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Noether's Theorem is perhaps the most beautiful mathematical theorem in physics •It states that: Every continuous symmetry (T) of the Lagrangian has a corresponding conservation lawpic.twitter.com/RrsS1QKm4A
Caveats (1/2) The astute reader will notice that technically Noether's theorem applies to invariance of the action, which is the spacetime integral over the Lagrangian density Hence, you will also need to make sure that the measure is invariant under the transformation as well
Caveat (2) Since surface terms do not affect the Euler Lagrange equations, the equations of motion are always trivially invariant upon adding a 4-divergence to the Lagrangian density
In Quantum Field Theories, however, Noether's theorem can be violated due to quantum effects! A symmetry at the classical level that is broken by quantum corrections is referred to as 'Anomalous' Anomalous symmetries are one of the many great predictions of the Standard Model!pic.twitter.com/Xku5q3MERl
The reason for this is because Noether's theorem relies on the Euler-Lagrange equations, which need not be satisfied at the quantum level!
If Noether’s theorem gets 1k likes I will lose my marbles (in a good way)
yes! the classical electrodynamics lagrangian has a U(1) symmetry which corresponds to the conservation of charge!
I could be wrong; however, I believe this form of the current assumes the Lagrangian density is only a functional of the field and the derivatives of the field. Again, if I’m not mistaken, the form can be generalized if the Lagrangian density depends on higher derivatives.
if there are higher derivatives of the field then they will show up when you take the partial with respect to the field derivatives .. in QFT this isn't an issue if we only consider renormalizable operators
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