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An example of a symmetry transformation for certain Lagrangians (notably the canonical complex scalar field Lagrangian) is multiplication of the fields by a complex phase. When we multiply the fields by a spacetime-dependent phase and demand that the Lagrangian is left unchanged under the transformation, then we must introduce a new gauge field, and this field is nothing but the photon field, which is a combination of the vector potential A and the scalar potential φ.

Furthermore, from the Lagrangian, we can derive the equations of motion for the fields, and in the example just mentioned, the equations are indeed the Maxwell equations. And since for every symmetry, there is a conserved quantity, the conserved quantity there is electric charge. Can you please explain why it turns out to be electric charge?

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    $\begingroup$ possible duplicate of Noether theorem and classical proof of electric charge conservation $\endgroup$
    – ACuriousMind ♦
    Commented Dec 12, 2014 at 0:06
  • $\begingroup$ Related: physics.stackexchange.com/q/2721/2451 $\endgroup$
    – Qmechanic ♦
    Commented Dec 12, 2014 at 0:08
  • $\begingroup$ Definitely not a duplicate IMO. Chat discussion can be found here $\endgroup$
    – Danu
    Commented Dec 12, 2014 at 0:09
  • $\begingroup$ Aha so when somebody answered the question in the link @acuriousmind posted, it was said change of phase would lead to this conserved quantity. Is multiplication by complex number like in this case considered as a change of phase? $\endgroup$
    – beyondtheory
    Commented Dec 12, 2014 at 0:37
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    $\begingroup$ Comment to the question (v7): Although the question formulation (v7) is strictly speaking not an exact duplicate of Noether theorem and classical proof of electric charge conservation, the answer will be essentially the same: The electric charge is the Noether charge that corresponds to the global gauge symmetry of the action, and hence conserved. $\endgroup$
    – Qmechanic ♦
    Commented Dec 12, 2014 at 12:55

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