Fermionic de Sitter representation tensor products

Determine the tensor products of fermionic unitary irreducible representations of the de Sitter isometry groups, extending the known tensor-product results for bosonic representations.

Background

The paper reviews tensor products of unitary irreducible representations for the de Sitter groups SO(1,2) and SO(1,3). It notes that the corresponding problem for fermionic representations has not yet been solved, despite its relevance for understanding how fermionic states combine in de Sitter quantum field theory.

The issue is especially pertinent because the paper gives an example suggesting that a gravitino Hilbert space may occur inside the tensor product of photon and electron Hilbert spaces, indicating nontrivial structure that remains to be characterized.

References

Understanding the tensor product for fermionic representations is to date an open problem.

— Notes on de Sitter space  (2609.19454 - Mühlmann, 16 Sep 2026) in Section 3.1, paragraph 'dS4 Representations'

As we will shortly explain, in Lorentzian signature the quantisation of the theory is not yet fully settled.

— Sphere path integrals for fermionic gauge fields  (2609.19275 - Baracco et al., 16 Sep 2026) in Section 2, “Lorentzian formulation,” subsection “Lorentzian formulation”