### Planets: 0001

Alpha piles himself up near the lake with his eyes closed, feeling the first rain after returning to Planet A. The raindrops, with average diameters of 10 centimeters, are products of relatively weak gravity and strong air resistance. Calmly they fall, and some choose to stay in Alpha's palm. In a distance behind the gentle curtain of the rain, the peach-colored horizon is hazy. The air is full of fragrances of methane and ethane.

When on Earth, Alpha liked to linger near the ports. The gasoline and rotten fish smell like the air on Planet A, relieving his Nostalgia. But now back here, he misses the aroma of fried chicken on Earth.

The drops beat the muffled drum in his heart. Alpha pulls back his arm into his body - "What is Beta doing right now?"

Shaking his head, Alpha leaves his baggage onshore and swims towards the center of the lake, alone.

### 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:   David Tong, Lectures on gauge theory, chapter 8. http://www.damtp.cam.ac.uk/user/tong/ga