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

## TL;DR

This paper constructs a non-separable Hilbert space where the inhomogeneous SL(2,C) acts unitarily, representing vectors as space-like surfaces in R^4 with an inner product based on surface area.

## Contribution

It introduces a novel geometric representation of the inhomogeneous SL(2,C) group using space-like surfaces in four-dimensional space.

## Key findings

- Unitary representation of inhomogeneous SL(2,C) constructed
- Vectors represented by space-like surfaces in R^4
- Inner product defined via surface area integral

## Abstract

We will construct a non-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) space-like surface in $\mathbb{R}^4$, for which a frame consisting of a time-like vector and a space-like vector, and 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.