I've got another homework problem that I have trouble with:
4.3 Exercise 3
Use the Feynman rules to show that the expansion in number of loops is an expansion in powers of $\hbar$. Hint: obviously you will have to reinstate the factors $\hbar$ into the expressions for the interaction vertex and the Feynman propagators! You will also need the rule we derived which states that the number of loops (which is the same as the number of undetermined momenta) is equal to the number of propagators in a diagram minus the number of interaction vertices plus one. [So $L = I – V + 1$]
So far I've got:
The factors for internal lines will be proportional to some power $\hbar^x$ (this must be the same power for fermion lines and photon lines, or otherwise the question isn't answerable). I think both factors
$$
\text{i}D_{\text{F}\alpha\beta}(k) = \text{i} \frac{-g_{\alpha\beta}}{k^2 + \text{i}\varepsilon}
$$
and
$$
\text{i}S_F(p) = \text{i}\frac{1}{\not{p}-m+\text{i}\varepsilon}
$$
need to be dimensionless, but I don't see how you could get them to be dimensionless (or even to be the same dimension) by just adding $c$'s and $\hbar$'s. Could someone actually write these expressions with the appropriate factors inserted?
The factors for vertices will be proportional to some power $\hbar^y$. But since the expression here is
$$
\text{i}e \gamma^\alpha
$$
which carries units of charge, I'm lost again.
Once that's solved, the total S-matrix amplitude of a diagram should be proportional to $(\hbar^x)^I (\hbar^y)^V = \hbar^{xI+yV}$. The question, again, cannot be answered properly unless it happens that $x = -y$ (which I'm therefore assuming will turn out to be the case), in which case you can use $L = I – V + 1$ to see that $\hbar^{xI+yV} = \hbar^{x(I-V)} = \hbar^{x(L-1)}$.
So I'm stuck until I find out how to properly deal with those units
Side question to anyone who's actually working in theoretical physics: are natural units really worth it? So far, for all the slight simplification of written expressions they allow, the dimensional confusion they create has caused me nothing but annoyance.
Best Answer
The easiest way to see how you can recover the powers of $\hbar$ is by looking at the path integral, which (in natural units) is given by \begin{align} \mathcal{Z}=\int \left[\mathcal{D}\phi\right]e^{iS}\end{align} with the action $S$. Going back to dimensionful units, $S$ will have dimension of an action, so the same as $\hbar$. To make the argument of the exponential dimensionless, we therefore need the path integral to be \begin{align} \mathcal{Z} = \int \left[\mathcal{D}\phi\right] e^{\frac{i}{\hbar}S},\end{align} so the action is rescaled by $\frac{1}{\hbar}$. This means that every internal propagator gets a factor of $\hbar$ (because it is the inverse of the kinetic term) and every vertex gets a factor of $\frac{1}{\hbar}$. You can probably continue from here on your own.
One thing to note is, that $D_F$ and $S_F$ don't need to have the same dimensions, as one of those has the dimensions of a bosonic field squared and the other those of a fermion field squared (remember that they are given as expectation values of time-ordered field products).