An unitary representation of inhomogeneous ${\rm SL}(2,\mathbb{C})$ using surfaces in $\mathbb{R}^4$

Adrian P. C. Lim

Published 2023-03-01, updated 2023-06-22Version 2

We will construct a separable Hilbert space for which the inhomogeneous ${\rm SL}(2,\mathbb{C})$ acts on it unitarily. Each vector in this Hilbert space is described by a (rectangular) surface in $\mathbb{R}^4$, for which a vector field is defined on it. The inner product on this Hilbert space is defined via a surface integral, which is associated with the area of the surface.