Charge Conservation: Understanding Klein-Gordon Equation

cjellison
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So, I was just introduced to the Klein-Gordon equation. I've been asked to derive the continuity equation for charge density and current density. I am having trouble understanding this. If I were to derive a continuity equation involving charge, doesn't this say that charge is conserved locally?

Obviously, I am confused. My current thinking says that charge cannot be locally conserved in quantum mechanics since things "jump" around and tunnel. However, I suppose I could also make the same argument about probability conservation---yet we do believe that probability is conserved in quantum mechanics.

Could someone elighten me with a general discussion on this topic?
 
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How about applying Noether's theorem in the classical fields and the current 4-vector is automatically conserved ?

Daniel.
 
I'm not disputing the result; I'm seeking an explanation as to why I should expect charge to be locally conserved in light of the fact the QM is a nonlocal theory. That fact that the result can be dervied mathematically, in a variety of ways, does not answer this question (at least in my humble opinion).

Looking forward to your response (and the responses of others as well).
 
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Charge is not locally conserved,but globally.It comes from a rigid global symmetry of the (electrically) charged KG field's Lagrangian action.

Incidentally,when coupling to the abelian gauge field,the electric charge conservation follows from gauge/local symmetry.

But for a free charged field,it's a global symmetry.

Daniel.

P.S.That "KG" is not Kevin Garnett,though under certain circumstances,TD stands for Tim Duncan and not Theory Development.
 
Even in Classical EM, the continuity eq. does not prove charge conservation locally, but only when integrated using the div theorem.
 
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