# Antiparticles in Non-Relativistic Quantum Mechanics
## 3 Quantum Bundles \(L^{\pm}_{\mathbb{C}}\)
Group \(\mathrm{U}(1)_{\mathsf{v}}\) acts on the space of \(\mathbb{C}^{2}\)-vectors in (41) by multiplying on the left by the matrix  
\[e^{\theta J}=\cos\theta+J\sin\theta= \begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix} \in\mathrm{SO}(2)_{\mathsf{v}}\;. \tag{46}\]  
For subspaces \(V^{\pm}\) in \(\mathbb{C}^{2}=V^{+}\oplus V^{-}\), we obtain  
\[\Psi(\theta)=e^{\theta J}\Psi=\Psi_{+}(\theta)+\Psi_{-}(\theta)=e^{\mathrm{i} \theta}\psi_{+}v_{+}+e^{-\mathrm{i}\theta}\psi_{-}v_{-}\;, \tag{47}\]  
which coincides with the definition in (40) of spaces \(V^{\pm}\) in \(L_{\mathbb{C}}^{\pm}\). Note that the action of the generator \(\partial_{\theta}\) of group \(\mathrm{U}(1)_{\mathsf{v}}\) on \(\Psi(\theta)\) has the form  
\[\partial_{\theta}\Psi(\theta)=J\Psi(\theta)=\mathrm{i}\Psi_{+}(\theta)- \mathrm{i}\Psi_{-}(\theta)\;, \tag{48}\]  
i.e., it is equivalent to the action of the generator \(J\) from (42). The column vectors in (44) are sections of the complex vector bundle  
\[L_{\mathbb{C}^{2}}:=L_{\mathbb{C}}^{+}\oplus L_{\mathbb{C}}^{-} \tag{49}\]  
with the structure group given in (46)-(48). The \(\mathbb{C}^{2}\)-bundle in (49) inherits its connection \(A_{\mathsf{vac}}\) and curvature \(F_{\mathsf{vac}}\) from connections and curvature (39) on \(L_{\mathbb{C}}^{\pm}\):  
\[A_{\mathsf{vac}}=AJ=\theta_{\mathbb{R}^{2}}J\quad\text{and}\quad F_{\mathsf{ vac}}=\mathrm{d}A_{\mathsf{vac}}=J\mathrm{d}\theta_{\mathbb{R}^{2}}=\omega_{ \mathbb{R}^{2}}J\;. \tag{50}\]  
The components of this connection are given in (35):  
\[A_{x}=\tfrac{1}{2}p\quad\text{and}\quad A_{p}=-\tfrac{1}{2}x\;. \tag{51}\]  
Accordingly, the covariant derivatives on \(L_{\mathbb{C}^{2}}\) have the form  
\[\
abla_{x}=\partial_{x}+A_{x}J=\partial_{x}+\tfrac{1}{2}pJ\quad\text{and}\quad \
abla_{p}=\partial_{p}+A_{p}J=\partial_{p}-\tfrac{1}{2}xJ\;. \tag{52}\]  
Note that \(A_{\mathsf{vac}}\in\mathrm{so}(2)_{\mathsf{v}}\) and, therefore, the connection \(A_{\mathsf{vac}}\) is compatible with the Hermitian metric  
\[\langle\Psi,\Psi\rangle=\Psi^{\dagger}\Psi \tag{53}\]  
**Operators \(\hat{p}\) and \(\hat{x}\)**. We consider quantum mechanics as a gauge theory of fields \(\psi_{\pm}\in\Gamma(L_{\mathbb{C}}^{\pm})\) with \(q_{\mathsf{v}}=\pm 1\) interacting with gauge fields \(A_{\mathsf{vac}}=\mathrm{i}q_{\mathsf{v}}\theta_{\mathbb{R}^{2}}\) defined on these bundles. We also use the bundle in (49) to describe \(\psi_{\pm}\) simultaneously as Sections (44), (47) of the bundle \(L_{\mathbb{C}^{2}}\). As has been noted more than once, the spaces \(\Gamma(L_{\mathbb{C}}^{\pm})\) of sections of bundles \(L_{\mathbb{C}}^{\pm}\) are too large and they need to be narrowed down to spaces of irreducible representations of CCR by imposing conditions  
\[X_{\pm}\psi_{\pm}=0\,,\quad\psi_{\pm}\in\Gamma(L_{\mathbb{C}}^{\pm})\;,\quad X _{\pm}\in\Gamma(\mathcal{T}_{\pm})\;,\quad\mathcal{T}_{\pm}\subset T^{\mathbb{ C}}\mathbb{R}^{2}\;, \tag{54}\]  
where \(X_{\pm}\) are vector fields from the subbundles \(\mathcal{T}_{\pm}\) of the complexified tangent bundle of the phase space \(T^{*}\mathbb{R}\) of oscillators. For real polarizations \(\mathcal{T}_{\pm}\), they are real subbundles of the tangent bundle \(T\mathbb{R}^{2}\) and we can consider polarization for sections of the bundle \(L_{\mathbb{C}^{2}}=L_{\mathbb{C}}^{+}\oplus L_{\mathbb{C}}^{-}\) since \(\mathcal{T}_{+}=\mathcal{T}_{-}\). In the two-dimensional case that we are considering, the real polarization is either the independence of sections \(\Psi\in\Gamma(L_{\mathbb{C}^{2}})\) from the momenta  
\[\partial_{p}\Psi=0\quad\Rightarrow\quad\Psi=\Psi(x,t)\, \tag{55}\]  
or their independence from the coordinates  
\[\partial_{x}\Psi=0\quad\Rightarrow\quad\Psi=\Psi(p,t). \tag{56}\]  
After imposing one of these conditions, we arrive at quantum mechanics in coordinate or momentum representation. Let us see how this works for the polarization in (55). Note that vector field \(\partial_{p}\) does not commute with the covariant derivatives in (52), which is unacceptable.