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

We will construct a non-separable Hilbert space for which the inhomogeneous $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ 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.