# Lab Notes for a Scientific Revolution (Physics)

## January 28, 2008

### Lab Note 3, Part 1: Yang Mills Theory, the Origin of Baryons and Confinement, and the Mass Gap

(You may download this Lab Note in a PDF file at: qcd-confinement-handout-10.pdf)

This is part 1 of a Lab Note dealing with the origin of baryons and confinement in Yang-Mills theory, and attempting to lay the foundation for a solution to the so-called “Mass Gap” problem.  I have organized this into eight brief, bite-sized sections.

1.  What Makes Yang-Mills Gauge Theory Different from an Abelian Gauge Theory like QED?

In an Abelian Gauge Theory such as QED, a field strength two-form $F={\tfrac{1}{2!}} F^{\mu \nu } dx_{\mu } \wedge dx_{\nu } =F^{\mu \nu } dx_{\mu } dx_{\nu }$ is expressed in terms of a potential one-form $A=A^{\mu } dx_{\mu }$ for a field of vector bosons, in this case photons, using the compact language of differential forms, as:

$F=dA$, (1.1)

where $dA=\partial ^{\mu } A^{\nu } dx_{\mu } \wedge dx_{\nu } =\left(\partial ^{\mu } A^{\nu } -\partial ^{\nu } A^{\mu } \right)\, dx_{\mu } dx_{\nu } \equiv \partial ^{[\mu } A^{\nu ]} dx_{\mu } dx_{\nu }$.

In Yang Mills theory, also known as non-Abelian gauge theory, there is an extra term in the field strength, and in particular, if the vector potential one-form is now $G=G^{\mu } dx_{\mu }$, then:

$F=dG+igG^{2}$, (1.2)

where $G^{2} =\left[G,G\right]={\tfrac{1}{2!}} \left[G^{\mu } ,G^{\nu } \right]dx_{\mu } \wedge dx_{\nu } =\left[G^{\mu } ,G^{\nu } \right]dx_{\mu } dx_{\nu }$, and g is the group “running charge” strength.

The only difference is the existence of this extra term $igG^{2}$! (more…)

Physical science, which is an enterprise designed to make sense of what we measure when observing natural phenomena, has long rested on the three “elemental dimensions” of length, time, and mass.  Various other physically-measurable quantities (velocity, acceleration, force, energy, momentum, power, etc.) are built out of well-known combinations of these three elemental dimensions.  With the recognition in his 1908 paper Space and Time that the Lorentz transformations of special relativity signaled that “space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality,” Minkowski began the integration of the space and time dimensions into what we now routinely think of as spacetime.  Then, in 1915, in a stunning marriage of observational physics to pure Riemannian geometry, the general theory of relativity came to explain the dynamics of gravitational behavior solely on the basis of geodesic worldline motion through a curved generalization of Minkowski’s spacetime geometry.  The promise of general relativity, that we might one day be able to understand all of physics solely on the basis of pure geometry, and that general relativity itself might be merely the first glimpse of an elegant geometric structure underlying all of physical reality, was later coined by Wheeler as the “geometrodynamic” program.  To date, the promise of this program is still largely unfulfilled, and one of the main reasons for this is that we still do not understand the rest masses of material bodies and elementary particles on a purely geometric foundation.   Notwithstanding the success of electroweak theory in predicting the $W^{\pm \mu } ,Z^{\mu }$ masses via spontaneous symmetry breaking, rest mass is, for all intents and purposes, a foreign object introduced, ad hoc, “by hand,” into spacetime.  Rest mass still stands apart from Minkowski’s spacetime.  One can draw spacetime diagrams and worldlines which show the motion of a given massive body through spacetime, but to specify complete information about this massive body, we must also associate with that worldline, a “number” which represents the magnitude of that mass when viewed at rest.  A worldline, by itself, omits this vital information about the material body.  In a gravitational field, for example, the worldlines of a golf ball and a bowling ball starting out in the same place with the same velocity vectors will be identical due to the equivalence of gravitational and inertial mass, and just knowing the worldline of each will tell us nothing about their difference in mass.  One must specify the mass as a separate parameter independent of the worldline.  From a geometrodynamic viewpoint, this is an unsatisfactory state of affairs, because we have to specify “worldline plus mass.”  It would be much preferred if we could speak merely of a body traveling along a worldline through spacetime, making no reference to its mass, and if we could deduce solely from our knowledge of this worldline, not only the path through spacetime, but also the mass of the body, and thereby the forces — if any —  acting on this body, simply and solely by knowing the worldline of its travel.  One would seek in this way to complete what Minkowski started, to arrive at a complete geometric union of the three elemental dimensions upon which we base all physical measurement: space and time and mass.  We seek to go from having to know about “worldline plus mass,” to having to know only about “worldline alone.”  We wish to understand particle worldlines in such a way that we can deduce the mass of a particle solely by knowing its worldline.  In this way, we can move one step closer to Wheeler’s dream of understanding all of physics as pure geometry — or to be precise — understanding all of the measurements we obtain in physics, including those of mass and energy — as measurements of geometric lengths and trajectories.   (more…)