A conceptual problem for non-commutative inflation and the new approach
  for non-relativistic inflationary equation of state
  In a previous paper, we connected the phenomenological non-commutative
inflation of Alexander, Brandenberger and Magueijo (2003) and Koh S and
Brandenberger (2007) with the formal representation theory of groups and
algebras and analyzed minimal conditions that the deformed dispersion relation
should satisfy in order to lead to a successful inflation. In that paper, we
showed that elementary tools of algebra allow a group like procedure in which
even Hopf algebras (roughly the symmetries of non-commutative spaces) could
lead to the equation of state of inflationary radiation. In this paper, we show
that there exists a conceptual problem with the kind of representation that
leads to the fundamental equations of the model. The problem comes from an
incompatibility between one of the minimal conditions for successful inflation
(the momentum of individual photons is bounded from above) and the group
structure of the representation which leads to the fundamental inflationary
equations of state. We show that such a group structure, although
mathematically allowed, would lead to problems with the overall consistency of
physics, like in scattering theory, for example. Therefore, it follows that the
procedure to obtain those equations should be modified according to one of two
possible proposals that we consider here. One of them relates to the general
theory of Hopf algebras while the other is based on a representation theorem of
Von Neumann algebras, a proposal already suggested by us to take into account
interactions in the inflationary equation of state. This reopens the problem of
finding inflationary deformed dispersion relations and all developments which
followed the first paper of Non-commutative Inflation.
