Real Quantum Theory Is Mostly a Question about Composition

Status: a self-study literature overview, not original research. References dated 2025–26 should be rechecked before external scholarly submission.

Replacing a complex Hilbert space by a real one is easy for a single isolated system: a $d$-dimensional complex state can be embedded in a $2d$-dimensional real space carrying an operator $J$ with $J^2=-I$. The difficult question is how independently prepared systems compose.

The usual real theory takes a real tensor product. Network experiments rule out that particular package of state space and composition rule. They do not, by themselves, prove that every real-valued reformulation is impossible. Recent proposals instead alter the composition rule, so the live disagreement is structural rather than a dispute over the experimental data.

There is also a useful algebraic trichotomy. Reconstruction results based on quantum logic lead naturally to Hilbert spaces over the associative real division algebras $\mathbb R$, $\mathbb C$, and $\mathbb H$. Choosing complex numbers therefore requires an additional physical principle. Candidate principles include local tomography, composition, and relativistic symmetry.

The linked overview follows these threads from Stueckelberg’s real-Hilbert construction through quaternionic quantum mechanics, network no-go results, and the contemporary composition-rule debate. It does not yet develop a $p$-adic theory; that remains a separate prospective direction and is not claimed as completed work.

Read the literature overview (PDF).