Basics: dimensional analysis

I decide to begin writing some back-of-the-envelope stuff in this blog; mostly very basic things.

Consider Dirac fermion coupled to $U(1)$ field.

$$\mathcal{L}=\bar{\psi}\gamma^\mu (\partial_\mu-iA_\mu) \psi - \frac{1}{e^2} F_{\mu\nu}F^{\mu\nu}. $$

In d spacetime dimensions, the first part in the kinetic term implies $[\partial]+2[\psi]=d$, leading to $[\psi]=(d-1)/2$. The second part leads to $2[\psi]+[A]=d$. Plugging in the value for $[\psi]$, we have $[A]=1$, then $[F]=2$.

Using this and plug back in the Maxwell term $2[F]-[e^2]=d$, we arrive at $[e^2]=4-d$.

When we are in 3+1d, the gauge coupling is thus dimensionless. For 2+1d, the gauge coupling has dimension $[e^2]=1$, so this theory is strongly coupled in the infrared. Also, the potential energy between two charges increases logarithmically with respect to the distance between them, which is a very mild form of confinement.

References: 

[1] David Tong, Lectures on gauge theory, chapter 8. http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html 

[2] Chat with Hao-Yu Sun.