Why is quantum mechanics complex?

Quantum mechanics is written with complex numbers. Probability amplitudes are complex, and the imaginary unit i appears in the Schrödinger equation. But is that essential, or just mathematical convenience?

The question is old and has two answers — and the difference between them only became sharp in 2021. This course walks through what is proven (machine-checked) and marks, honestly, where the proof ends and the experimental result begins.

For an isolated system, real numbers suffice. Any complex Hilbert space of dimension n can be rewritten as a real space of dimension 2n by replacing i with the matrix J = [[0,−1],[1,0]], which satisfies J² = −I — a real "square root of −1" in dimension 2.

This complexification preserves what matters: states, effects, and Born-rule probabilities. For decades, real and complex QM looked operationally indistinguishable for a single system. The proof beside this lesson shows that this copy of ℂ inside the real 2×2 matrices is faithful (injective).

If real and complex coincide for one system, how do we tell them apart? One clue is the number of parameters of a state: real needs d(d+1)/2 numbers, complex d², quaternionic 2d²−d.

That number is a fingerprint. Local tomography — reconstructing the whole from measurements on the parts — only works when the composite parameters are the product of the parts. The proof beside this lesson shows this condition selects exactly the complex case: real QM is not locally tomographic.

There is a sharp algebraic difference. Ask: how many ±1-outcome observables can all pairwise anticommute? In the complex case, three — the Pauli matrices σx, σy, σz. In the real 2×2 case, at most two.

The third anticommuting one is σy, the one carrying the i. It is the dimension real lacks. The proof beside this lesson formalizes both claims. But note: this root underlies, and does not prove, the experimental separation in the next lesson.

Everything changes in networks with independent sources. Renou et al. (Nature, 2021) showed there are correlations reachable by complex QM that no real QM reproduces — a bilocal Bell inequality where the real ceiling falls below the complex value. In 2022 the experiment was done (photonics and superconductors), ruling out real QM as a physical theory.

Honesty: this result is known and experimental, and it involves optimization (SDP) that we did NOT formalize. What we formalized is the foundation — single-system complexification (lesson 2) and the algebraic root (lesson 4). This lesson frames Renou; it does not re-derive it — which is why it has no "view proof": pretending otherwise would be dishonest.