# Antiparticles in Non-Relativistic Quantum Mechanics
## 1 Introduction and Summary  
**Prequantum gauge theory. ** Let us consider an even-dimensional manifold \(X\) with a symplectic two-form \(\omega_{X}\) (\(={\rm d}\theta_{X}\) locally) as the phase space of Hamiltonian mechanics. A non-relativistic classical particle is a point in space \(X\) moving along a trajectory in \(X\) given by a Hamiltonian vector field \(V_{H}\) generated by a function \(H\) (Hamiltonian) with an evolution parameter \(t\). In the approach of geometric quantization (see e.g., [1, 2]), which is a neat mathematical reformulation of the canonical quantization procedure, the transition from \((X,\omega_{X})\) to quantum mechanics is carried out through the following:  
* The introduction of the principal U(1)-bundle \(P(X,{\rm U}(1)_{\
u})\) over \(X\) and the associated complex line bundle \(L_{\
u}\to X\) with connection \(A_{\rm vac}={\rm i}\,\theta_{X}\) (we use the natural units with \(\hbar=c=1\)) and curvature \(F_{\rm vac}={\rm i}\,\omega_{X}\);
* The introduction of polarization \(\mathcal{T}\) on \(X\), i.e., integrable Lagrangian subbundle of the complexified tangent bundle \(T^{\mathbb{C}}X\) of \(X\). Both \(A_{\rm vac}\) and \(F_{\rm vac}\) take values in the Lie algebra \({\rm u}(1)_{\
u}={\rm Lie}{\rm U}(1)_{\
u}\). The abbreviations \"\(\omega\)\" and \"\(\rm vac\)\" here mean \"vacuum\" since \(\theta_{X}\) and \(\omega_{X}\) have no sources and define a symplectic structure on \(X\). The polarization \(\mathcal{T}\) makes it possible to impose on sections \(\psi\) of the bundle \(L_{\mathbb{C}}^{+}:=L_{\
u}\) the condition \({\rm d}_{\mathcal{T}}\psi=0\) of independence from one \"half\" of the coordinates on \(X\). The space \(\mathcal{H}\) of polarized sections of \(L_{\mathbb{C}}^{+}\) is the quantum Hilbert space and functions \(\psi\in\mathcal{H}\) are the \"wavefunctions\" of quantum mechanics. Accordingly, the bundle \(L_{\mathbb{C}}^{+}\) with a polarization \(\mathcal{T}\) is a quantum bundle encoding the basic quantum mechanical (QM) information. By definition, a non-relativistic quantum particle is a polarized section \(\psi_{+}\in\Gamma(L_{\mathbb{C}}^{+},\mathcal{T})\) of the complex line bundle \(L_{\mathbb{C}}^{+}\) over \(X\). **QM as gauge theory. ** The above two steps (bundle \(L_{\mathbb{C}}^{+}\) polarization \(\mathcal{T}\)) towards the introduction of quantum mechanics do not encounter obstacles because it is simply the introduction of a bundle over \(X\) with some conditions on \(X\) and \(L_{\mathbb{C}}^{+}\), for example, the holomorphicity condition. Moreover, it is not difficult to show that covariant derivatives in \(L_{\mathbb{C}}^{+}\) can be chosen so that they coincide with the standard operators of coordinates and momenta. Having covariant derivative \(\
abla_{A}\) acting on polarized sections \(\psi_{+}\) of the bundle \(L_{\mathbb{C}}^{+}\), one can construct quantum Hamiltonians as is customary in differential geometry. However, geometric quantization, like any other quantization scheme, also attempts to define a map:  
* Functions \(f\in C^{\infty}(X)\ \to\ \text{operators}\ \hat{f}\) on a Hilbert space \(\mathcal{H}\), which should satisfy some axioms [1,2]. But this does not work because of the problem with operator ordering, inconsistency with polarization and many other problems. For this reason we do not use either the map \(f\mapsto\hat{f}\) or the metaplectic correction, but keep only the first two steps and consider quantum mechanics as the theory of fields \(\psi_{+}\in\Gamma(L^{+}_{\mathbb{C}},\mathcal{T})\), which are acted upon by covariant derivatives \(\
abla_{A}\) with canonical connection \(A_{\text{vac}}=\mathrm{i}\,\theta_{X}\). **Antiparticles and quantum charges.