Similarities between projective quantum fields and the Standard Model

Many homogeneous, four-dimensional space-time geometries can be considered within real projective geometry, which yields a mathematically well-defined framework for their deformations and limits without the appearance of singularities. Focussing on generalized unitary transformation behavior, projective quantum fields can be axiomatically introduced, which transform smoothly under geometry deformations and limits. Connections on the related projective frame bundles provide gauge fields with gauge group $\mathrm{PGL}_5\mathbb{R}$. For Poincar\'e geometry, on operator level only $\mathrm{P}(\mathrm{GL}_2\mathbb{R}\times \mathrm{GL}_3\mathbb{R})\cong \mathbb{R}_{\neq 0}\times \mathrm{PGL}_2\mathbb{R}\times \mathrm{PGL}_3\mathbb{R}$ gauge bosons can interact non-trivially with other projective quantum fields from the non- to ultra-relativistic limits. The corresponding propagating, complexified gauge bosons come with the Standard Model gauge group $G_{\mathrm{SM}}=(\mathrm{U}(1)\times \mathrm{SU}(2)\times \mathrm{SU}(3))/\mathbb{Z}_6$. Physical scale transformations can act as global gauge transformations and their spontaneous breaking can lead to masses for the projective quantum fields including the $\mathrm{SU}(2)$ gauge bosons. Projective quantum fields, which are irreducible both with respect to the Lie algebra $\mathfrak{pgl}_5\mathbb{R}$ and the Poincar\'e group, form Dirac fermions and $G_{\mathrm{SM}}$ gauge bosons interact with them similar to the Standard Model. For homogeneous, curved Lorentz geometries a gauge group similar to the gauge group of metric-affine gravity appears.