# Phenomenologies in Hypersphere Soliton and String Photon Models
## 4 SPM Predictions
Note that, similar to the phonon associated with massive particle lattice vibrations, the photon is massive so that we can have three polarization directions: two transverse directions, as in the massless photon case, and an additional longitudinal one. Keeping this argument in mind, we find that there exist two transverse modes \(z_{1}\) and \(z_{2}\) associated with the photon vibrations on \(z_{1}\)-\(z_{2}\) cross sectional disk, in addition to one longitudinal mode \(z_{3}\). We thus have the transverse mode in \(z_{1}\) direction to yield the eigenfunction for the ground state, with an arbitrary phase constant \(\phi_{1}\) similar to \(\phi_{2}\) and \(\phi_{3}\) discussed above,  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]3Note that, as in the case of massless photons, \(z_{1}\) mode oscillates with the same frequency \(\omega\) that \(z_{2}\) mode does. Note also that the above solutions \(z_{i}\) satisfy their endpoint conditions at \(\sigma=0\) and \(\sigma=\pi\)  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]4  
The energy eigenvalues in the ground states in (76)-(78) are then given by  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]5  
Exploiting the energies in (80), we arrive at the vibrational energy of the open string ground state  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]6  
In the SPM, the classical energy of the string is given by \(E^{rot}\) in (70), while the corresponding quantum mechanical energy is given by \(E^{\text{vib}}\) in (81). Now, we define the total energy of the string configuration as a function of \(\omega\)  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]7  
Note that we have already removed the translational degree of freedom, by considering the string observer residing on the photon center of mass in the SPM. Variating the energy \(E\) with respect to \(\omega\), we find the minimum value condition for \(E\) at \(\omega=\omega_{\gamma}\)  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]8  
to yield the intrinsic frequency \(\omega_{\gamma}\) for the photon [13]  
\[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]9  
which is greater than the baryon intrinsic frequencies, as shown in \[\frac{\partial p_{\mu}}{\partial\tau}+\frac{\partial\pi_{\mu}}{\partial\sigma }=0, \tag{63}\]0 Note that we have related the HSM and SPM, in terms of the intrinsic frequencies of the baryons and photon, both of which are the extended objects. Next, we consider the photon radius as a half of the open string length. Exploiting the photon intrinsic frequency \(\omega_{\gamma}\) in (84), we obtain the photon radius [13]  
\[\frac{\partial p_{\mu}}{\partial\tau}+\frac{\partial\pi_{\mu}}{\partial\sigma }=0, \tag{63}\]1  
which is \(21\%\) of the proton magnetic charge radius \(\langle r^{2}\rangle_{M}^{1/2}(\text{proton})=0.80\) fm as shown in \[\frac{\partial p_{\mu}}{\partial\tau}+\frac{\partial\pi_{\mu}}{\partial\sigma }=0, \tag{63}\]2  
Next, even though up to now we have investigated the stringy photon model in \(D=3+1\) dimensional spacetime [13], we parenthetically discuss the bosonic string theory in the critical dimension \(D=26\).