Srednicki — Quantum Field Theory · Problem 50.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spinor-helicity-bra-ket-notation

Problem

Consider a bra-ket notation for twistors, ∣p]=u−(p)=v+(p),∣p⟩=u+(p)=v−(p),[p∣=u‾+(p)=v‾−(p),⟨p∣=u‾−(p)=v‾+(p).(50.33)\begin{aligned} |p] &= u_-(\mathbf{p}) = v_+(\mathbf{p}) , \nonumber\\ |p\rangle &= u_+(\mathbf{p}) = v_-(\mathbf{p}) , \nonumber\\ [p| &= \overline{u}_+(\mathbf{p}) = \overline{v}_-(\mathbf{p}) , \nonumber\\ \langle p| &= \overline{u}_-(\mathbf{p}) = \overline{v}_+(\mathbf{p}) . \tag{50.33} \end{aligned} We then have ⟨k∣ ∣p⟩=⟨k p⟩,[k∣ ∣p]=[k p],⟨k∣ ∣p]=0,[k∣ ∣p⟩=0.(50.34)\begin{aligned} \langle k|\, |p\rangle &= \langle k\, p\rangle , \nonumber\\ [k|\, |p] &= [k\, p] , \nonumber\\ \langle k|\, |p] &= 0 , \nonumber\\ [k|\, |p\rangle &= 0 . \tag{50.34} \end{aligned} a) Show that −p̸=∣p⟩[p∣+∣p]⟨p∣,(50.35)-\not{p} = |p\rangle[p| + |p]\langle p| , \tag{50.35} where pp is any massless four-momentum. b) Use this notation to rederive eqs. (50.28--50.30).

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