Charge Operators & Electric Charge - Wikipedia

[Superfluid universe]

In the wikipedia article about Charge, it says that when the symmetry group is a Lie group, then the charge operators correspond to the simple roots of the root system of the Lie algebra. So for U(1) group, how do the simple roots show that the charge in question is the electric charge?

 

[PeterDonis]

So for U(1) group, how do the simple roots show that the charge in question is the electric charge?

I'm not sure what you mean. Electric charge is defined as the charge associated with the U(1) symmetry of the electromagnetic interaction.

 

[Superfluid universe]

Hello, thank you for replying to me. :)
Well, i mean how does the charge "correspond" to the simple roots then?

 

[PeterDonis]

how does the charge "correspond" to the simple roots

Have you read through the Wikipedia articles linked to in the one about charge? These in particular seem relevant:

https://en.wikipedia.org/wiki/Weight_(representation_theory)#Highest-weight_module

https://en.wikipedia.org/wiki/Root_system

This is pretty advanced math and you should not expect it to seem obvious (at least, not by any non-mathematician's definition of "obvious").

 

[Superfluid universe]

Thanks for replying to me, Peter. I get that it is complicated, but is there any way of explaining it a bit simplified? At least what they mean by "correspond to".

 

[PeterDonis]

is there any way of explaining it a bit simplified?

I don't know that there is.

At least what they mean by "correspond to".

Note that the article says the charge operators correspond to simple roots, not the charge quantum numbers. The charge quantum numbers (i.e., things like $-1$ for the electron) correspond to weights of representations. Not that that necessarily simplifies things, but it should make clear that there are two different correspondences involved, because there are two different concepts associated with "charge", the operators (things that act on states) and the quantum numbers (the numbers you get when you act on states with charge operators).

 

[Superfluid universe]

"Note that the article says the charge operators correspond to simple roots, not the charge quantum numbers."

Ah, i thought they used "charge operator" and "charge" as synonyms. Thanks for engaging with me.

 

[Superfluid universe]

Peter, you meantioned quantum numbers. https://en.wikipedia.org/wiki/Quantum_number May I ask if there are more quantum numbers than stated in this article? Even hypothetical ones.

 

[PeterDonis]

May I ask if there are more quantum numbers than stated in this article?

Look at the section of the article titled "How many quantum numbers exist?".

 

[Superfluid universe]

Yes, i read that paragraph. But since they didn't give any examples of other quantum numbers, i asked you. :)