Ψ i µ Ψ i ig 2 A aµ � a j i Ψ j S 51 and hence J

Ψ i µ ψ i ig 2 a aµ ? a j i ψ j s 51 and hence j

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Unformatted text preview: Ψ i = ∂ µ Ψ i + ig 2 A aµ ( λ a ) j i Ψ j , (S . 51) and hence J a µ = − ∂ L mat ∂A a µ = − D ν Ψ ∗ i × ∂D ν Ψ i ∂A a µ − ∂D ν Ψ ∗ i ∂A a ν × D µ Ψ i = − ig 2 parenleftBig D ν Ψ ∗ i × δ µ ν ( λ a ) j i Ψ j − δ µ ν Ψ ∗ j ( λ a ) i j × D ν Ψ i parenrightBig = − ig 2 parenleftBig D µ Ψ † λ a Ψ − Ψ † λ a D µ Ψ parenrightBig = − g Im parenleftBig Ψ † λ a D µ Ψ parenrightBig . (S . 52) To combine these N 2 − 1 currents into an hermitian traceless matrix J µ = 1 2 λ a J aµ we 10 need a lemma: For any row vector Ψ † and any column vector Ψ ′ , summationdisplay a ( Ψ † λ a Ψ ′ ) × 1 2 λ a = Ψ ′ ⊗ Ψ † − (Ψ † Ψ ′ ) N × 1 N × N (S . 53) where in the first term on the RHS Ψ ′ ⊗ Ψ † denotes an N × N matrix ( Ψ ′ ⊗ Ψ † ) j i = Ψ ′ i Ψ ∗ j ⋆ while the second term subtracts the trace part of the first term. Applying this lemma to column/row vectors Ψ † and D µ Ψ or D µ Ψ † and Ψ, we obtain J µ = summationdisplay a − ig 2 parenleftBig D µ Ψ † λ a Ψ − Ψ † λ a D µ Ψ parenrightBig × 1 2 λ a = ˜ J µ − tr( ˜ J µ ) N × 1 (S . 54) where ˜ J µ are hermitian but not traceless matrices ˜ J µ ( x ) = − ig 2 parenleftBig Ψ ⊗ D µ Ψ † − D µ Ψ ⊗ Ψ † parenrightBig . (S . 55) Under a local SU ( N ) symmetry Ψ( x ) → U ( x )Ψ( x ) , D µ Ψ( x ) → U ( x ) D µ Ψ( x ) , Ψ † ( x ) → Ψ † ( x ) U † ( x ) , D µ Ψ † ( x ) → D µ Ψ † ( x ) U † ( x ) . (S . 56) Consequently, the tensor product D µ Ψ ⊗ Ψ † transforms covariantly, parenleftBig D µ Ψ( x ) ⊗ Ψ † ( x ) parenrightBig → U ( x ) parenleftBig D µ Ψ( x ) ⊗ Ψ † ( x ) parenrightBig U † ( x ) (S . 57) — indeed, in components parenleftBig D µ Ψ( x ) ⊗ Ψ † ( x ) parenrightBig j i ≡ D µ Ψ i ( x ) × Ψ ∗ j ( x ) → U k i ( x ) D µ Ψ k ( x ) × Ψ ∗ ℓ U † j ℓ ≡ parenleftBig U ( x ) parenleftBig D µ Ψ( x ) ⊗ Ψ † ( x ) parenrightBig U † ( x ) parenrightBig j i (S . 58) ⋆ The tensor product Ψ ′ ⊗ Ψ † is the finite-matrix analogy of the Hilbert-space operator | Ψ ′ )( Ψ | . In contrast, Ψ † Ψ ′ is just a number, analogous to the Dirac product ( Ψ | Ψ ′ ) . 11 — and likewise parenleftBig Ψ( x ) ⊗ D µ Ψ † ( x ) parenrightBig → U ( x ) parenleftBig Ψ( x ) ⊗ D µ Ψ † ( x ) parenrightBig U † ( x ) . (S . 59) Therefore, the ˜ J µ ( x ) matrices transform in the similar manner, ˜ J µ ( x ) → U ( x ) ˜ J µ ( x ) U † ( x ) , (S . 60) which in turn makes the current matrices J µ ( x ) transform covariantly according to eq. (3.3): J µ ( x ) = ˜ J µ ( x ) − tr( ˜ J µ ) N × 1 → U ( x ) ˜ J µ ( x ) U † − tr( ˜ J µ ) N × 1 = U ( x ) J µ ( x ) U † ( x ) . (S . 61) In other words, the currents J aµ ( x ) transform into each other as members of an adjoint multiplet, Q . E . D ....
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