# Antiparticles in Non-Relativistic Quantum Mechanics
## 1 Introduction and Summary
** The logic of gauge theories tells us that, if there is a complex line bundle \(L^{+}_{\mathbb{C}}\), then we should also consider the complex conjugate bundle \(L^{-}_{\mathbb{C}}:=\overline{L^{+}_{\mathbb{C}}}\) and associate the charges \(q_{\mathsf{v}}=1\) and \(q_{\mathsf{v}}=-1\) with the structure group \(\mathrm{U}(1)_{\mathsf{v}}\) of these bundles, which leads to the interpretation of sections of these bundles as particles and antiparticles. Observables are introduced not through the mapping \(f\to\hat{f}\), but using a metric on \(X\) and covariant derivatives \(\
abla_{A}\), which automatically gives the Weyl ordering. Connection \(A_{\text{vac}}=-\mathrm{i}\theta_{X}\) and curvature \(F_{\text{vac}}=-\mathrm{i}\omega_{X}\) of the bundle \(L^{-}_{\mathbb{C}}\) have opposite signs compared to those in the bundle \(L^{+}_{\mathbb{C}}\). A non-relativistic quantum antiparticle is a polarized section \(\psi_{-}\in\Gamma(L^{-}_{\mathbb{C}},\mathcal{T}^{*})\) of the complex line bundle \(L^{-}_{\mathbb{C}}\). In Hamiltonian mechanics, this corresponds to a point in the symplectic manifold \((X,-\omega_{X})\) with the evolution parameter \(-t\). The fibers of the principal bundle \(P(X,\mathrm{U}(1)_{\mathsf{v}})\) are circles \(S^{1}\) parametrized by elements \(\mathrm{e}^{\mathrm{i}\theta}\in\mathrm{U}(1)_{\mathsf{v}}\subset\mathbb{C}\), where \(\mathbb{C}\) is a typical fiber of the associated complex line bundle \(L^{+}_{\mathbb{C}}\) with a coordinate \(\psi_{+}=\rho_{+}\mathrm{e}^{\mathrm{i}\theta}\in\mathbb{C}\) on the fiber. A circle \(S^{1}\) is a closed curve on \(\mathbb{C}\) around the point \(\psi_{+}=0\), and it has the winding number \(q_{\mathsf{v}}=1\). The complex conjugate bundle \(L^{-}_{\mathbb{C}}\) has on its fibers a complex coordinate \(\psi_{-}=\rho_{-}\mathrm{e}^{-\mathrm{i}\theta}\) and the winding number \(q_{\mathsf{v}}=-1\). In general, the winding number of a curve in \(\mathbb{C}\) is the total number of times that the curve travels counterclockwise around a fixed point for \(q_{\mathsf{v}}=r\) and clockwise for \(q_{\mathsf{v}}=-s\), \(r,s=0,1,\ldots\) For \(S^{1}\subset\mathbb{C}\), these curves correspond to one-dimensional representations of the group \(\mathrm{U}(1)\) and \(\mathbb{C}^{*}\), defined by the homomorphisms  
\[f_{q_{\mathsf{v}}}:\ S^{1}\to S^{1}\quad\text{with}\quad f_{q_{\mathsf{v}}}( \mathrm{e}^{\mathrm{i}\theta})=\mathrm{e}^{\mathrm{i}q_{\mathsf{v}}\theta} \quad\text{for}\ \ q_{\mathsf{v}}\in\pi_{1}(S^{1})=\mathbb{Z}\, \tag{1}\]  
and  
\[\psi_{+}\to\psi_{+}^{r}\quad\text{for}\ \ q_{\mathsf{v}}=r\geq 0\quad\text{and} \quad\psi_{-}\to\psi_{-}^{s}\quad\text{for}\ q_{\mathsf{v}}=-s\leq 0. \tag{2}\]  
The first mapping in (2) defines the tensor product of bundles \((L^{+}_{\mathbb{C}})^{\otimes r}\) with \(q_{\mathsf{v}}=r\), and the second mapping in (2) defines the tensor product of bundles \((L^{-}_{\mathbb{C}})^{\otimes s}\) with \(q_{\mathsf{v}}=-s\). We call \(q_{\mathsf{v}}\) a _quantum charge_; sections of bundles with \(q_{\mathsf{v}}\geq 0\) are _particles_, and those with \(q_{\mathsf{v}}\leq 0\) are _antiparticles_. Sections with \(q_{\mathsf{v}}=0\) are neutral particles; these are sections of the diagonal subbundle in the bundle \(L^{+}_{\mathbb{C}}\oplus L^{-}_{\mathbb{C}}\). We emphasize that all quantum particles have a quantum charge, including those that have zero electric charge. **Problem statement. ** In classical Hamiltonian mechanics and the corresponding non-relativistic quantum mechanics (QM), there is no concept of antiparticles. These theories describe only particles, and the number of particles \(q_{\mathsf{v}}=r\) is given as initial data and cannot change (particles can be created and destroyed only in quantum field theory). In Newtonian mechanics, non-relativistic particles are described by a phase space \(X=T^{*}\mathbb{R}^{3}=\mathbb{R}^{3}\times\mathbb{R}^{3}\) parametrized by coordinates \(x\in\mathbb{R}^{3}\) and momenta \(p\in\mathbb{R}^{3}\).