FR02-A · Different clocks, the same experiment

Local Phase & Wilson Loops

Which phase changes alter an interference experiment? Changing the phase convention at each point changes the link phases with it. The phase accumulated around a closed loop stays fixed, so the interference probabilities stay the same. Changing the enclosed flux changes that loop phase and moves the signal between the two outputs.

Turn a local phase clock and watch the neighboring links compensate. Then change the enclosed flux and watch the two output ports exchange brightness.

Wilson loop $W=e^{0.70\pi i}$

Port 1 probability $0.206$

Port 2 probability $0.794$

Selected covariant bond phase $\delta_{AB}=0.175\pi$

Change the convention or change the flux

What really changes?

Drag a gold clock, then move the Enclosed flux phase slider. Which operation changes the output probabilities?

Enclosed flux phase

\Phi

0.70\pi

Choose a local clockLocal gauge rotation

\chi

0.50\pi

Model and keyboard controls

This is a static U(1) link model with an ideal coherent, equal-arm combiner. Gold clocks show local reference-field phases; purple clocks show oriented link phases. The links have no dynamical gauge-field equation, and the output probabilities are ideal model predictions.

Gold clocks are unit-amplitude reference fields. The two arms transport the source phase separately to C, where an ideal combiner gives two output probabilities.

Focus the scene: up and down select a clock; left and right rotate it. These operations change the local phase convention.

Ideal model prediction

A full flux cycle brings the output back

$P_1=\frac{1+\cos\Phi}{2},\qquad P_2=1-P_1$

Compare phases at the combiner

\varphi_{\rm upper}=-0.65\pi,\quad\varphi_{\rm lower}=0.05\pi

Both arrival phases use the local convention at C. Rotating the clock at C shifts both by the same amount, so their difference stays fixed.

Read the course: gauge fields and effective theories

Start with the essentials

Focus question: Which phase changes alter an interference experiment?
One-sentence intuition: A local change of phase convention rotates a site and its adjoining links together; the phase around a closed loop survives.

Core mathematical model

Changing the local phase convention

\begin{aligned}\phi_i&=e^{i\theta_i},\quad U_{ij}=e^{ia_{ij}}\\\theta_i&\mapsto\theta_i+\chi_i\\a_{ij}&\mapsto a_{ij}+\chi_i-\chi_j\end{aligned}

The gold clock is a unit-amplitude reference field. A link maps the field at its endpoint into the phase convention at its start. Rotating one site therefore changes the two adjoining links with opposite signs.

Comparing two sites and going around the loop

\begin{aligned}\delta_{ij}&=\arg(\phi_i^*U_{ij}\phi_j)\\&=\theta_j-\theta_i+a_{ij}\pmod{2\pi}\\W&=U_{AB}U_{BC}U_{CD}U_{DA}\\&=e^{i\Phi}\end{aligned}

The local rotations cancel in both expressions. Clockwise links run A to B to C to D to A. Traversing a link backward uses its complex conjugate. The displayed bond phase is the principal angle.

The two ideal output probabilities

\begin{aligned}\varphi_{\rm upper}&=\theta_A-a_{AB}-a_{BC}\\\varphi_{\rm lower}&=\theta_A+a_{DA}+a_{CD}\\P_1&=\frac{1+\cos\Phi}{2}\\P_2&=1-P_1\end{aligned}

Both arms have equal amplitude and dynamical phase. Transport along a forward link uses its conjugate, consistent with the endpoint convention above. At the balanced combiner, amplitudes add at one port and subtract at the other. The reference fields at intermediate sites are a separate phase-comparison probe, not a propagated wave.

Common difficulties

A clock angle needs a comparison rule

Typical misconceptionChanging one phase angle must change the experiment.

Better mental modelA phase convention also changes the links used to compare sites. Interference depends on the relative phase after both arms reach the same site.

A Wilson loop contains more information than one intensity

Typical misconceptionThe two output intensities determine the sign of the flux.

Better mental modelThe intensities depend on the cosine, so reversing the flux leaves them unchanged. The imaginary part of the Wilson loop changes sign. A phase-sensitive reference would distinguish the orientations.

Run the experiment

01
Rotate one clock
Predict whether the output will change. Drag the gold clock at B or use Local gauge rotation.
What to observe: The two adjoining purple clocks rotate in opposite directions. The covariant bond phase, Wilson loop and both output probabilities stay fixed.
02
Change the flux
Compare Equal output with The bright port goes dark. Then move Enclosed flux phase continuously between them.
What to observe: A quarter of a full flux cycle gives equal output probabilities. At half a cycle, destructive interference closes port 1 and sends every particle to port 2.
03
Bring the flux around a full cycle
Set the flux phase to zero, then to a full turn. Save a model snapshot at each setting.
What to observe: The local bond phases can differ, but the Wilson loop and the output probabilities return to their initial values. A complete phase turn has the same complex phase as zero.

Local Phase & Wilson Loops

A full flux cycle brings the output back