Geology is, of course, the central science: after all, humanity's center of mass sits firmly within our planet's rocky interior. These notes explore some of the amazing phenomena that sculpt the landscapes around us. We begin by studying the subatomic constituents of matter; from there, we compose toward minerals, formations, and continents.
To do geology, we must locate events in space and time. So let us write $\mathcal{S}$ for space-time. We model each type of subatomic particle as a section $\psi$ of some vector bundle $\pi:E\to\mathcal{S}$ over $\mathcal{S}$. The Lagrangian density is $$ \bar{\psi} (D - m) \psi $$ for some appropriate differential operator $D$. Both the fibers and the base of the bundle enjoy some important symmetries.
Our space-time $\mathcal{S}$ enjoys rotation symmetries $SO(3)$. We insist that the Lagrangian density be invariant under rotation; in particular, the mass term must be invariant and hence $SU(2)$ must act on each fiber. The fiber $F$ will be some irreducible representation of $SU(2)$ and hence of $\mathfrak{so}(3)$. Let us classify these irreps. We note that $J_x, J_y, J_z \in \mathfrak{so}(3)$ form a basis, where each represents right-handed rotation along its associated axis. Since $x$-rotation swings the $y$-axis toward the old $z$-axis, we have $[J_x, J_y]=+J_z$. In general, $[J_i,J_j]=\epsilon_{ijk} J_k$. For a fixed irreducible representation $V$, $J_z$ has some eigenvector $v$ with $$ J_z v = \lambda v $$ We note that $$ J_z (J_x v) = J_x (J_z v) + [J_z, J_x] v = (\lambda J_x + J_y) v $$ and likewise $$ J_z (J_y v) = J_y (J_z v) + [J_z, J_y] v = (\lambda J_y - J_x) v $$ Therefore, $$ J_z (J_x \pm i J_y) v = (\lambda \mp i ) (J_x \pm i J_y) v $$ By irreducibility, $J_z$ has an eigenbasis $v_{\lambda + m i}$ indexed by its integrally-spaced eigenvalues for $a\leq m \leq b$. By compactness of $SU(2)$, $\lambda$ is imaginary, without loss $0$. In fact, the most symmetrical situation that $a+b=0$ is always true. Indeed, Let $\boxed{\pm} = (J_x \mp i J_y)/\sqrt{2}$ so that $[\boxed{+}, \boxed{-}] = i J_z$ and $[i J_z, \boxed{\pm}] = \boxed{\pm}$. Then one sees by repeated application of those commutators that $$ 0 = \boxed{-} \boxed{+}^{b-a+1} v_{ai} = -(b-a+1)(b+a)/2 $$ so $a+b=0$. In particular, we have $-b\leq m \leq b$ integrally spaced for $2b$ a non-negative half-integer. We have thus classified (and constructed) all the complex representations of $SU(2)$. The smallest non-trivial representations have $b=1/2$ and $b=1$. These are inhabited, for instance, by the electron and by the triplet state of the heavy Hydrogen nucleus.
$SO(3)$ acts on $\mathcal{S}$. In fact, this action('s complexification) is $Hom(\pi_{1/2}, \pi_{1/2})$, which is two-times-two dimensional and splits into a one-dimensional $\pi_0$ summand (acting on time) and a three-dimensional $\pi_1$ summand (acting on space).
In the case where $\psi$ is a section of $\pi_{1/2}$, we may thus interpret the Lagrangian density precise as: $$ \bar{\psi}_c (\sigma^{c\mu}_{b} D^b_{a\mu} - m I^c_a ) \psi^{a} $$ where $\sigma$ converts from $Hom(\pi_{1/2}, \pi_{1/2})$ to space-time vectors.
The field $\pi_b:E\to \mathcal{S}$ enjoys a connection $\nabla$ that preserves magnitudes on the fibers and commutes with the $SU(2)$ action. The connection induces the $D$ that we invoke in the Lagrangian density $$ \bar{\psi} (\sigma D - m) \psi $$ Let us write $D$ more explicitly by comparing it to some trivialization of $\pi_b$ (that respects magnitudes and commutes with the $SU(2)$ action). The connection form is a one-form $-i e A$ valued in $\mathfrak{u}(2b)$ by norm-preservation and in $i\mathcal{R} = \mathfrak{u}(1) \subseteq \mathfrak{u}(2b)$ because, by irreducibility, nothing but phases commutes with the action of $SU(2)$. In short, we may write $$ D^b_{a\mu} = I^b_a (\partial_\mu - i e A_\mu) $$ where $A$ is a real-valued one-form. Likewise, the curvature form is a two-form $-i F$ valued in $\mathfrak{u}(1)$. In daily life, $\nabla$ is nontrivial in that $F\neq 0$.
It is a calculational convenience to choose a trivialization so that we may write $\nabla$ in terms of $A$, but doing so involves arbitrary choices on which physics does not depend. Still, the connection restricted to topologically nontrivial regions of $\mathcal{S}$ contains more information than the curvature (due to holonomy), and physics depends on this information (the Aharonov–Bohm effect). Therefore, $A$ has too much information, $F$ has too little information, and $\nabla$ is just right as a physical model of the connection.
We promote the connection $\nabla$ to a field with its own dynamics determined by a kinetic term $- F_{\mu\nu} F^{\mu\nu}$. Adopting traditional constants of proportionality, the resulting Lagrangian density is $$ \mathcal{L} = \bar{\psi} (\sigma D_e - m) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} $$
The amplitude to go from a field configuration at time $t$ to another at time $t^\prime$ is an integral over the space $\mathcal{C}$ of interpolating field configurations: $$ \int_{\mathcal{C}} \exp\left(-i \int_{\mathcal{S}} \mathcal{L}/\hbar\right) $$ In the free theory where $e=0$, we may solve exactly. Let us work in momentum space.