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 law
•It states that: Every continuous symmetry (T) of the Lagrangian has a corresponding conservation law
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
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
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
also the math just looks cool
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!
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!
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)
