In this article
- A calculable benchmark in the dual model
- The small-fluctuation mass
- Deriving the kink tension
- A dimensionless example and its quantum limit
- A second harmonic changes the ratio
- What a temporal source prediction would require
Kink tension, mass, and higher harmonics
Explain the classical dual-photon relation between kink tension, stiffness, and mass in the specified model. Discuss why additional harmonics change the relation and require an error analysis.
01A calculable benchmark in the dual model
Once the effective dual-photon action is specified, one can derive an exact classical relation between the potential curvature and the energy of a winding profile. This provides a useful benchmark for normalization and correction analysis. It does not depend on treating the Single Monad as a literal magnetic object. In the SMM/DTT program, the relevance is methodological: a proposed temporal source should produce a definite effective action whose consequences can be computed and checked. The leading sine-Gordon action is such an action, but its coefficients are matching inputs until the source construction determines them.
02The small-fluctuation mass
Take gradient stiffness κ>0 and potential A(1−cosσ), A>0. Near a minimum, the potential is Aσ²/2 plus higher powers. With canonical variable χ=√κσ, the quadratic energy has mass parameter mσ²=A/κ. This is a classical curvature mass. It is not automatically the quantum spectral gap or the mass measured in every gauge-invariant channel. The distinction is particularly important because some physical operators may couple to multi-particle or symmetry-neutral sectors rather than directly to one dual-photon excitation.
03Deriving the kink tension
For a planar profile σ(z) joining adjacent lifts 0 and 2π, the energy per area is the integral of κ(σ′)²/2+V(σ). Completing the elementary square gives a lower bound ∫₀²π√(2κV(s))ds. A monotone profile saturates it when σ′=√(2V/κ). For the cosine potential, integration yields the familiar profile and tension. The two endpoints are equivalent lifts of the periodic vacuum; the construction does not assert two distinct physical vacua at θ=0.
σ(z)=4 arctan exp[mσ(z−z₀)].
Tcl=8√(κA)=8κmσ; Tcl/(κmσ)=8.
04A dimensionless example and its quantum limit
Set y=mσ(z−z₀). The winding profile runs smoothly from zero to 2π, and its normalized energy density is 4sech²y. Its integral is eight, making an exact numerical check of any discretized classical solver. Quantum fluctuations can renormalize both the mass and tension. Rescaling the three-dimensional action shows the loop-counting parameter mσ/κ; the leading semiclassical treatment requires it to be small after matching. Enlarging the Wilson loop suppresses boundary effects but does not remove a quantum correction proportional to its area.
05A second harmonic changes the ratio
Add B(1−cos2σ) and define η=B/A. For η>−1/4 the adjacent lifted minima remain separated by a positive barrier. The curvature mass becomes mη²=(A/κ)(1+4η), while the tension becomes 8√(κA)I(η), with I(η)=∫₀¹√(1+4ηu²)du. Expanding gives T/(8κmη)=1−4η/3+O(η²). Thus the number eight is not a universal topological constant independent of the action. The draft supplies a controlled remainder for bounded |η|<1/4, and the expansion loses uniformity near the vanishing-curvature endpoint.
T/(8κmη)=I(η)/√(1+4η)=1−4η/3+O(η²).
06What a temporal source prediction would require
A source model could predict a relation among κ, A and B, producing a testable corrected ratio. It must compute those coefficients and the competing derivative and loop corrections in a common prescription. Choosing B to match a desired tension would be calibration, not a prediction from inner dynamics. The six nested levels may constrain allowed winding operators, but they do not by themselves fix their weights. This article therefore provides an exact classical target, a sensitivity calculation and a clear separation between the shape of a topological profile and the physical quantum observables of a matched gauge theory.
KEEP THE CONNECTION
Key ideas
- The leading classical cosine model gives Tcl=8κmσ exactly.
- Higher harmonics change that ratio in a calculable way.
- Quantum mass, tension and source-matched coefficients require further work.
Scope of the argument. Exact classical variational statements for the displayed effective action, with quantum and gauge matching treated separately.
Sources & further reading
- Exact SU(2) Center-Selection Thresholds for a Finite-Cycle One-Loop Potential, seventh revisionyang_mills_holonomy_monopoles_v7(1).pdf, §§3–10, pp.8–32; gap and matching audit, pp.76–81
Source editions and catalogue entries
Working manuscript
Exact SU(2) Center-Selection Thresholds for a Finite-Cycle One-Loop Potential
Draft file: yang_mills_holonomy_monopoles_v7(1).pdf
Working manuscript. Article-level references identify the consulted sections and the scope of the interpretation. No public URL was supplied for this exact manuscript version.
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