# Antiparticles in Non-Relativistic Quantum Mechanics
## 3 Quantum Bundles \(L^{\pm}_{\mathbb{C}}\)
Calculating commutator of the vector fields in (36), we obtain the curvature  
\[\mathcal{F}_{xp}=[\
abla_{x},\
abla_{p}]=F_{xp}\partial_{\theta}=\omega_{xp} \partial_{\theta}=-\partial_{\theta} \tag{38}\]of the connection \(A_{\mathsf{vac}}\). Note that fields interacting with \(A_{\mathsf{vac}}\) depend on \(\theta\) as \(\exp(\mathrm{i}q_{\
u}\theta)\), \(q_{\
u}\in\mathbb{Z}\), and, for the cases of quantum charges \(q_{\
u}=\pm 1\) that we are considering here, we obtain  
\[\
abla_{x}=\partial_{x}\pm\mathrm{i}A_{x}\,\quad\
abla_{p}=\partial_{p}\pm \mathrm{i}A_{p}\quad\text{and}\quad\mathcal{F}_{xp}=\pm\mathrm{i}\omega_{xp}= \mp\mathrm{i}\,. \tag{39}\]  
We call the number \(q_{\
u}\) the quantum charge; it distinguishes between particles (\(q_{\
u}\geq 0\)) and antiparticles (\(q_{\
u}\leq 0\)), with \(q_{\
u}=0\) corresponding to neutral particles. Quantum charge \(q_{\
u}\) is related to the winding number in (1). In (39), we have \(A_{\mathsf{vac}}=\mathrm{i}\theta_{\mathbb{R}^{2}}\) for \(q_{\
u}=1\) and \(A_{\mathsf{vac}}=(\mathrm{i}\theta_{\mathbb{R}^{2}})^{*}=-\mathrm{i}\theta_{ \mathbb{R}^{2}}\) for \(q_{\
u}=-1\). **Complex line bundle \(L^{\pm}_{\mathbb{C}}\)**. The fibers of the principal bundle in (34) over points \(x\in\mathbb{R}^{2}\) are groups \(\mathrm{U}(1)_{\
u}\). Let us consider complex one-dimensional vector spaces \(V^{\pm}\) on which the group \(\mathrm{U}(1)_{\
u}\) acts according to the rule \(V^{\pm}\
i\psi_{\pm}\mapsto\exp(\pm\mathrm{i}\theta)\psi_{\pm}\in V^{\pm}\). We associate with \(P(\mathbb{R}^{2},\mathrm{U}(1)_{\
u})\) the complex line bundles  
\[L^{\pm}_{\mathbb{C}}=P\times_{\mathrm{U}(1)_{\
u}}V^{\pm}=\left\{P\times V^{ \pm}\
i(p,\psi_{\pm})\sim(pg_{\pm}^{-1},g_{\pm}\psi_{\pm})\in P\times V^{\pm} \right\}=\mathbb{R}^{2}\times V^{\pm}\, \tag{40}\]  
where \(g_{\pm}=\exp(\pm\mathrm{i}\theta)\). The sign \"\(\sim\)\" means equivalence under the action of the group \(\mathrm{U}(1)_{\
u}\) on the direct product \(P\times V^{\pm}\) of the space \(P\) and \(V^{\pm}\). Spaces \(V^{\pm}\) are introduced as follows. Let us consider two-dimensional columns  
\[\Psi= \begin{pmatrix}\psi^{1}\\ \psi^{2}\end{pmatrix} \in\mathbb{C}^{2} \tag{41}\]  
with complex components \(\psi^{1},\psi^{2}\). These columns are acted upon by matrix  
\[J= \begin{pmatrix}0&-1\\ 1&0\end{pmatrix} \,, \tag{42}\]  
which is the generator of group \(\mathrm{U}(1)_{\
u}\). In the space of \(\mathbb{C}^{2}\)-vectors in (41), we introduce a basis of eigenvectors of the matrix \(J\):  
\[Jv_{\pm}=\pm\mathrm{i}v_{\pm}\ \Rightarrow\ v_{\pm}=\frac{1}{\sqrt{2}} \begin{pmatrix}1\\ \mp\mathrm{i}\end{pmatrix} ,\quad v_{-}=v_{+}^{*}\,\quad v_{\pm}^{\dagger}v_{\pm}=0\,\quad v_{\pm}^{ \dagger}v_{\mp}=0\, \tag{43}\]  
where \"\(\ast\)\" means complex conjugation. These vectors \(v_{\pm}\) are basis vectors in the spaces \(V^{\pm}\), i.e., \(\mathbb{C}^{2}=V^{+}\oplus V^{-}=\mathbb{C}\oplus\mathbb{C}\). Now, Vector (41) can be expanded in \(V^{\pm}\)-parts:  
\[\Psi= \begin{pmatrix}\psi^{1}\\ \psi^{2}\end{pmatrix} =\Psi_{+}+\Psi_{-}=\psi_{+}v_{+}+\psi_{-}v_{-}\in V^{+} \oplus V^{-}\quad\text{with}\quad\psi_{\pm}=\frac{1}{\sqrt{2}}\,(\psi^{1}\pm \mathrm{i}\psi^{2}). \tag{44}\]  
These \(\psi_{\pm}\) are complex coordinates on fibers \(V^{\pm}\) of the bundles \(L^{\pm}_{\mathbb{C}}\) in (40). Note that \(\psi^{1},\psi^{2}\) are complex; therefore, in the general case, \(\psi_{-}\) is not complex conjugate to \(\psi_{+}\) despite the fact that \(v_{-}=\tilde{v}_{+}\equiv v_{+}^{*}\). We introduce a Hermitian structure on the bundles \(L^{\pm}_{\mathbb{C}}\) by equipping fibers \(V^{\pm}\) with the Hermitian inner product:  
\[\langle\psi_{\pm},\psi_{\pm}\rangle=\Psi^{\dagger}_{\pm}\Psi_{\pm}=\psi^{*}_{ \pm}\psi_{\pm} \tag{45}\]  
It is obvious that Metric (45) is invariant under the action \(\psi_{\pm}\mapsto g_{\pm}\psi_{\pm}\) of the group \(\mathrm{U}(1)_{\
u}\) with \(g_{\pm}=\exp(\pm\mathrm{i}\theta)\in\mathrm{U}(1)_{\
u}\). **Complex vector bundle \(L_{\mathbb{C}^{2}}\)**.