# Antiparticles in Non-Relativistic Quantum Mechanics
## 4 Complex Polarizations  
**Dolbeault operators. ** We considered bundles \(L^{\pm}_{\mathbb{C}}\) with anti-Hermitian connections \(A_{\mathsf{vac}}\) and Automorphisms (57)-(59), (62) that transform covariant derivatives with this connections into operators \(\hat{x}\) and \(\hat{p}\) in the irreducible coordinate representation in (60) and the momentum representation in (62)-(64). Now, we describe how complex polarizations \(\mathcal{T}_{\pm}\subset T^{\mathbb{C}}\mathbb{R}^{2}\) are introduced. For them, particles \(\Psi_{+}\in L^{+}_{\mathbb{C}}\) are holomorphic functions of the complex coordinate \(z_{+}\) from (12) (Segal-Bargmann representation [11; 12; 13]) and antiparticles \(\Psi_{-}\in L^{-}_{\mathbb{C}}\) are holomorphic functions of the complex coordinate \(z_{-}\) on \(\mathbb{R}^{2}\) given in (20) (antiholomorphic in \(z_{+}\) since \(z_{-}=z_{+}\)). To define holomorphic structures in the bundles \(L^{\pm}_{\mathbb{C}}\), we introduce the Dolbeault operators  
\[\partial_{L^{\pm}_{\mathbb{C}}}=\mathrm{d}\mathbf{z}_{\pm}\bigg{(}\frac{\partial} {\partial\mathbf{z}_{\pm}}+\frac{z_{\pm}}{2w^{2}}\bigg{)}\;, \tag{66}\]  
and impose the conditions  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]  
on sections \(\Psi_{\pm}\) of the bundles \(L^{\pm}_{\mathbb{C}}\). These are conditions for holomorphic polarization, and their solutions are the functions  
\[\Psi_{\pm}=\psi_{\pm}(z_{\pm},t)v^{\mathrm{c}}_{\pm}\quad\text{with}\quad v^{ \mathrm{c}}_{\pm}=e^{-z\overline{z}/2w^{2}}v_{\pm}\;, \tag{68}\]  
where \(z\overline{z}=z_{+}\mathbf{\bar{z}}_{+}=z_{-}\mathbf{\bar{z}}_{-}=\frac{1}{2}\left(x^ {2}+w^{4}p^{2}\right)\). Note that \(z_{-}=\mathbf{\bar{z}}_{+}\) and the operator \(\tilde{\partial}_{L^{-}_{\mathbb{C}}}\) is complex conjugate to the operator \(\tilde{\partial}_{L^{+}_{\mathbb{C}}}\), but the functions \(\psi_{+}(z_{+},t)\) and \(\psi_{-}(z_{-},t)\) in the general case are not related by complex conjugation. **Covariant derivatives. ** Note that the basis vectors \(v_{\pm}\) of the complex line bundles \(L^{\pm}_{\mathbb{C}}\) define the Hermitian Metrics (45) and (53) in fibers. These are Hermitian bases of Hermitian bundles. At the same time, the basis vectors \(v^{\mathrm{c}}_{\pm}\) in (68) define in \(L^{\pm}_{\mathbb{C}}\) the complex bases associated with the principal bundle  
\[P(\mathbb{R}^{2},\mathrm{GL}(1,\mathbb{C})_{\mathsf{v}})\to\mathbb{R}^{2} \tag{69}\]  
having the structure group \(\mathrm{GL}(1,\mathbb{C})_{\mathsf{v}}=\mathbb{C}^{*}\supset\mathrm{U}(1)_{ \mathsf{v}}\), and the previously considered bundle \(P(\mathbb{R}^{2},\mathrm{U}(1)_{\mathsf{v}})\) is a Hermitian subbundle [14] in Bundle (69). The function  
\[\tilde{\varphi}_{0}=\exp\Bigl{(}-\frac{z\overline{z}}{2w^{2}}\Bigr{)} \tag{70}\]  
in (68) is an element of the group \(\mathrm{GL}(1,\mathbb{C})_{\mathsf{v}}\) that defines a mapping of Hermitian bases \(v_{\pm}\) into holomorphic bases \(v^{\mathrm{c}}_{\pm}\) along which the holomorphic sections in (68) of the bundles \(L^{\pm}_{\mathbb{C}}\) are decomposed. In the covariant derivatives in (39), we initially use anti-Hermitian connections \(A_{\mathsf{vac}}=\pm\mathrm{i}\theta_{\mathbb{R}^{2}}\).