Srednicki — Quantum Field Theory · Problem 60.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spinor-helicity-polarization-vectors

Problem

a) Show that p⋅ε+(k;q)=⟨q p⟩[p k]2 ⟨q k⟩,p⋅ε−(k;q)=[q p]⟨p k⟩2 [q k].(60.34)\begin{aligned} p\cdot\varepsilon_+(k;q) &= \frac{\langle q\,p\rangle [p\,k]}{\sqrt{2}\,\langle q\,k\rangle} , \tag{60.34} \\ p\cdot\varepsilon_-(k;q) &= \frac{[q\,p]\langle p\,k\rangle}{\sqrt{2}\,[q\,k]} . \end{aligned} Use this result to show that k⋅ε±(k;q)=0,(60.36)k\cdot\varepsilon_\pm(k;q) = 0 , \tag{60.36} which is required by gauge invariance, and also that q⋅ε±(k;q)=0.(60.37)q\cdot\varepsilon_\pm(k;q) = 0 . \tag{60.37} b) Show that ε+(k;q)⋅ε+(k′;q′)=⟨q q′⟩[k k′]⟨q k⟩⟨q′ k′⟩,ε−(k;q)⋅ε−(k′;q′)=[q q′]⟨k k′⟩[q k][q′ k′],ε+(k;q)⋅ε−(k′;q′)=⟨q k′⟩[k q′]⟨q k⟩[q′ k′].(60.38)\begin{aligned} \varepsilon_+(k;q)\cdot\varepsilon_+(k';q') &= \frac{\langle q\,q'\rangle [k\,k']}{\langle q\,k\rangle\langle q'\,k'\rangle} , \tag{60.38} \\ \varepsilon_-(k;q)\cdot\varepsilon_-(k';q') &= \frac{[q\,q']\langle k\,k'\rangle}{[q\,k][q'\,k']} , \\ \varepsilon_+(k;q)\cdot\varepsilon_-(k';q') &= \frac{\langle q\,k'\rangle [k\,q']}{\langle q\,k\rangle [q'\,k']} . \end{aligned} Note that the right-hand sides of eqs. (60.38) and (60.39) vanish if q′=qq'=q, and that the right-hand side of eq. (60.40) vanishes if q=k′q=k' or q′=kq'=k.

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