lecture10 (1) - Lecture X Non-Abelian Gauge Theory and Strong Interaction Introduction The generalization of an Abelian gauge theory to an non-Abelian

# lecture10 (1) - Lecture X Non-Abelian Gauge Theory and...

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Lecture X: Non-Abelian Gauge Theory and Strong Interaction Introduction The generalization of an Abelian gauge theory to an non-Abelian gauge theory is straightforward. However, its application in particle physics, which was first made by Yang and Mills, has a revolutionary impact. It means that gauge symmetry is not something accidentally appearing in EM field theory, but a guiding principle for describing fundamental interactions or constructing their theories. Non-Abelian Gauge Theory For a single Dirac spinor filed ψ , the general Lagrangian with non-Abelian gauge symmetry is given by L = ¯ ψ ( i γ μ D μ - m ) ψ - 1 4 X a F aμν F a μν = ψ ( i γ μ μ - m ) ψ - g X a A a μ J - 1 4 X a F aμν F a μν (1) Here a runs over all gauge fields. In this Lagrangian, “g” is gauge coupling which is an analogue of “e” in the EM theory. “ t a ” is a generator which is an analogue of electric charge number “q” in the EM theory. J = ¯ ψγ μ t a ψ (2) is the gauge current of the gauge field A a μ . The gauge transformation is defined as ψ (1 - i X a θ a ( x ) t a + O ( θ 2 )) ψ A a μ A a μ - 1 g μ θ a ( x ) - X b,c f abc A b μ θ c ( x ) (3) Again, the gauge fields are massless in an non-Abelian gauge theory, because their mass terms are not invariant under gauge transformation. What is the difference between an Abelian and a non-Abelian gauge theory? Let’s make a comparison.

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