Charge Conservation: Understanding Klein-Gordon Equation

[cjellison]

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?

 

[dextercioby]

How about applying Noether's theorem in the classical fields and the current 4-vector is automatically conserved ?

Daniel.

 

[cjellison]

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).

 

[dextercioby]

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.

 

[Meir Achuz]

Even in Classical EM, the continuity eq. does not prove charge conservation locally, but only when integrated using the div theorem.