# Filling the Mass Gap: Chromodynamic Symmetries ... Filling the Mass Gap: Chromodynamic Symmetries,...

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J. R. Yablon

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Filling the Mass Gap: Chromodynamic Symmetries, Confinement Properties, and Short-Range Interactions of Classical and Quantum

Yang-Mills Gauge Theory

Jay R. Yablon Schenectady, New York 12309

jyablon@nycap.rr.com

November 7, 2013

Abstract: We show how SU(3)C chromodynamics, which is the theory of strong interactions, is a corollary theory emerging naturally from the combination of nothing other than Maxwell / Weyl gauge theory with Yang-Mills theory. In the process, we show not only the emergence from the Maxwell / Yang-Mills combination of all that is to be expected from SU(3)C chromodynamics, but additionally, we show how the observed baryons containing three colored quarks in the ground state are the magnetic charges of Yang-Mills gauge theory and how these magnetic charges naturally confine their quarks and gluons but do pass mesons in order to interact. That is, we explain quark and gluon confinement and how it is that strong interactions are mediated by mesons but not gauge fields. Additionally, we demonstrate how the inherent non-linearity of Yang-Mills theory may be used to solve the “mass gap” problem and yield a nuclear interaction that is short range notwithstanding its being based on massless gluon gauge fields. We further demonstrate the origin of “chiral symmetry breaking” in strong interactions. We find that the non-linear nature of Yang-Mills theory contains a recursive aspect which provides a useful tool for solving the Yang-Mills path integral in order to exactly, analytically arrive at quantum Yang- Mills theory. As a result of further developing Weyl’s original geometric view of gauge theory, we uncover a classical field equation unifying gravitational theory with Weyl’s gauge theory including both its Maxwell / Abelian and Yang-Mills variants, at the level of the Einstein equation for gravitation. Finally, we use the recursive aspects of Yang-Mills theory to develop and solve an exact, closed recursive path integral for Quantum Yang-Mills Theory and thereby prove the existence of a non-trivial quantum Yang–Mills theory on R4 for any simple gauge group G. PACS: 12.38.Aw; 12.40.-y; 14.20.-c; 14.40.-n

J. R. Yablon

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Contents 1. Introduction ............................................................................................................................. 3

2. Classical Yang-Mills Theory: Three Equivalent Viewpoints .................................................. 6

3. The Field Equations and Configuration Space Operator of Classical Yang-Mills Theory ... 10

4. The Magnetic Field Equation of Classical Yang-Mills Theory, and its Apparent Confinement Properties ................................................................................................................ 14

5. The Yang-Mills Perturbation Tensor: A Fourth View of Yang-Mills ................................... 19

6. Hermann Weyl’s Gauge Theory and Gravitational Curvature: A Fifth, Geometric View of Yang-Mills .................................................................................................................................... 21

7. The Classical Gravitational Field Equation for Yang-Mills Gauge Theory, Inclusive of Maxwell’s Electrodynamics.......................................................................................................... 28

8. The Configuration Space Inverse of the Electric Charge Field Equation of Classical Yang- Mills Theory.................................................................................................................................. 31

9. Populating Yang-Mills Monopoles with Fermions, and the Recursive Nature of the Yang- Mills: A Sixth View of Yang-Mills which may Aid in the Quantum Path Integration of Yang- Mills Theory.................................................................................................................................. 37

10. The Mass Gap Solution ......................................................................................................... 43

11. Populating Yang-Mills Monopoles with Fermions to Reveal that Yang-Mills Monopoles have the Chromodynamic and Confinement Symmetries of Baryons and Emit and Absorb Objects with the Chromodynamic Symmetries of Mesons ........................................................... 50

12. Chiral Symmetry Breaking .................................................................................................... 57

13. Quantum Yang-Mills Theory ................................................................................................ 60

14. Conclusion ............................................................................................................................. 70

References ..................................................................................................................................... 71

J. R. Yablon

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1. Introduction In this paper we study the strong “chromodynamic” interactions for which the Yang- Mills gauge group is (3)CSU . But contrary to how chromodynamic interactions are commonly

approached, we make no a priori supposition about Yang-Mills SU(3)C being the theory of strong interactions. We simply postulate that Maxwell’s U(1)em electrodynamics is a correct theory of nature and that any other non-gravitational interactions have the exact same form as electrodynamics with the sole exception that they employ gauge groups SU(N) with all spacetime derivatives µ∂ in the Maxwell Lagrangian and the classical field equations including those operating on gauge fields and on the field strength replaced by D iGµ µ µ µ∂ → = ∂ − , and so are non-Abelian versions of Maxwell’s electrodynamics.

Starting from this view, we show how chromodynamics in the form of an SU(3)C gauge theory need not be posited at all, but emerges entirely as a corollary theory based on positing Maxwell gauge theory with Yang-Mills extension as the underlying, fundamental theory. But in the process, extending beyond the pedagogical utility of this viewpoint, we not only uncover SU(3)C chromodynamics in its usual expected form, but we also come upon baryons and show them to be the magnetic monopoles of these Yang-Mills extensions of Maxwell. We further find out how and why interactions between observed strong particle states such as protons and neutrons are mediated by mesons, we develop certain important connections to gravitational Riemannian geometry, and we solve the Yang Mills mass gap and confinement problems.

In laying out the “Yang-Mills and Mass Gap” problem which the present paper solves, Jaffe and Witten point out at page 3 of [1] that:

“. . . for QCD to describe the strong force successfully, it must have at the

quantum level the following three properties, each of which is dramatically different from the behavior of the classical theory: 1) It must have a “mass gap;” namely there must be some constant 0∆ > such that every excitation of the vacuum has energy at least ∆ . (2) It must have “quark confinement,” that is, even though the theory is described in terms of elementary fields, such as the quark fields, that transform non-trivially under SU(3), the physical particle states—such as the proton, neutron, and pion—are SU(3)-invariant. (3) It must have “chiral symmetry breaking,” which means that the vacuum is potentially invariant (in the limit, that the quark-bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.”

They further proceed to state that:

“The first point is necessary to explain why the nuclear force is strong but short-ranged; the second is needed to explain why we never see individual quarks; and the third is needed to account for the ‘current algebra’ theory of soft pions that was developed in the 1960s.”

They then continue (emphasis added, original references renumbered):

J. R. Yablon

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“Both experiment – since QCD has numerous successes in confrontation with experiment – and computer simulations . . . have given strong encouragement that QCD does have the properties [short range, confinement and chiral symmetry breaking] cited above. These properties can be seen, to some extent, in theoretical calculations carried out in a variety of highly oversimplified models (like strongly coupled lattice gauge theory, see, for example, [2]). But they are not fully understood theoretically; there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD, as opposed to a severely simplified truncation of it.”

Moving past a statement of the problem to how the mass gap might be solved, Jaffe and

Witten later proceed to survey a wide variety of methods used “to show the existence of quantum fields on non-compact configuration space” and specifically to demonstrate that “relativistic, nonlinear quantum field theories exist.” On page 12 of [1], they finally observe that:

“One view of the mass gap in Yang–Mills theory suggests that it could arise from the quartic potential (A ^ A)2 in the action, where F = dA + gA ^ A, see [3], and may be tied to curvature in the space of connections, see [4].”

This is the view of the Yang-Mills mass gap that will be developed here and used to solve this problem. It is in accord Occam’s Razor as restated by Einste

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