Conserved Currents From Noether's Theorem
For more detailed explanations and derivations, see Noether's theorem.If a field φ(r, t), which can be a tensor field of any order describing (say) gravitational, electromagnetic, or even quantum fields, is varied by a small amount;
then Noether's theorem states the Lagrangian L (also Lagrangian density ℒ for fields) is invariant (doesn't change) under a continuous symmetry (the field is a continuous variable):
This leads to conserved current densities in a completely general form:
because they satisfy the continuity equation
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Derivation of conserved currents from Lagrangian density The variation in is:
next using the Euler-Lagrange equations for a field:
the variation is:
where the product rule has been used, so we obtain the continuity equation in a different general form:
So the conserved current (density) has to be:
Integrating Jμ over a spacelike region of volume gives the total amount of conserved quantity within that volume:
Read more about this topic: Continuity Equation
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