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

TL;DR

This paper identifies a conceptual inconsistency in non-commutative inflation models related to group representations and proposes alternative mathematical frameworks to resolve it, impacting the formulation of inflationary equations.

Contribution

It reveals a fundamental problem with existing group-based representations in non-commutative inflation and suggests modifications using Hopf algebras or Von Neumann algebras.

Findings

Identified a conceptual incompatibility in current inflation models.

Proposed two alternative mathematical approaches to fix the problem.

Highlighted implications for the consistency of inflationary physics.

Abstract

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.