# Antiparticles in Non-Relativistic Quantum Mechanics
## 2 Classical Harmonic Oscillator
To indicate the sign of this winding number, we rewrite solution (17) as  
\[z_{+}=e^{\mathrm{i}\omega t}z_{0}\quad\text{with}\quad z_{0}=\frac{1}{\sqrt{2}} \left(x_{0}-\mathrm{i}w^{2}p_{0}\right)=\rho_{0}e^{\mathrm{i}\varphi_{0}},\quad w ^{2}p_{0}=\frac{v_{0}}{\omega}\;, \tag{18}\]  
where \(v_{0}\) is the initial velocity. Let us now make the substitution \(t\to-t\) in (11). This is equivalent to replacing \(p\to-p\) and changing the sign of the complex structure on \(\mathbb{R}^{2}\):  
\[J_{p}^{x}=\frac{1}{w^{2}}\ \to\ J_{p}^{x}=-\frac{1}{w^{2}}\quad\text{and}\quad J _{x}^{p}=-w^{2}\ \to\ J_{x}^{p}=w^{2}\;, \tag{19}\]  
and therefore obtaining a new complex coordinate  
\[z_{-}=\frac{1}{\sqrt{2}}\left(x+\mathrm{i}w^{2}p\right)\,. \tag{20}\]  
In terms of \(z_{-}\), Equation (11) and its solutions have the form  
\[z_{-}=-\mathrm{i}\omega z_{-}\quad\Rightarrow\quad z_{-}=e^{-\mathrm{i} \omega t}z_{0}^{\prime}\quad\text{with}\quad z_{0}^{\prime}=\frac{1}{\sqrt{2}} \left(x_{0}^{\prime}+\mathrm{i}w^{2}p_{0}^{\prime}\right)=\rho_{0}e^{-\mathrm{ i}\varphi_{0}^{\prime}}\;, \tag{21}\]  
where \(z_{0}^{\prime}\) does not necessarily coincide with \(z_{0}\). This solution has \(q_{l}=-1\), since it describes movement along \(S^{1}\) clockwise. Thus, the reversal of time corresponds to a change \(J\to-J\) in the sign of the complex structure, a change in the sign \(\omega_{\mathbb{R}^{2}}\to-\omega_{\mathbb{R}^{2}}\) and a replacement of \(q_{l}=1\) (particle) with \(q_{l}=-1\) (antiparticle). On the plane \((x^{1},x^{2})=(x,-w^{2}p)\), we have  
\[ \begin{split}& x(t)=x_{0}\cos\omega t+v_{0}\frac{\sin\omega t}{ \omega}\;,\quad p_{0}=mv_{0}\;,\\ & p(t)=p_{0}\cos\omega t-m\omega x_{0}\sin\omega t\;,\end{split}  \tag{22}\]  
where \(x_{0},p_{0}\) are initial data for \(q_{l}=1\). For \(q_{l}=-1\), we have (22) with \(t\to-t\) and \((x_{0},p_{0})\to(x_{0}^{\prime},p_{0}^{\prime})\). Thus, we obtain  
\[ \begin{split}&\left(z_{+}=e^{\mathrm{i}\omega t}z_{0},\ \omega_{\mathbb{R}^{2}},\ J,\ q_{l}=1\right)=\text{particle}\\ &\left(z_{-}=e^{-\mathrm{i}\omega t}z_{0}^{\prime},\ -\omega_{\mathbb{R}^{2}},\ -J,\ q_{l}=-1\right)=\text{antiparticle}\end{split}  \tag{23}\]  
Note that exactly the same difference as between \(z_{+}\) and \(z_{-}\) exists for vortices and antivortices on the plane \(\mathbb{R}^{2}\); they are also associated with complex conjugate spaces and are characterized by integer winding numbers with \(q_{l}>0\) and \(q_{l}<0\) (see, e.g., [8]). In conclusion, we note that, in (23), we consider a solution \(z_{+}\) with positive and \(z_{-}\) with negative frequency, as is customary in mathematics. In physics, it is often accepted the other way around, and, to do this, one simply needs to swap the complex and complex conjugate fields and solutions. **Symplectic reduction. ** The complex structure in (16) can be associated with the vector field  
\[\mathcal{J}=J_{b}^{a}x^{b}\partial_{a}=x^{1}\partial_{2}-x^{2}\partial_{1}=w^{2} p\partial_{x}-\frac{x}{w^{2}}\,\partial_{p}=\mathrm{i}(z\partial_{z}-z \partial_{z})=\partial_{\varphi}\;. \tag{24}\]  
This vector field defines the group \(\mathrm{U}(1)\) of transformations of complex coordinates \(z\) and \(\bar{z}\):  
\[\mathrm{U}(1)\
i g=e^{\mathrm{i}a\partial_{\varphi}}:\quad gz=e^{\mathrm{i}a} z\quad\text{and}\quad g\bar{z}=e^{-\mathrm{i}a}\bar{z}\;, \tag{25}\]  
corresponding to rotations on the plane \(\mathbb{R}^{2}\) with winding numbers \(q_{l}=1\) and \(q_{l}=-1\), respectively. At the same time, comparing (10) and (24), we see that  
\[V_{H}=\omega\mathcal{J}\;\Rightarrow\;\dot{z}_{+}=\omega\mathcal{J}(z_{+}) \quad\text{and}\quad\dot{z}_{-}=\omega\mathcal{J}(z_{-})\;, \tag{26}\]  
i.e., the Hamiltonian of the oscillator has a geometric origin. This leads to a geometric interpretation of both equations in (26) and their solutions. To see the geometry behind (26), we identify the dual space \(\mathfrak{u}(1)^{*}\) of the Lie algebra \(\mathfrak{u}(1)\cong\,\mathrm{Lie}\,\mathrm{U}(1)\) with the generator in (24) with the space \(\mathbb{R}\).