Milne-like Spacetimes and their Symmetries

Abstract

When developing a quantum theory for a physical system, one determines the system's symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincar\'e group, $\mathbb{R}^4 \rtimes \text{O}(1,3)$, and the irreducible unitary representations are interpreted as elementary particles which determine the particle's mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be $\mathbb{R} \times \text{O}(1,3)$ and classify their irreducible unitary representations. Again they represent particles with mass and spin. Unlike the classification for the Poincar\'e group, we do not obtain any faster-than-light particles. The factor $\mathbb{R}$ corresponds to cosmic time translations. These generate a mass Casimir operator which yields a Lorentz invariant Dirac equation on Milne-like spacetimes. In fact it's just the original Dirac equation multiplied by a conformal factor $\Omega$. Therefore many of the invariants and symmetries still hold. We offer a new interpretation of the negative energy states and propose a possible solution to the matter-antimatter asymmetry problem in our universe.