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Quantum Field Theory

Yang-Mills Theory

The Yang-Mills (Figure 03i) theory was originally devised to discribe the strong interaction between nucleons (neutrons and protons, now recognize as not elementary) mediating by gauge bosons - the pion in this case. It was also involved in the early attempt to formulate a
Yang Mills Vertex for Gluon gauge theory for the weak interaction. It is a generalization of the U(1) gauge theory for QED with non-abelian group (for the internal space). It fails to account for the mass of the gauge bosons in the interaction. However, it provides a good framework for QCD in strong interaction as its mediating bosons are massless. Since non-abelian gauge bosons can be emitted and absorbed from the gauge bosons themselves (see Figure 03j), they create anti-shielding for the quark anti-quark pairs in the virtual particle cloud (the Yukawa cloud). The color charge becomes weaker at smaller probing distance, and leads to asymptotic freedom. Conversely, at larger distances, the color charge increases,

Figure 03i Yang-Mills [view large image]

Figure 03j Gluon Vertex

so that the quarks tends to bind more tightly together giving rise to quark confinement, which is the flip side of asymptotic freedom.

The mathematical formulation starts with the local gauge transformation of the fermion field:

' = exp(i (x))

where (x) is a function of x denoting the phase angles in the internal space, and represents three 2x2 non-commuting matrices (generators) for the SU(2) group; and eight 3x3 non-commuting matrices for the SU(3) group. In other word, this kind of transformation is non-abelian.

The problem with the construction is that the derivatives of the fermion field are not covariant under this transformation. It can be shown that the formulation is invariant only if the derivative is replaced by the covariant derivative:

D = + i aba

where the space-time component is labeled by the index , the index "a" is for the phase angles, is the coupling constant, and
b = aba represents the new fields (the gauge bosons) introduced to render an invariant theory under the local gauge transformation. Then the Lagrangian density takes the form:

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