Building the language of Lie groups and their representations from the ground up — generators, structure constants, representations, Casimir operators, and the exponential map.

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.

Decomposing the fluctuations of massive vectors and massless 2-forms to determine their physical degrees of freedom and observational signatures.

Applying the cosmological principle to massive vectors and massless 2-forms to see if they can drive the expansion of the Universe.

A deep dive into gauge symmetry, equations of motion, constraints, and degrees of freedom for massive vectors and massless 2-forms.

An introduction to using higher-rank tensor fields like vectors and 2-forms as dark energy candidates, going beyond the scalar field quintessence.

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.