Understanding the Relationship Between Ward Identity and Charge Conservation

[Bobhawke]

There are two things about this identity that I don't understand:

1. Why is it equivalent to a statement of charge conservation?
2. Wikipedia claims that it is like a quantum version of the classical noether's theorem. In what sense is this true?

Thanks

 

[blechman]

WT identity says (VERY roughly) that if you replace a polarization vector with a momentum then the amplitude vanishes.

Remembering that momentum = derivative, this is saying that if you do a gauge transformation on the vector boson, then the gauge transformation part does not contribute to physical amplitudes.

This is the same statement as gauge invariance.

This is the same statement as conservation of charge.

 

[xepma]

A more general statement is that a Ward Identity is a reflection of the (continious) symmetries present in the system. You can, in some sense, view them as restrictions on the correlation functions.

Noether's theorem is a similar statement for classical systems. Furthermore, continious symmetries are associated with conserved charges (= Noether's theorem).

So, given a continuous symmetry of the system you can ask: what is the implication on the correlation functions, i.e. the physical observables? The answer is that they are restricted by the associated Ward identities.