A Minimal Metaplectic Loop Interferometer
Date: 17 October 2026 Branches: B07 and B09 Claim status: exact quantum-mechanical observable for specified symplectic paths; no physical braid implementation selected
1. The missing observable
The braid and Hopf branches repeatedly produced signs that disappear from Jordan density matrices. A sign becomes measurable only by comparing two coherent paths.
The minimal construction uses two paths in \(Sp(4,\mathbb R)\) with the same classical endpoint.
2. Two loops
Let \(\gamma_0\) be the constant loop at \(I_4\).
Let \(\gamma_1\) rotate one canonical plane through \(2\pi\):
Both end at
They therefore have identical classical action on the phase-space endpoint and on every Jordan relation there.
3. Metaplectic lifts
For one oscillator mode,
At \(2\pi\),
Thus the lifts obey
The sign is scalar on the entire internal state space because these are two lifts of the identity endpoint, not lifts of \(-I_4\), whose action is parity.
4. Interference protocol
Introduce a coherent path label and prepare
Transport branch \(0\) along \(\gamma_0\) and branch \(1\) along \(\gamma_1\). The final state is
Recombining in the basis
switches the output deterministically:
This is a genuine phase-sensitive observable even though the classical Jordan endpoint is unchanged.
5. Relation to the braid center
The existing braid representation gives
The squared center is therefore the appropriate closed classical operation for a scalar metaplectic sign.
But its endpoint does not determine the homotopy class of the physical path. To identify \(z^2\) with \(\gamma_1\), one must specify:
- how the braid motion varies continuously in \(Sp(4,\mathbb R)\);
- its Maslov/winding class;
- a coherent state space on which the two histories can interfere;
- a recombination operation.
Without these, the braid representation alone predicts no interferometer output.
6. Relation to the Hopf sign
The helicity Berry loop also gives a \(-1\) state holonomy. The two signs can be compared experimentally only if a canonical map identifies:
- the momentum-space helicity loop;
- the one-plane symplectic loop;
- their coherent path labels and dynamical phases.
Matching \(-1\) algebraically is insufficient. Matching interferometric phase after subtracting dynamical contributions would be meaningful.
7. Stop-everything criteria
Advance: a physical braid or transport protocol selects the odd-winding loop and predicts the output-port switch independently of adjustable dynamical phases.
Pin: the endpoint representation is known but its path class is not physically fixed.
Falsify the bridge: every admissible realization of the squared braid center has even Maslov parity, or decoherence prevents coherent comparison.
8. Certificate
The metaplectic interferometer certificate verifies the symplectic loop, identical endpoints, opposite metaplectic lifts and deterministic interference probabilities.