Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 3.3

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

Physics — QFT · energy-momentum-tensor-ambiguities

Problem

Ambiguities in the energy-momentum tensor. (a) If you add a total derivative to the Lagrangian

\mathcal{L} \to \mathcal{L} + \partial_\mu X^\mu

, how does the energy-momentum tensor change? (b) Show that the total energy

Q = \int \mathcal{T}_{00}\, d^3x

is invariant under such changes. (c) Show that

T_{\mu\nu} \neq T_{\nu\mu}

is not symmetric for

\mathcal{L} = -\frac{1}{4}F^2_{\mu\nu}

. Can you find an

X^\mu

so that

T_{\mu\nu}

is symmetric in this case?