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
The model pairs a mobile rank-m carrier with a stationary dual. A positive penalty Δ favors their singlet at a specified loop position, and hopping t transports the mobile carrier around four positions. A holonomy matrix W enters through its eigenangles. The free cycle eigenvalues are λak = 2t cos[(θa + 2πk)/4], and a rank-one resolvent equation determines the physical ground state.
The calculation uses Δ as its energy unit. Rank two explores eigenangles (θ,−θ). At rank three, choose either the balanced slice (θ,−θ,0) or the full diagonal family (θ,φ,−θ−φ). Varying both angles covers SU(3) conjugacy classes, with the permutation and periodic identifications explained in the two-angle domain guide. The φ input and family selector do not enter the rank-two calculation. The positive root lies above every free eigenvalue and is found in a guaranteed analytic bracket. [Exact singlet results, v20, Section 2]
Run the calculation
Exact singlet-loop response
Compare the exact energy shift of the declared finite singlet model with its leading weak-hopping loop term. Change the carrier rank, hopping and holonomy.
The singlet penalty Δ is the energy unit. Hopping is t/Δ; displayed energies and gaps are divided by Δ. Angles are radians.
Exact energy shift 0.001072956 Δ; finite-model gap 0.6784925 Δ.
- Exact E(W) − E(I), in Δ
- 0.00107295645
- Leading loop term, in Δ
- 0.00136030346
- Exact excitation gap, in Δ
- 0.678492492
- Uniform finite-model bound, in Δ
- 0.08333333
SU(3): diag(exp(iθ),exp(−iθ),1). The selected eigenangles are 1.2, -1.2, 0 radians. The second SU(3) angle is inactive for this selection.
| Quantity | Value |
|---|---|
| Ground energy E(W)/Δ | -0.07849249223 |
| Identity ground energy E(I)/Δ | -0.07956544869 |
| Root z/Δ | 1.078492492 |
| Secular residual | 6.66e-16 |
| Real trace Re Tr W | 1.72471551 |
| Imaginary trace Im Tr W | 0 |
| Loop cost m − Re Tr W | 1.275284 |
| Exact Hessian κ̂ₘ/Δ | 0.0008392853737 |
| Weak-hopping κloop/Δ | 0.001066666667 |
| Absolute approximation error / Δ | 0.0002873470028 |
For W = exp(iX) close to identity, the convention is E(W) − E(I) = ½κ̂ₘ Tr X² + O(X⁴). Along the balanced slice, Tr X² = 2θ². The exact Hessian is positive when hopping is nonzero; it vanishes at t = 0.
Values across the displayed angle family
| θ / rad | Exact shift / Δ | Leading shift / Δ | Gap / Δ |
|---|---|---|---|
| 0 | 0 | 0 | 0.6795654 |
| 0.3141593 | 0.00008217053 | 0.0001044128 | 0.6794833 |
| 0.6283185 | 0.0003208137 | 0.0004074304 | 0.6792446 |
| 0.9424778 | 0.0006930287 | 0.0008793915 | 0.6788724 |
| 1.256637 | 0.001162952 | 0.001474097 | 0.6784025 |
| 1.570796 | 0.001685051 | 0.002133333 | 0.6778804 |
| 1.884956 | 0.002208403 | 0.00279257 | 0.677357 |
| 2.199115 | 0.002681607 | 0.003387275 | 0.6768838 |
| 2.513274 | 0.003057878 | 0.003859236 | 0.6765076 |
| 2.827433 | 0.003299802 | 0.004162254 | 0.6762656 |
| 3.141593 | 0.003383226 | 0.004266667 | 0.6761822 |
Equations and conventions used
(Δ/4m) Σa,k(z − λak)−1 = 1; E = Δ − z; gap = z − λ* ≥ Δ/(4m).
E(W) − E(I) ≈ [2t⁴/(mΔ³)] [m − Re Tr W].
A finite model with a unique ground state for the stated ranks and penalty. Its finite-dimensional excitation gap is not the four-dimensional Yang–Mills mass gap. At rank three, choose the balanced slice or the full diagonal two-angle family diag(exp(iθ),exp(iφ),exp(−i(θ+φ))). At rank two, the family selector and φ do not enter the SU(2) 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
- Choose the balanced slice and weak hopping t/Δ = 0.1, then inspect the exact and leading curves. Both vanish at identity because the plotted quantity subtracts E(I).
- Increase hopping to 0.5. The leading term remains a well-defined expression, but it is a less accurate approximation to the exact response.
- Set t/Δ = 0. The model has E = 0, gap Δ and no loop response.
- Use a small nonzero θ and compare [E(W) − E(I)]/θ² with the displayed exact Hessian coefficient.
At rank three and t/Δ = 0.1, the identity energy is approximately −0.01999231644Δ, the identity gap is 0.81999231644Δ, and κ̂₃ is approximately 0.00006277560191Δ. These provide reproducible checkpoints independent of a measured physical coupling.
Interpret the result
The exact result has E = Δ − z and, for the stated ranks, gap = z − λ* ≥ Δ/(4m). Its local expansion is E(exp(iX)) − E(I) = ½κ̂mTr X² + O(X⁴). Along the balanced slice Tr X² = 2θ², so the coefficient of θ² is κ̂m. In the full two-angle SU(3) chart, a near-identity representative instead has Tr X² = θ²+φ²+(θ+φ)². The leading weak-hopping expression is 2t⁴[m − Re Tr W]/(mΔ³).
A finite-model gap is an exact result about this Hamiltonian. It does not establish the gap of a quantized gauge field, the infinite-volume vacuum or the four-dimensional continuum theory. The exact-model derivation and Hessian guide explain the result’s scope and the extension to independent rank-three and rank-two blocks.
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.