# Antiparticles in Non-Relativistic Quantum Mechanics
## 5 Harmonic Oscillators and Orbifolds  
**Solutions**. We discuss only solutions \(\psi_{+}\) of Equation (86) in the Segal-Bargmann representation since, for \(\psi_{-}\), everything is similar. We also move on to dimensionless coordinate \(z^{\prime}=z/w\) and omit the prime in \(z^{\prime}\) in the formulae below. We have a discrete set of solutions  
\[\Psi_{+}(n)=(\pi n!)^{-\frac{1}{2}}\big{(}e^{i\omega t}z\big{)}^{n}v_{+}^{c}(t )\,\quad v_{+}^{c}(t)=e^{i\omega t/2}v_{+}^{c}=e^{-z\bar{z}/2}e^{i\omega t/2}v_{+} \tag{96}\]  
with the energy  
\[E_{n}=\hbar\omega(q_{l}+\tfrac{1}{2}q_{\
u})=\hbar\omega(n+\tfrac{1}{2})= \hbar\omega n+\tfrac{1}{2}\,\hbar\omega\,\ n=0,1,\ldots, \tag{97}\]  
(we return [PERSON]'s constant to the expressions for the energy levels), where the term \(\hbar\omega n\) is the energy of rotating surface  
\[\mathbb{C}/\mathbb{Z}_{n}=\Big{\{}(z,\psi_{n}(z)=z^{n})\Big{\}}\subset L_{ \mathbb{C}}^{+}\, \tag{98}\]  
and the term \(\tfrac{1}{2}\,\hbar\omega\) is the energy of the rotating basis vector \(v_{+}^{c}\) of fibers \(\mathbb{C}\) of the bundle \(L_{\mathbb{C}}^{+}\). Note that the squared modulus of Function (96) is the Husimi \(Q\)-function:  
\[Q_{n}(x,p)=\Psi_{+}^{\dagger}(n)\Psi_{+}(n)=\frac{1}{\pi n!}\,(zz)^{n}e^{-z \bar{z}}\, \tag{99}\]  
which is a quasiprobability distribution in phase space. From the point of view of gauge theory \((L_{\mathbb{C}}^{+},A_{\text{vac}},\Psi_{+})\), this function is the quantum charge density of section \(\Psi_{+}(n)\) of the bundle \(L_{\mathbb{C}}^{+}\rightarrow\mathbb{R}^{2}\). **Cross-sections of \(L_{\mathbb{C}}^{+}\)**. Thus, we consider the bundle \(L_{\mathbb{C}}^{+}\) with projection \(\pi\):  
\[\pi:\ L_{\mathbb{C}}^{+}=\mathbb{R}^{2}\times\mathbb{C}\cong\mathbb{C}^{2}\ \overset{\mathbb{C}}{\rightarrow}\ \mathbb{R}^{2}\cong\mathbb{C}\, \tag{100}\]and Solution (96) to Equation (86). Recall that the graph of a function \(f:X\to Y\) can be identified with a function \(\sigma\) taking value in the Cartesian product,  
\[\sigma:\ X\to X\times Y\,,\quad\sigma(x)=(x,f(x))\in X\times Y\,. \tag{101}\]  
In our case, we consider a section \(\sigma_{n}\) of Bundle (100),  
\[\sigma_{n}:\ \mathbb{C}\to\mathbb{C}^{2}=L_{\mathbb{C}}^{\perp}\,,\quad\sigma_{n} (z)=(z,\psi_{n}(z))=(z,z^{n})\, \tag{102}\]  
included in Solution (96), where the function  
\[\psi_{n}:z\to z^{n} \tag{103}\]  
defines the graph in \(L_{\mathbb{C}}^{\perp}\). This graph is a one-dimensional complex surface \(\mathbb{C}/\mathbb{Z}_{n}\) in \(L_{\mathbb{C}}^{\perp}\), and Solution (96) describes a standing wave on this surface. **Orbifold \(\mathbb{C}/\mathbb{Z}_{n}\)**. Recall that Map (103) with \(n\geq 2\) is the branched covering of degree \(n\), where \(z=0\) is the branch point. For \(n=1\), the map \(\psi_{1}\) is the identity. Surface (102) in \(L_{\mathbb{C}}^{+}\) is the orbifold \(\mathbb{C}/\mathbb{Z}_{n}\), where \(\mathbb{Z}_{n}=\mathbb{Z}/n\mathbb{Z}\) is the cyclic group of order \(n\), generated by an element \(\zeta\) with \(\zeta^{n}=1\), i.e., \(\zeta\) is \(n\)-th root of unity. Thus, we have the projection  
\[\psi_{n}:\ \mathbb{C}\to\mathbb{C}/\mathbb{Z}_{n}\, \tag{104}\]  
and \(\mathbb{C}\) is the total space of this bundle. Function (103) is an ordinary function and its inverse  
\[\psi_{n}^{-1}:\ \mathbb{C}/\mathbb{Z}_{n}\to\mathbb{C} \tag{105}\]  
is a multivalued function  
\[E_{n}=\hbar\omega(q_{l}+\tfrac{1}{2}q_{\
u})=\hbar\omega(n+\tfrac{1}{2})= \hbar\omega n+\tfrac{1}{2}\,\hbar\omega\,\ n=0,1,\ldots, \tag{97}\]0  
where, abusing this notation, we denote the complex coordinate on \(\mathbb{C}/\mathbb{Z}_{n}\) by \(\psi_{n}\). \(\mathbb{C}/\mathbb{Z}_{n}\) **as a cone. ** The orbifold \(\mathbb{C}/\mathbb{Z}_{n}\) is a metric cone of \(S^{1}/\mathbb{Z}_{n}\). Recall that the metric cone over the circle \(S^{1}\) is Euclidean space \(C(S^{1})=\mathbb{C}\backslash\{0\}\) and, hence, may in fact be continued non-singularly at the cone tip \(z=0\).