# Why doesn't Noether thm produce exactly the stress-energy tensor?

1. Mar 1, 2012

### roomzeig

In classical field theory, use noether theorem to compute conserved currents for electromagnetic Lagrangian.
$\mathcal{L} = \frac{1}{4}F_{\mu\nu}F_{\mu\nu}, \quad F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$
For arbitrary translational symmetries, the Noether conserved current evaluates to:
$T^{\mu \nu}=-F^{\mu \sigma}\eta^{\nu\lambda} \partial_\lambda A_\sigma + \tfrac{1}{4} \eta^{\mu\nu} F_{\alpha \beta} F^{\alpha \beta}$
which is almost but not equal to the classical enm stress energy tensor:
$\hat T^{\mu \nu}=-F^{\mu \sigma}\eta^{\nu\lambda} F_{\lambda \sigma} + \tfrac{1}{4} \eta^{\mu\nu} F_{\alpha \beta} F^{\alpha \beta}$
with the difference being a total divergence:
$K^{\mu\nu} \equiv \hat T^{\mu\nu} - T^{\mu\nu} = \partial_\sigma \left( F^{\mu \sigma} A^{\nu} \right)$

I understand that the difference does not change global conservation etc, but why is it that the Noether procedure does not produce a *symmetric* classical stress energy tensor? is there a physical meaning of K^{\mu\nu}? if \hat T^{\mu nu} is interpreted, on the grounds of ENM, as stress-energy, what is T^{\mu\nu}?

Thanks.

2. Apr 18, 2016