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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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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Absolutely!
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i see .. that makes sense. in a lorentz invariant theory however the time derivatives always have to come with a spatial derivative
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The Euler-Lagrange equation looks a little different for higher derivative theories. In QFT, higher derivatives can be a used for intermediate regularization procedures. (But my brain is a little rusty) Again, super picky...so I apologize for the confusion.
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oh you mean higher time derivatives? yea i do recall something like that from my CM course
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