Assumptions, results and research status
An explicit finite mathematical example with declared inputs, conventions and scope.
Assumptions
- The operator, state family or representations described on this page are supplied as model inputs.
- Every exported quantity uses the units and conventions printed beside the calculation.
Results
- The default example, tables and equations remain readable without JavaScript.
- With JavaScript, changing valid inputs recomputes the declared model and enables numerical downloads.
Open questions
- What additional source dynamics and physical matching would turn this example into a prediction of a specified physical system?
The declared model
This explorer starts from two SU(2) matrices, X = exp(iaσx) and Y = exp(ibσy). The four links along 0 → 1 → 2 → 3 → 0 are X, Y, X† and Y†, so the loop is W = Y†X†YX. The target index is first: U10 carries a state from vertex 0 to vertex 1. This makes the product order unambiguous.
Each vertex also receives a frame matrix. Transforming an edge multiplies it by the target frame and the inverse source frame. Internal frames cancel when the transformed links are composed, leaving W′ = g0Wg0†. The displayed covariance error compares the two sides as full matrices, while the trace error checks the invariant scalar. [Temporal completion, v20]
Run the calculation
Local frames and a non-Abelian loop
Change two physical link angles and four locally chosen frames. Matrix entries can change while the trace of the closed loop stays fixed.
All angles are radians; matrices, traces and probabilities are dimensionless.
Loop trace 0.490691. Frame-change trace error 0.
- Normalized loop trace
- 0.490691
- Trace after frame changes
- 0.490691
- Trace difference
- 0
- Matrix covariance error
- 2.01e-16
| Readout | Value | Definition |
|---|---|---|
| Wilson cost | 0.509309 | 1 − Re Tr W/2 |
| Interferometer dark-port probability | 0.2546545 | (1 − Re Tr W/2)/2 |
| Forced singlet-return probability | 0.2407776 | |Tr W/2|² |
The first-link matrix changes by Frobenius norm 0.8381477. At W = −I the interferometer dark port has probability one while the forced-return probability is also one: these are distinct readouts.
Inspect the loop and a changed link
| Entry | Original complex value | Transformed complex value |
|---|---|---|
| 1, 1 | 0.49069 + 0.47984i | 0.49069 + 0.47984i |
| 1, 2 | -0.40416 + 0.60467i | -0.72718 − 0.013385i |
| 2, 1 | 0.40416 + 0.60467i | 0.72718 − 0.013385i |
| 2, 2 | 0.49069 − 0.47984i | 0.49069 − 0.47984i |
| Entry | Original complex value | Transformed complex value |
|---|---|---|
| 1, 1 | 0.76484 + 0i | 0.82438 − 0.45036i |
| 1, 2 | 0 + 0.64422i | -0.16439 + 0.30092i |
| 2, 1 | 0 + 0.64422i | 0.16439 + 0.30092i |
| 2, 2 | 0.76484 + 0i | 0.82438 + 0.45036i |
Equations and conventions used
Re Tr W / 2 = 1 − 2 sin²a sin²b.
An exact four-link SU(2) example. Local frame changes relabel the same loop. Its energy and physical dynamics are not determined by this kinematic calculation.
Downloads include the chosen inputs, units and model scope. Values use standard floating-point arithmetic. Read the source and its version notes.
Try these comparisons
- Keep a and b fixed and change γ or δ. The first-link entries should change, while the normalized loop trace stays fixed up to floating-point roundoff.
- Restore the default example, then change a. You have changed the specified configuration, so the invariant trace can change.
- Choose the commuting example a = 0. The loop becomes identity and the Wilson cost is zero.
- Choose the central-loop preset a = b = π/2. Compare the interference dark port with the forced singlet-return probability, and explain why the same return probability can occur at both W = I and W = −I.
Open the matrix table to see why gauge invariance means invariant predictions rather than invariant entries. The finite covariance and trace discrepancies shown near zero are numerical roundoff, not a detected physical effect.
Interpret the result
For this family, Re Tr W/2 = 1 − 2sin²a sin²b. The Wilson cost 1 − Re Tr W/2 lies between zero and two. The interferometer dark-port probability is half that cost; the forced singlet-return probability instead squares the trace amplitude. They answer different experimental questions.
The calculation supplies transport kinematics and readout formulas. A field equation, energy operator or apparatus that implements a particular reference comparison requires further construction. The observables article explains the distinct readouts, while parallel transport derives the endpoint and connection laws.
Reproduce or extend the calculation
Download JSON to retain the inputs, units, model scope and full computed results. Download CSV for the displayed numerical family or ledger; its opening rows record the input settings and definitions. The tool uses local browser calculations and ordinary floating-point arithmetic.
For an independent check, use the worked example in the exercise collection and the linked source manuscript. A useful extension states which input or assumption changes and which comparison would test the change. See the research collaboration page for focused starting questions.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.