Relating the different "notions of charges" in field theory

Question

Thinking back to my lectures of QFT/Classical field theory, I am getting confused about the different things called charges:

• First of all, we have the notion of Noether charge, i.e. the integral of the first component of the Noether current $J^\mu$: $$Q=\int_\Sigma J^0.$$ It is associated to the the invariance of the action under a symmetry, through Noether theorem. This one is well defined.

• Then, we have the elementary charge $e$ in the sense of a coupling constant, like in QED for example. It's just a number regulating the intensity of the interactions between the gauge field and matter. If one computes the Noether current associated to the $U(1)$ invariance of QED, one gets $J^\mu=\bar{\psi}\gamma^\mu\psi$, meaning that the Noether charge is the integral of $\bar{\psi}\gamma^0\psi$.

• Finally, we have the charge appearing in the action of a field under the action of a symmetry group, for example we often say that if $\phi$ transforms as $\phi\mapsto e^{iq\alpha}\phi$ under $U(1)$ then it has charge $q$. This notion of charge is therefore more a choice of the representation in which $\phi$ is.

How are those notions of charges related to each other? Do they have something in common?

Since they are all called charges, I guess that are all conserved in some way. For the Noether charges it's clear but not for the other two.

Additional troubles: the second notion of charge runs, i.e. changes with the energy scale, not the other ones.

Answer 1

The 3 charges you mentioned are essencially the same.

There is a 'Noether charge theorem' which states that the Noether charge should be the quantum generator of the transformation (Weinberg Chapter 7): Under transformation $$\phi(x)\mapsto \phi(x) + \mathcal{F}(x)$$ we have a Noether charge $Q$, and it satisfies: $$ \begin{align} [Q,\phi(y)] &= \int d^3x. [\frac{\delta L}{\delta \partial_0 \phi} \mathcal{F}, \phi(y)]\\ &= [\int d^3x. \mathcal{P} \mathcal{F}, \phi(y)]\\ &= \int d^3x. [\mathcal{P}(x), \phi(y)] \mathcal{F}(x)\\ &= \int d^3x. \delta^{(3)}(x-y) \mathcal{F}(x)\\ &= \mathcal{F}(y) \end{align} $$ where $\mathcal{P} = \frac{\delta L}{\delta \partial_0 \phi} $ is the canonical momentum. And we used canonical commutation relation $[\mathcal{P}(x), \phi(y)] = \delta^{(3)}(x-y)$.

So, the Noether charge is the same as the generator of the representation for this transformation! This relates your 1 & 3, for example, $$ [Q, \phi(x)] = q \phi(x) $$ $Q$ is the generator of the 1-dimensional $u(1)$ representation, and $q$ is the real number that labels it.

For your 2, the coupling constant appears in the Noether charge $Q$: $$ Q = \int d^3x. e\overline{\psi}\gamma^0 \psi $$ or say the coupling constant is proportional to the Noether charge. So these 3 charges are the same notion: the generator of Lie algebra representation.

For your last sentense, it is not relavent to this question. This is when you do renormalization, you choose diffenrent Z's so you have different charges, different representations......