Charge Conservation: Understanding Klein-Gordon Equation

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Discussion Overview

The discussion revolves around the concept of charge conservation in the context of the Klein-Gordon equation, particularly focusing on the derivation of the continuity equation for charge density and current density. Participants explore the implications of quantum mechanics on local versus global charge conservation, as well as the role of symmetries in these concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the local conservation of charge in quantum mechanics, suggesting that phenomena like tunneling imply nonlocal behavior.
  • Another participant proposes applying Noether's theorem to argue that the current 4-vector is conserved in classical fields.
  • A different participant questions the expectation of local charge conservation in quantum mechanics, emphasizing that mathematical derivations do not address the underlying conceptual issues.
  • One participant asserts that charge is not locally conserved but globally conserved, attributing this to a global symmetry in the Lagrangian action of the charged Klein-Gordon field.
  • Another participant notes that in classical electromagnetism, the continuity equation does not demonstrate local charge conservation, but rather conservation when integrated over a volume using the divergence theorem.

Areas of Agreement / Disagreement

Participants express differing views on the nature of charge conservation, with some arguing for local conservation and others asserting it is only global. The discussion remains unresolved, with multiple competing perspectives presented.

Contextual Notes

There are limitations in the discussion regarding assumptions about local versus global conservation, the implications of quantum mechanics, and the definitions of symmetries involved. These aspects remain unresolved and are subject to interpretation.

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).
 
Last edited:
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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