Local vs. global charge conservation

[kexue]

Is it correct that theories such as the free complex scalar field or the free Dircac field with their global U(1) symmetry give rise to only globally conserved charges (a globally conserved Noether charge)? If so, how can that be shown?

Also, is it somewhat correct to say that the main reason for gauging a global symmetry, i.e. turning it into a local symmetry, is turning the globally conserverd charge into locally conserved one?

thank you

 

[blechman]

Is it correct that theories such as the free complex scalar field or the free Dircac field with their global U(1) symmetry give rise to only globally conserved charges (a globally conserved Noether charge)? If so, how can that be shown?

Also, is it somewhat correct to say that the main reason for gauging a global symmetry, i.e. turning it into a local symmetry, is turning the globally conserverd charge into locally conserved one?

thank you

I'm not sure you're using the word "global" and "local" correctly. If you have a GLOBAL symmetry, you have a LOCAL conservation law:

$$\partial_\mu J^\mu=0$$

This follows from a standard derivation of Noether's Theorem in your favorite textbook or on Wikipedia.

When you gauge a symmetry, thus making it a LOCAL symmetry, you still get the local conservation law, but you also introduce gauge fields (like the photon) coupling to your fermion or scalar. THAT is why you "gauge" the symmetry.

Hope that helps.

 

[kexue]

Well, I own a book, 'Moonshine beyond the monster' by Terry Gannon where on page 268 the author says that a global symmetry implies conservation of a global charge, whereas a gauge symmetry implies local conservation of charge. But as you point out every other textbook says that a global symmetry gives a local conservation law. That confused me.

Also, why then gauging a symmetry is necessary and so overly important is not clear at all to me. What is gained by making a global symmetry local?

 

[blechman]

Never heard of that book, but the statement about "local conservation law" and "local symmetry" doesn't work for me.

A gauge symmetry is necessary for many reasons. Probably the biggest reason is that it is the only way we know of to write down a consistent, Lorentz-invariant local theory of interacting spin-1 particles (photon, W, Z, gluon, ...). By promoting a global symmetry to a local symmetry, you have to introduce "electromagnetism" and its various generalizations (weak nuclear force, strong nuclear force, etc).