We derive the Faddeev–Popov identity using a discrete warm-up, insert it into the path integral, extract the gauge-fixed photon propagator, and ask: what other gauge-fixing terms are allowed?

Why you can't just write down the path integral for a gauge theory and call it a day — the operator you can't invert, the orbits you can't avoid, and the geometric picture that makes gauge fixing click.

Checking if Bose-Einstein Condensation can occur in 1D and 2D for free particles using periodic boundary conditions — and why the density of states is the key.

Deriving the Mandelstam variables s, t, u for e+e- → μ+μ- scattering in the center-of-mass frame — from kinematics to Lorentz-invariant cross sections.

Deriving the virial expansion for a classical gas of hard spheres — from the Mayer f-function to excluded volume and higher-order corrections.