---
title: "Local Phase & Wilson Loops Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/gauge-links/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gauge-links/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

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](https://sandboxphysics.com/stories/gauge-effective-theory/)

Physics tutorial

## Local Phase & Wilson Loops

**Background** A phase is an angle on a clock face. Comparing clocks at different sites requires a rule for carrying a phase between them. The oriented links supply that rule.

**Why it matters** The two arms meet at the same output. Their phase difference decides the probability of a particle leaving through either port.

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 misconception** Changing one phase angle must change the experiment.

**Better mental model** A 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 misconception** The two output intensities determine the sign of the flux.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Use the closed-loop phase to tell a change of phase convention from a physical change of flux.

## Related experiments and articles

Gauge fields and effective theories · Reading path

### Gauge symmetry, effective theories and anomalies

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Can changing local phase scales change a particle’s motion?**](https://sandboxphysics.com/stories/gauge-effective-theory/local-phase-links/)
2. [Article **Why does an ordinary derivative miss the same physical state?**](https://sandboxphysics.com/stories/gauge-effective-theory/covariant-derivative-ward/)
3. [Article **Why can the comparison rule interact with itself?**](https://sandboxphysics.com/stories/gauge-effective-theory/yang-mills-higgs/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)