# Antiparticles in Non-Relativistic Quantum Mechanics
## 3 Quantum Bundles \(L^{\pm}_{\mathbb{C}}\)  
**Gauge theory \(\Rightarrow\) QM. ** Repeating what was said in the introduction, we emphasize that we are not engaged in quantization in the spirit of the Dirac program [10] and are not considering the mapping \(f\mapsto\hat{f}\) of functions \(f\) on phase space into quantum operators \(\hat{f}\). Instead, we consider a gauge theory on phase space described by the set \((L^{+}_{\mathbb{C}},A_{\mathsf{vac}},\mathcal{T})\), where the connection \(A_{\mathsf{vac}}\) defines the canonical commutation relations (CCR), and the polarization \(\mathcal{T}\) defines the Hilbert space on which the CCRs are irreducibly realized. Note that the connection \(A_{\mathsf{vac}}\) on \(L^{+}_{\mathbb{C}}\) is given up to an automorphism of the bundle \(L^{+}_{\mathbb{C}}\) and, by choosing different automorphisms, we can obtain coordinate, momentum, holomorphic or antiholomorphic representations that are unitarily equivalent by virtue of the Stone-von Neumann theorem. We show how all these representations are obtained from the choice of polarization \(\mathcal{T}\) and automorphism from the group \(\mathrm{Aut}(L^{+}_{\mathbb{C}})\) for the case of harmonic oscillators. All observables are introduced only through the operators of covariant derivatives and a metric on phase space. The field \(A_{\mathsf{vac}}\in\mathsf{u}(1)_{\mathsf{v}}\) enters in these covariant derivatives and determines the interaction of particles with vacuum. **Principal bundle \(P(\mathbb{R}^{2},\mathsf{U}(1)_{\mathsf{v}})\). ** Let us consider a Newtonian particle of mass \(m\) in one-dimensional space \(\mathbb{R}\). On its phase space \(T^{*}\mathbb{R}=\mathbb{R}^{2}\), the symplectic structure in (7), the complex structure in (12)-(16) and the metric in (33) are given. This particle is a harmonic oscillator if we choose the Hamiltonian of the form in (9). This Hamiltonian defines the vector field in (10), which has the geometric meaning of Generator (24) of the group SO(2)\(\cong\)U(1) acting on \(\mathbb{R}^{2}\cong\mathbb{C}\) by the rotations in (25). A particle oscillating in space \(\mathbb{R}\) corresponds to a point \(z_{+}(t)\) on phase space \(\mathbb{R}^{2}\) moving in a circle (28) with a winding number \(q_{l}=1\). Antiparticles are described by a trajectory \(z_{-}(t)\) in \(\mathbb{R}^{2}\) with a winding number \(q_{l}=-1\). The difference between particles and antiparticles is associated with orientation on circles in phase space and orientation on the time axis. To describe quantum harmonic oscillators, it is first necessary to define a principal bundle \(P(\mathbb{R}^{2},\mathsf{U}(1)_{\mathsf{v}})\) with structure group \(\mathsf{U}(1)_{\mathsf{v}}\) and connection \(A_{\mathsf{vac}}\). To do this, consider the space  
\[P(\mathbb{R}^{2},\mathsf{U}(1)_{\mathsf{v}})=\mathbb{R}^{2}\times S^{1}\quad \text{with}\quad S^{1}\cong\mathsf{U}(1)_{\mathsf{v}} \tag{34}\]  
and introduce on it a one-form \(A_{\mathsf{vac}}=\mathcal{A}_{x}\mathrm{d}x+\mathcal{A}_{p}\mathrm{d}p\) along \(\mathbb{R}^{2}\) whose components are vector fields along \(S^{1}\),  
\[\mathcal{A}_{x}=A_{x}\partial_{\theta}\,\quad\mathcal{A}_{p}=A_{p}\partial_{ \theta}\,\quad A=A_{x}\mathrm{d}x+A_{p}\mathrm{d}p=\theta_{\mathbb{R}^{2}}=\tfrac{1}{ 2}\,p\mathrm{d}x-\tfrac{1}{2}\,x\mathrm{d}p. \tag{35}\]  
Here, \(0\leq\theta<2\pi\) is a coordinate on \(S^{1}\). Then, we introduce, on the space in (34), vector fields  
\[\
abla_{x}=\partial_{x}+A_{x}\partial_{\theta}\,\quad\
abla_{p}=\partial_{p} +A_{p}\partial_{\theta}\quad\text{and}\quad\
abla_{\theta}=\partial_{\theta} \tag{36}\]  
and dual one-forms  
\[\Xi^{x}=\mathrm{d}x\,\quad\Xi^{p}=\mathrm{d}p\quad\text{and}\quad\Xi^{ \theta}=\mathrm{d}\theta-A_{x}\mathrm{d}x-A_{p}\mathrm{d}p. \tag{37}\]  
Vector fields in (36) and one-forms in (37) form a frame and coframe on \(P(\mathbb{R}^{2},\mathsf{U}(1)_{\mathsf{v}})\) as a manifold.