# Solution to "Yang-Mills Mass Gap" Millennium Prize Problem

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Abstract: We show the existence of the Yang-Mills “mass gap” by 1) highlighting the condition that we call boundary heaviness in 3-dimensional Gaussian manifolds, and 2) showing how this condition yields “point particles” when the manifolds are analyzed on background spaces of constant positive curvature.

I. The Problem

Yang-Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on ℝ^4 and has a mass gap Δ > 0.

II. The Solution

We imagine that there are finite, n-dimensional manifolds (see note a) that are non-trivial, smooth, and vanish at their boundaries. If the manifolds are defined, smoothly and bijectively, over a closed, unbounded background space of constant positive curvature (an n-sphere), then they must be “one-point compactified” at their boundaries.

The “compactified boundary points” will be set to x[0], i.e. the “initial” locations. For each x[0], there is a natural “final” location, x[f], at the polar opposite point of the (n-spherical) background space. A vector x is any interpolated sequence of locations that span between x[0] and x[f] along a given longitude. The plots of the measured quantities that are associated with each vector x are designated as “spectra.”

We imagine (n-1)-spheres of some arbitrarily large radius r, which are centered around each point in a given x. Characteristic of each sphere is a hypervolume which we will designate as its “mass,” m.

We now consider how the masses change when r→0. The question of the “mass gap” can be framed by way of comparing the values of the different masses in x to each other. This comparison may be done for each r. In other words: How are the slopes of an individual “mass spectrum” affected as r→0?

For simplicity, we now imagine the manifolds to be perfectly Gaussian in their “unwrapped” aspects (see note b).

First, for n=1, it can be trivially shown that there is no mass gap because mass(x[0])/mass(x[f])→0 as r→0, and the slope of the spectrum approaches +infinity. We designate this condition: boundary lightness.

Next, for n=2, it can be shown that all masses (x[0] to x[f]) are approximately constant, and that the slope of the spectrum is approximately 0. In other words, the spectrum appears to be a horizontal line that can be brought arbitrarily close to the vacuum state as r→0, and so a mass gap cannot be demonstrated. We designate this condition: boundary neutrality.

Finally, for n=3, the situation is effectively the reverse of n=1, because mass(x[0])/mass(x[f])→infinity as r→0, and the slope of the spectrum approaches -infinity. In other words, the more that one tries to bring the absolute magnitude of the mass of x[0] down to 0, the more that it “stands out” from the rest of the masses in x, and appears as an isolated “point particle,” (see note c) and a mass gap has been demonstrated. We designate this condition: boundary heaviness.

III. Theory Construction

Various theories may be constructed by considering the manifolds to be oscillators of various “harmonic shapes” which may each vibrate at various frequencies and with various intensities. With respect to the background space, they may each have distinct coordinate locations and orientations. Constraints may be applied to restrict their vibrational “energies,” and to impose certain conditions of symmetry and/or smoothness on the linear combination of the entire set of manifolds. We consider these to be the essential variables of any non-trivial Yang-Mills theory.

The full theory of Schrödinger may perhaps be employed in order to bring everything into “compliance” with the usual notation of canonical quantum mechanics. The resulting system may have some similarities to what is called “the universal wavefunction,” albeit with no need for a metaphysics of “collapse” or of “many worlds.”

IV. Conclusion

The above outline may not be sufficiently detailed or rigorous to qualify as a formal proof, but it is our considered opinion that no essential concepts are missing from it.

V. Glossary of Terms

ℝ^4: the ambient space that is needed to embed the n-spherical “vacuum state” when n=3.

G: the “gauge group,” which is the metric that allows us to assign the radii r that circumscribes each point in x and to calculate the magnitudes of the differential hypervolumes (the “masses”) which are bounded by 1) the (n-1)-spheres around the “sides,” 2) the hypersurfaces of the manifolds at the “top,” and 3) the hypersurface of the vacuum state at the “bottom.”

Δ: the infinitesimal “step down” that is found, when n=3, in a spectrum of infinitely negative slope, as r→0.

VI. TODO

• Historical background, esp. of the physics
• Philosophical explanations, esp. of why math has the “right” to construct physical theories
• Examples with calculations, graphs and/or dynamic visualizations
• Try to not worry too much if these results might usher in a Kuhnian paradigm shift

Notes

a. The individual manifolds are the “quanta” of any Yang-Mills theory
b. In a flat (rather than curved) background space, the manifolds would appear to be perfectly Gaussian.
c. In other words, a classical “atom in the void.”