How do representations of an isometry group correspond to degrees of freedom/entropy in a system?

Question

To put the question into context: I am currently writing my bachelors thesis on de Sitter space, specifically, $dS_4$. I am trying to show that while the horizon entropy is finite, the isometry group $SO(4,1)$ has infinite dimensional representations which leads to the idea of infinite degrees of freedom.

I have been able to find quite some articles discussing this issue; it remains unclear to me why representations (and specifically, unitary irreducible representations of infinite dimension) dictate degrees of freedom in a system. Most QFT courses include a chapter on representations of groups, but solely aimed at explaining the link between particles as representations.

The question, in short, how does one show that a specific Lie group dictates degrees of freedom in a physical system through QFT?

Answer 1

The entropy is simply $$ S = k_B\cdot \ln N $$ where $N$ is the number of macroscopically indistinguishable microstates. If they transform as a representation of a group, then $N$ is the dimension of this representation.

The relevant Hilbert space must be a unitary representation of the isometry groups: the isometries have to preserve the sesquilinear norm of the vectors. However, $SO(d-1,1)$ has no unitary finite-dimensional representation.

Incidentally, a quantum deformation of $SO(d-1,1)$ does typically have finite-dimensional unitary representations.

The term "number of degrees of freedom" refers to the dimension of the configuration space or phase space (which one depends on the context) if there is a clear space of this sort (either in a classical theory or in a quantum theory with a preferred classical limit); more generally, it is a vague synonym of the entropy $S$, usually expressed up to the renormalization by a purely numerical factor of order one.

Because the $N$ for the representation of the de Sitter symmetry group were infinite (needed for unitarity), de Sitter space would formally have infinitely many degrees of freedom. However, holography suggests that the entropy is actually bounded by $S=A/4G$ where $A$ is the area of the de Sitter cosmic horizon. Effectively, the number of relevant microstates should be finite but they can't transform as a clear representation of the isometry group. This suggests some intrinsic vagueness in the description of quantum gravity in de Sitter space (note that thermal radiation is coming from the cosmic horizon and its precise microstate is probably undetermined). A lot of papers have been written about these issues, especially a decade ago. However, as far as I know, no clear or provable results beyond the guesses above have been found.