# Antiparticles in Non-Relativistic Quantum Mechanics
## 2 Classical Harmonic Oscillator  
**Symplectic structure. ** In classical mechanics, a simple harmonic oscillator is a particle of mass \(m\) under the influence of a restoring (attractive) force \(F=-m\omega^{2}x\) and the equation of motion  
\[\dot{x}+\omega^{2}x=0\, \tag{6}\]  
where \(x\in\mathbb{R}\) is a coordinate, \(\dot{x}:=\mathrm{d}x/\mathrm{d}t\), \(\omega\) is the frequency and \(t\) is the evolution parameter. Let us consider the phase space \(T^{*}\mathbb{R}=\mathbb{R}^{2}\) of an oscillator with coordinate \(x\) and momentum \(p\), and define on it a symplectic 2-form  
\[\omega_{\mathbb{R}^{2}}=\mathrm{d}\theta_{\mathbb{R}^{2}}=\mathrm{d}p\wedge \mathrm{d}x\quad\text{for}\ \ \theta_{\mathbb{R}^{2}}=\tfrac{1}{2}\,p\mathrm{d}x-\tfrac{1}{2}\,x\mathrm{d}p \tag{7}\]  
with a bivector  
\[\omega_{\mathbb{R}^{2}}^{-1}=\partial_{x}\wedge\partial_{p} \tag{8}\]inverse to the 2-form in (7). We define a function (Hamiltonian) on \(\mathbb{R}^{2}\):  
\[H=\frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}\, \tag{9}\]  
and associate with it the Hamiltonian vector field  
\[V_{H}=\omega_{\mathbb{R}^{2}}^{-1}(\mathrm{d}H)=\partial_{p}H\partial_{x}- \partial_{x}H\partial_{p}=\frac{p}{m}\partial_{x}-m\omega^{2}x\partial_{p}. \tag{10}\]  
Then, Equation (6) can be rewritten in the Hamiltonian form  
\[\dot{x}=V_{H}(x)=\frac{p}{m}\quad\text{and}\quad\dot{p}=V_{H}(p)=-m\omega^{2}x= F\, \tag{11}\]  
where \(F\) is the restoring force. **Complex structure. ** We introduce complex coordinates on \(T^{*}\mathbb{R}=\mathbb{R}^{2}\),  
\[z=\frac{1}{\sqrt{2}}\left(x-\mathrm{i}w^{2}p\right)\quad\text{and}\quad\dot{ z}=\frac{1}{\sqrt{2}}\left(x+\mathrm{i}w^{2}p\right) \tag{12}\]  
with derivatives  
\[\partial_{z}=\frac{1}{\sqrt{2}}\left(\partial_{x}+\frac{\mathrm{i}}{w^{2}} \partial_{p}\right)\quad\text{and}\quad\partial_{\dot{z}}=\frac{1}{\sqrt{2}} \left(\partial_{x}-\frac{\mathrm{i}}{w^{2}}\partial_{p}\right)\,. \tag{13}\]  
Here, \(w\in\mathbb{R}^{+}\) is a length parameter defined as  
\[w^{2}=\frac{1}{m\omega} \tag{14}\]  
so that \([w^{2}p]=[\mathrm{length}]=[x]\). (The dependence of all quantities on [PERSON]'s constant \(\hbar\) is not important for us, so we use the natural units with \(\hbar=c=1\).) Note that \(\partial_{x}\) and \(\partial_{p}\) form the basis of the tangent space \(V=\mathbb{R}^{2}\) to \(T^{*}\mathbb{R}\), and on it we can introduce an endomorphism \(J\in\mathrm{End}(V)\) defined by formulae  
\[J(\partial_{z})=\mathrm{i}\partial_{z}\quad\text{and}\quad J(\partial_{z})=- \mathrm{i}\partial_{z}. \tag{15}\]  
It is easy to see that  
\[J\left(\frac{\partial}{\partial x^{a}}\right)=J_{a}^{b}\,\frac{\partial}{ \partial x^{b}}\quad\text{with}\quad J_{2}^{1}=-1,\ J_{1}^{2}=1\ \Rightarrow\ J= \begin{pmatrix}0&-1\\ 1&0\end{pmatrix} \,, \tag{16}\]  
where \(x^{1}:=x\) and \(x^{2}:=-w^{2}p\). For matrix \(J\in\mathrm{End}(V)\), condition \(J^{2}=-\mathbb{1}_{2}\) is satisfied and \(J\) is called a complex structure on \(\mathbb{R}^{2}\), so that \((\mathbb{R}^{2},J)\cong\mathbb{C}\) is a complex space with coordinate \(z\). From (15), it also follows that \((\mathbb{R}^{2},-J)=\mathbb{C}\) with the complex conjugate coordinate \(\tilde{z}\), i.e., complex conjugation is equivalent to replacing \(J\to-J\) in (16). **Harmonic oscillators. ** In terms of the complex coordinate \(z\) on \(\mathbb{R}^{2}\), Equation (11) and its solutions have the form  
\[\dot{z}=\mathrm{i}\omega z\ \Rightarrow\ z=\mathrm{e}^{\mathrm{i}\omega t}z_{0}\, \tag{17}\]  
where \(z_{0}\) specifies the initial values of \(x\) and \(p\). Note that \(z_{0}=0\) gives a trivial solution \(z=0\) and, therefore, the space \(\mathbb{R}^{2}\backslash\{0\}=\mathbb{C}\backslash\{0\}\) is usually considered as the phase space of the oscillator. We identify this solution with a particle that has a quantum charge \(q_{J}=1\)coinciding with the winding number of the circle \(e^{\mathrm{i}\tau}\) with \(\tau=\omega t\) in (17).