Apparent Geometry from the Quantum Mechanics of $Sp(8,\mathbb{C})$

Restricting attention to kinematics, we develop the $C^{\ast }$-algebraic quantum mechanics of $Sp(8,\mathbb{C})$. The non-compact group does double duty: it furnishes the quantum Hilbert space through induced representations, and it spawns the quantum $C^{\ast }$-algebra through a crossed product construction. The crossed product contains operators associated with the lie algebra of $Sp(8,\mathbb{C})$ whose spectra can be interpreted as a ${\mathrm{d}\mathrm{i}\mathrm{m}}_{\mathbb{C}}=20$ non-commutative phase space with a dynamical, commutative ${\mathrm{d}\mathrm{i}\mathrm{m}}_{\mathbb{C}}=10$ configuration subspace and an internal $U(4,\mathbb{C})$ symmetry. The construction realizes quantization without first passing through the classical domain, and it exhibits apparent geometry.