Which-Path Detectors

Why interference disappears when transport bookkeeping is forced to branch

Claim: Which-path detectors destroy interference by breaking transport coherence.
Not by measurement, observation, or wavefunction collapse.

What Changes When a Detector Is Added

Path A — Transport Geometry View

Primitive: Transport closure

Effect of detector: Introduces a branch in transport history

Interference requires closure: the ability to recombine transport corridors.

A which-path detector tags a corridor with an irreversible interaction.

$$ \phi_1 \rightarrow (\phi_1, D) \quad \phi_2 \rightarrow (\phi_2, \varnothing) $$

Closure is now impossible because transport states no longer match.

The cosine-squared law collapses to a sum:

$$ I = I_1 + I_2 $$
↓

Path B — Phase Statistics View

Primitive: Phase density conservation

Effect of detector: Phase becomes conditionally correlated

Each emission samples a phase $\theta$.

The detector correlates $\theta$ with an external record.

Marginalizing over detector states removes cross terms:

$$ \langle e^{i(\phi_1-\phi_2)} \rangle_D = 0 $$

The interference term vanishes statistically.

↓

Agreement Condition

$$ I_{\text{geometry}} = I_{\text{statistics}} = I_1 + I_2 $$

Both paths predict identical loss of fringes.

Why “Observation” Is the Wrong Word

The effect is purely transport-mechanical.

Delayed Choice Revisited

Adding or removing the detector changes transport bookkeeping at the point of interaction.

No future choice alters past motion.

Transport-First Summary