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

Abstract

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 $\mathrm{dim}_{\mathbb{C}}=20$ non-commutative phase space with a dynamical, commutative $\mathrm{dim}_{\mathbb{C}}=10$ 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.