In this post, quantum mechanics falls under what is traditionally called "first quantization". This is in contrast to quantum field theory which traditionally falls under "second quantization".
In textbook quantum mechanics, a system $A$ with states $\{S_i\}$ is represented by the subset of unit trace and positive-definite vectors in the Banach algebra of trace-class linear operators over a Hilbert space, denoted $\overline{\mathcal{B}}^+(\mathcal{H}) \subset \mathcal{B}(\mathcal{H})$. The dynamics of the system are generated by a Hamiltonian $H \in \mathcal{B}(\mathcal{H})$ via a dynamic relation called the von Neumann equation \begin{equation} \frac{\partial}{\partial t} \rho = -\frac{i}{\hbar}[H, \rho], \quad \rho \in \overline{\mathcal{B}}^+(\mathcal{H}). \end{equation} We observe that traditionally the Hamiltonian may include terms we call classical potential terms. A canonical example of a classical field term is the Coulomb interaction term in the basic hydrogen atom Hamiltonian: \begin{equation} H_{hyd} := -\frac{\hbar^2}{2m}\nabla^2 - \frac{e^2}{4\pi\epsilon_0r}. \end{equation} Another example of a classical potential is the Chern-Simons potential defined as follows. Consider a $\mathfrak{g}$-valued connection $1$-form $\omega \in \mathfrak{g} \otimes \Omega^1(P)$ where $P$ is a principal bundle. A Chern-Simons potential $A$ is the pullback of $\omega: P \to \mathfrak{g} \otimes T^*P$ via a global section $\sigma: M \to P$ satisfying the equation of motion obtained via stationizing an action containing at least the following terms \begin{equation} S = \frac{k}{8\pi^2}\int_M \kappa\left( A \stackrel{\wedge}{,} dA + \frac{1}{3} [A \stackrel{\wedge}{,} A] \right), \quad k \in \mathbb{R}\backslash\{0\} \end{equation} where $\kappa$ is the Killing form with domain extended from $\mathfrak{g}$ to $\mathfrak{g}$-valued $1$-forms.
I have only ever seen Chern-Simons talked about in the context of (classical or) quantum field theory, not in quantum mechanics. Can one write down a quantum mechanics Hamiltonian that includes a Chern-Simons potential? Can this potential be coupled with particles?
