# Noether current

by spookyfish
Tags: classical action, classical fields, current, noether, noether currents, noether theorem
 P: 20 Hi, I read about Noether's theorem, which states that if, under a continuous transformation, the Lagrangian is changed by a total derivative $\delta \cal L = \partial_\mu F^\mu$ then there is a conserved current $$j^\mu = \frac{\partial \cal L}{\partial(\partial_\mu \phi)}\delta \phi - F^\mu$$ However, I have seen in a different place the formulation that if the action is invariant, then the conserved quantity is: $$\frac{\partial \cal L}{\partial(\partial_\mu \phi)}\delta \phi - T^{\mu \nu}\delta x_\nu$$ where $T^{\mu \nu}$ is the energy-momentum tensor. Is the second formulation equivalent to the first? or is it a particular case
 Sci Advisor Thanks P: 2,096 Neither of both formulae is the most general case of a symmetry and Noether's theorem but special cases. The first case is the symmetry of the action under a variation of the field and unvaried space-time coordinates, where the Lagrangian changes by a total four-gradient, which means that the action is invariant. The second case is a symmetry under a more general transformation, where the space-time coordinates and fields are changed under the transformation and the Lagrangian is invariant. Of course a symmetry is still present also under such transformations, if the Lagrangian changes by a total four-gradient. Then the Noether current is $$\frac{\partial}{\partial (\partial_{\mu} \phi)} \delta \phi - T^{\mu \nu} \delta x_{\nu}-F^{\mu}.$$ You find this derived in some detail in my quantum-field theory manuscript: http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

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