procedure in whicheven Hopf algebras (roughly the symmetries of non-commutative spaces) couldlead to the equation of state of inflationary radiation. In this paper, we showthat there exists a conceptual problem with the kind of representation thatleads to the fundamental equations of the model. The problem comes from anincompatibility between one of the minimal conditions for successful inflation(the momentum of individual photons is bounded from above) and the groupstructure of the representation which leads to the fundamental inflationaryequations of state. We show that such a group structure, althoughmathematically allowed, would lead to problems with the overall consistency ofphysics, like in scattering theory, for example. Therefore, it follows that theprocedure to obtain those equations should be modified according to one of twopossible proposals that we consider here. One of them relates to the generaltheory of Hopf algebras while the other is based on a representation theorem ofVon Neumann algebras, a proposal already suggested by us to take into accountinteractions in the inflationary equation of state. This reopens the problem offinding inflationary deformed dispersion relations and all developments whichfollowed the first paper of Non-commutative Inflation.

A conceptual problem for non-commutative inflation and the new approach for non-relativistic inflationary equation of state