Srednicki — Quantum Field Theory · Problem 36.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · lagrangians-spinor-fields

Problem

Consider a field φA(x)\varphi_A(x) in an unspecified representation of the Lorentz group, indexed by AA, that obeys U(Λ)−1φA(x)U(Λ)=LAB(Λ)φB(Λ−1x).(36.66)U(\Lambda)^{-1}\varphi_A(x)U(\Lambda) = L_A{}^B(\Lambda)\varphi_B(\Lambda^{-1}x) . \tag{36.66} For an infinitesimal transformation, LAB(1+δω)=δAB+i2δωμν(Sμν)AB.(36.67)L_A{}^B(1+\delta\omega) = \delta_A{}^B + \tfrac{i}{2}\delta\omega_{\mu\nu}(S^{\mu\nu})_A{}^B . \tag{36.67} a) Following the procedure of section 22, show that the energy-momentum tensor is Tμν=gμνL−∂L∂(∂μφA)∂νφA.(36.68)T^{\mu\nu} = g^{\mu\nu}\mathcal{L} - \frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi_A)}\partial^\nu\varphi_A . \tag{36.68} b) Show that the Noether current corresponding to a Lorentz transformation is Mμνρ=xνTμρ−xρTμν+Bμνρ,(36.69)\mathcal{M}^{\mu\nu\rho} = x^\nu T^{\mu\rho} - x^\rho T^{\mu\nu} + B^{\mu\nu\rho} , \tag{36.69} where Bμνρ≡−i∂L∂(∂μφA)(Sνρ)ABφB.(36.70)B^{\mu\nu\rho} \equiv -i\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi_A)}(S^{\nu\rho})_A{}^B\varphi_B . \tag{36.70} c) Use the conservation laws ∂μTμν=0\partial_\mu T^{\mu\nu} = 0 and ∂μMμνρ=0\partial_\mu \mathcal{M}^{\mu\nu\rho} = 0 to show that Tνρ−Tρν+∂μBμνρ=0.(36.71)T^{\nu\rho} - T^{\rho\nu} + \partial_\mu B^{\mu\nu\rho} = 0 . \tag{36.71} d) Define the improved energy-momentum tensor or Belinfante tensor Θμν≡Tμν+12∂ρ(Bρμν−Bμρν−Bνρμ).(36.72)\Theta^{\mu\nu} \equiv T^{\mu\nu} + \tfrac{1}{2}\partial_\rho(B^{\rho\mu\nu} - B^{\mu\rho\nu} - B^{\nu\rho\mu}) . \tag{36.72} Show that Θμν\Theta^{\mu\nu} is symmetric: Θμν=Θνμ\Theta^{\mu\nu} = \Theta^{\nu\mu}. Also show that Θμν\Theta^{\mu\nu} is conserved, ∂μΘμν=0\partial_\mu\Theta^{\mu\nu} = 0, and that ∫d3x Θ0ν=∫d3x T0ν=Pν\int d^3x\,\Theta^{0\nu} = \int d^3x\,T^{0\nu} = P^\nu, where PνP^\nu is the energy-momentum four-vector. (In general relativity, it is the Belinfante tensor that couples to gravity.) e) Show that the improved tensor Ξμνρ≡xνΘμρ−xρΘμν(36.73)\Xi^{\mu\nu\rho} \equiv x^\nu\Theta^{\mu\rho} - x^\rho\Theta^{\mu\nu} \tag{36.73} obeys ∂μΞμνρ=0\partial_\mu\Xi^{\mu\nu\rho} = 0, and that ∫d3x Ξ0νρ=∫d3x M0νρ=Mνρ\int d^3x\,\Xi^{0\nu\rho} = \int d^3x\,\mathcal{M}^{0\nu\rho} = M^{\nu\rho}, where MνρM^{\nu\rho} are the Lorentz generators. f) Compute Θμν\Theta^{\mu\nu} for a left-handed Weyl field with L\mathcal{L} given by eq. (36.2), and for a Dirac field with L\mathcal{L} given by eq. (36.28).

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