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
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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
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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
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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
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The reason for this is because Noether's theorem relies on the Euler-Lagrange equations, which need not be satisfied at the quantum level!
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If Noether’s theorem gets 1k likes I will lose my marbles (in a good way)
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...and if anyone here might be interested in a heavily simplified version that can be used to teach at least some aspects of Noether's theorem to undergraduates or advanced high school students:https://ui.adsabs.harvard.edu/abs/2019AmJPh..87..141P/abstract …
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I remember that my physics prof derived it in the second semester of my undergraduate studies in a (from a mathematical point of view) whacky but understandable way he stitched together from a few different sources. Was the most beautiful thing I learned in the physics courses.
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