Understanding Noether's Theorem in Quantum Field Theory - K. Huang's Explanation

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SUMMARY

K. Huang's explanation of Noether's Theorem in Quantum Field Theory emphasizes that under a continuous infinitesimal transformation, the change in the Lagrangian density, represented as δL=∂^{\mu}W_{\mu}(x), must be Lorentz-invariant. This specific form is crucial because it ensures that the action integral remains unaffected, as it represents a four-divergence that vanishes. The discussion clarifies why alternative forms without derivatives are not permissible in this context.

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  • Basic grasp of differential calculus and four-divergence
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jtceleron
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This question is from K. Huang, Quantum Field Theory: from operators to path integrals.

He says that, under a continuous infinitesimal transformation,
\phi(x)->\phi(x)+\delta\phi
the change of the Lagrangian density must be in the from
\deltaL=∂^{\mu}W_{\mu}(x)

It is easily understood that this quantity must be Lorentz-invariant, but why should only be this form, not others (e.g. without derivatives).
 
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because only in this way,it can not affect the action integral.It is 4 divergence,so it will vanish.
 

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