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

## TL;DR

This paper develops a $C^t$-algebraic quantum mechanics framework for the non-compact group $Sp(8,c)$, revealing an apparent geometry and a non-commutative phase space structure directly from quantum principles.

## Contribution

It introduces a novel $C^t$-algebraic approach to quantum mechanics based on $Sp(8,c)$, avoiding classical pre-quantization and uncovering geometric structures.

## Key findings

- Constructs a quantum Hilbert space via induced representations.
- Defines a non-commutative phase space with 20 complex dimensions.
- Identifies a 10-dimensional classical configuration subspace.

## 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.