Srednicki — Quantum Field Theory · Problem 2.8

Lorentz Invariance — Mark Srednicki

Physics — QFT · lorentz-invariance

Problem

a) Let Λ=1+δω\Lambda = 1 + \delta\omega in eq. (2.26), and show that [φ(x),Mμν]=Lμνφ(x),(2.29)[\varphi(x), M^{\mu\nu}] = \mathcal{L}^{\mu\nu}\varphi(x) , \tag{2.29} where Lμν≡ℏi(xμ∂ν−xν∂μ).(2.30)\mathcal{L}^{\mu\nu} \equiv \frac{\hbar}{i}(x^\mu \partial^\nu - x^\nu \partial^\mu) . \tag{2.30} b) Show that [[φ(x),Mμν],Mρσ]=LμνLρσφ(x)[[\varphi(x), M^{\mu\nu}], M^{\rho\sigma}] = \mathcal{L}^{\mu\nu}\mathcal{L}^{\rho\sigma}\varphi(x). c) Prove the Jacobi identity, [[A,B],C]+[[B,C],A]+[[C,A],B]=0[[A,B],C] + [[B,C],A] + [[C,A],B] = 0. Hint: write out all the commutators. d) Use your results from parts (b) and (c) to show that [φ(x),[Mμν,Mρσ]]=(LμνLρσ−LρσLμν)φ(x).(2.31)[\varphi(x), [M^{\mu\nu}, M^{\rho\sigma}]] = (\mathcal{L}^{\mu\nu}\mathcal{L}^{\rho\sigma} - \mathcal{L}^{\rho\sigma}\mathcal{L}^{\mu\nu})\varphi(x) . \tag{2.31} e) Simplify the right-hand side of eq. (2.31) as much as possible. f) Use your results from part (e) to verify eq. (2.16), up to the possibility of a term on the right-hand side that commutes with φ(x)\varphi(x) and its derivatives. (Such a term, called a central charge, in fact does not arise for the Lorentz algebra.)

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