Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 3.2

Classical field theory — Schwartz M.D. z-lib.org

Physics — QFT · lorentz-currents-noether

Problem

Lorentz currents. (a) Calculate the conserved currents KμναK_{\mu\nu\alpha} associated with (global) Lorentz transformations xμ→Λμνxνx_\mu \to \Lambda_{\mu\nu} x_\nu. Express the currents in terms of the energy-momentum tensor. (b) Evaluate the currents for L=12ϕ(□+m2)ϕ\mathcal{L} = \frac{1}{2}\phi\left(\square + m^2\right)\phi. Check that these currents satisfy ∂αKμνα=0\partial_\alpha K_{\mu\nu\alpha} = 0 on the equations of motion. (c) What is the physical interpretation of the conserved quantities Qi=∫d3x K0i0Q_i = \int d^3x\, K_{0i0} associated with boosts? (d) Show that dQidt=0\frac{dQ_i}{dt} = 0 can still be consistent with i∂Qi∂t=[Qi,H]i\frac{\partial Q_i}{\partial t} = [Q_i, H]. Thus, although these charges are conserved, they do not provide invariants for the equations of motion. This is one way to understand why particles have spin, corresponding to representations of the rotation group, and not additional quantum numbers associated with boosts.

Browse the problem library