# Antiparticles in Non-Relativistic Quantum Mechanics
## 4 Complex Polarizations
In complex coordinates \(z_{\pm}\), we have \[\
abla_{z_{\pm}}=\frac{1}{\sqrt{2}}\bigg{(}\
abla_{x}\pm\frac{\mathrm{i}}{w^{2}} \,\
abla_{p}\bigg{)}=\partial_{z_{\pm}}+\frac{\underline{z}_{\pm}}{2w^{2}}\;, \quad\
abla_{\underline{z}_{\pm}}=\frac{1}{\sqrt{2}}\bigg{(}\
abla_{x}\mp \frac{\mathrm{i}}{w^{2}}\,\
abla_{p}\bigg{)}=\partial_{\underline{z}_{\pm}}- \frac{z_{\pm}}{2w^{2}} \tag{71}\]  
which implies that  
\[A_{z_{\pm}}=\frac{\underline{z}_{\pm}}{2w^{2}}\quad\text{and}\quad A_{z_{\pm}} =-\frac{\underline{z}_{\pm}}{2w^{2}}\;. \tag{72}\]  
Using Function (70) as an automorphism of the bundle \(L^{\pm}_{\mathbb{C}}\), we obtain the following components of the connection in the holomorphic bases \(v^{\varepsilon}_{\pm}\):  
\[A^{\Phi_{0}}_{z_{\pm}}=A_{z_{\pm}}+\phi_{0}^{-1}\partial_{z_{\pm}}\phi_{0}=0\;,\quad A^{\Phi_{0}}_{z_{\pm}}=A_{z_{\pm}}+\phi_{0}^{-1}\partial_{z_{\pm}}\phi_ {0}=-\frac{z_{\pm}}{w^{2}} \tag{73}\]  
\[\Rightarrow\;\
abla^{\Phi_{0}}_{z_{\pm}}=\frac{\partial}{\partial z_{\pm}} \quad\text{and}\quad\
abla^{\Phi_{0}}_{z_{\pm}}=\frac{\partial}{\partial \underline{z}_{\pm}}-\frac{z_{\pm}}{w^{2}}\;. \tag{74}\]  
Under this automorphism, the Dolbeault operators in (66) transform into ordinary \(\bar{\partial}\)-operators  
\[\tilde{\partial}^{\Phi_{0}}_{L^{\pm}_{\mathbb{C}}}=\mathrm{d}\underline{z}_{ \pm}\partial_{z_{\pm}} \tag{75}\]  
with partial derivatives \(\partial_{z_{\pm}}\). They commute with the covariant derivatives in (74). Therefore, connections (73) are holomorphic but not Hermitian. **Ladder operators. ** From (71), we see that the annihilation and creation operators for bundles \(L^{\pm}_{\mathbb{C}}\) have the form  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]0  
when they act on sections \(\Psi_{\pm}\) from (68). After Transformations (73)-(75), they take the usual form  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]1  
when acting on holomorphic functions \(\psi_{\pm}(z_{\pm},t)\) of \(z_{\pm}\). The scalar product of such functions has the form  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]2  
as it should be in the Segal-Bargmann representation [11; 12; 13]. **Complex \(\Rightarrow\) real. ** Note that Real Polarization (55) and Transformations (57)-(59) to the coordinate representation can be obtained as the limiting case of Complex Polarization (67):  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]3  
It is not difficult to verify that Operators (76) on such functions reduce to operators  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]4  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]5  
i.e., to standard ladder operators in coordinate representation. Similarly, Real Polarization (56) and Transformations (62)-(64) to the momentum representation can be obtained as another limit of Complex Polarization (67):  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]6Operators (76) on the functions in (82) are reduced to ladder operators \(a_{\pm}\), \(a_{\pm}^{\pm}\) in momentum representation. Here, we do not write out their explicit form. **Covariant Laplacians. ** Having considered real and complex polarizations, we move on to defining Hamiltonians for harmonic oscillators with \(q_{\
u}=\pm 1\). To do this, we introduce covariant Laplacians,  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]7  
acting on the polarized sections in (68) of the bundles \(L^{\pm}_{\mathbb{C}}\). Substituting the explicit form in (71) of covariant derivatives into (83), we obtain  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]8  
We can now introduce natural geometric Hamiltonians  
\[\tilde{\partial}_{L^{\pm}_{\mathbb{C}}}\Psi_{\pm}=0 \tag{67}\]9  
defined on functions \(\psi_{\pm}(z_{\pm},t)\). Here, we used the connection in (14) between \(w^{2}\) and \(m\omega\). **Schrodinger equations. ** Recall that \(z_{-}=z_{+}\) and \(\Psi_{\pm}\) are sections of complex conjugate bundles \(L^{\pm}_{\mathbb{C}}\).