mod the subgroup of diagonal matrices. Each of those tensors can be taken to its rest frame, where the electric and magnetic parts ${\boldsymbol{e}}$, ${\boldsymbol{m}}$ are *parallel unit vectors*, by a boost, and go back with the inverse boost, specified by *a third vector, orthogonal to those unit vectors*; thus $\Sigma$ can be viewed as the tangent bundle to two copies of the unit sphere in 3 space - its *base* $\Sigma_1$.

Irreducible representations at a point of $\Sigma_1$ identify with *Schrödinger’s $p, q$ in $2$ degrees of freedom*. The fibers are therefore the C\* algebras of the Heisenberg relations in 2 degrees of freedom - the algebra of all compact operators on a fixed infinite dimensional separable Hilbert space.

The continuous field can be shown to be trivial. Thus the C\* algebra $\mathcal E$ of Quantum Spacetime is identified with the tensor product of the continuous functions vanishing at infinity on $\Sigma$ and the algebra of compact operators.

The mathematical generalization of points are pure states. *Optimally localized states* minimize $$\Sigma_\mu(\Delta_\omega q_\mu)^2;$$ the minimum being $2$, reached by states concentrated on $\Sigma_1$, at each point coinciding (if optimally localized at the origin) with the *ground state of the harmonic 