Let's have Dirac spinor Ψ(x), which formally corresponds to (0,12)⊕(12,0) representation of the Lorentz group.
What representation is true for Ψ(x)Ψ+(x′)? I expect something like [(12,0)⊕(0,12)]⊗[(0,12)⊕(12,0)]= =(12,0)⊗(12,0)⊕(12,0)⊗(0,12)⊕(0,12)⊗(0,12)= =[(0,0)⊕(1,0)]⊕(12,12)⊕[(0,0)⊕(0,1)], but I'm not sure.
Also I know that [Ψ(x),Ψ+(y)]+=i(iγμ∂μ+m)γ0Dm(x−y), where Dm(x−y) is a lorentz scalar function, so formally (2) doesn't coinside with (1). How to compare it with (1)?
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