Quantum Field Theory. Note 12

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Quantum Field Theory. Note « 12 »

The standard model (Moulders,chapter 12)

Yang-Mills theories have an internal symmetry of the Lagrangian under a group transformation

ψ → exp(iα^aI^a)ψ .

The generators I^a of the group satisfy the Lie algebra

[I^a,I^b]=iC^c_abI^c .

If structure constants C^c_ab are not zero the theory is called non-Abelian.

The Yang-Mills Lagrangian

L=iψγ^μD_μψ - mψψ ,
D_μ=∂_μ-igB_μ , B_μ≡ B^a_μI^a

is invariant if the transformation rule for the gauge fields B_μ is chosen

B^a_μ → B^a_μ+¹/_gD_μα^a

The generalization of the electro-magnetic tensor F_μν is the invariant tensor G_μν

G_μν = i[D_μ,D_ν] = ∂_μB_ν - ∂_νB_μ - ig[B_μB_ν],

and the generalization of the electro-magnetic Lagrangian is

L_B = -¹/₄G^a_μνG^aμν = -¹/₂TrG_μνG^μν

Exercises

Check that the tensor G_μν = i[D_μ,D_ν] is gauge invariant.
Check that the trace Tr(G_μνG^μν) is a real scalar (can be used as a Lagrangian density).