# Charge Conservation: Understanding Klein-Gordon Equation

• cjellison
In summary, the conversation revolved around the derivation of the continuity equation for charge density and current density, and the confusion surrounding the conservation of charge in quantum mechanics. The participants discussed the role of Noether's theorem and global symmetry in explaining charge conservation, as well as the concept of local versus global conservation. Ultimately, it was concluded that charge is not locally conserved, but rather globally conserved in quantum mechanics.
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?

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.

## 1. What is charge conservation?

Charge conservation is a fundamental law in physics that states that the total electric charge in a closed system remains constant over time. This means that charge cannot be created or destroyed, only transferred or redistributed.

## 2. Why is understanding Klein-Gordon equation important in charge conservation?

The Klein-Gordon equation is a relativistic wave equation that describes the behavior of spinless particles, such as the charge of a particle. Understanding this equation is important because it provides a mathematical framework for understanding how charge is conserved in particle interactions.

## 3. How does the Klein-Gordon equation relate to special relativity?

The Klein-Gordon equation is a relativistic equation, meaning that it takes into account the effects of special relativity, such as time dilation and length contraction. This is important because it allows for a more accurate description of charge conservation in high-speed particle interactions.

## 4. Can the Klein-Gordon equation be applied to all particles?

No, the Klein-Gordon equation is specifically designed for spinless particles. It cannot be applied to particles with spin, such as electrons, which require a different equation known as the Dirac equation.

## 5. Are there any limitations to charge conservation?

While charge conservation is a fundamental law of physics, there are some situations where it may appear to be violated. For example, in certain quantum systems, charge can spontaneously appear or disappear due to fluctuations in energy. However, overall charge conservation is still upheld in these systems, as the total charge remains constant on average.

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