# Phenomenologies in Hypersphere Soliton and String Photon Models
## 3 Intrinsic Frequencies of Baryons
In order to investigate the gauge fixing of the first-class Hamiltonian \(\tilde{H}^{\prime}\) in (35), we introduce two canonical sets of ghost and anti-ghost fields together with auxiliary fields \((\mathcal{C}^{i},\mathcal{P}_{i})\), \((\mathcal{P}^{i},\mathcal{C}_{i})\), \((\mathcal{N}^{i},\mathcal{B}_{i})\), \((i=1,2)\), which satisfy the super-Poisson algebra, \(\{\mathcal{C}^{i},\mathcal{P}_{i}\}=\{\mathcal{P}^{i},\mathcal{C}_{i}\}=\{ \mathcal{N}^{i},\mathcal{B}_{i}\}=\delta_{i}^{i}\). Here, the super-Poisson bracket is defined as \(\{A,B\}=\frac{\delta A}{\delta q}|\frac{\delta B}{\delta p}|_{l}-(-1)^{\eta_{ A}\eta_{B}\frac{\delta B}{\delta q}|}\frac{\delta A}{\delta p}|_{l}\), where \(\eta_{A}\) denotes the number of fermions, called the ghost number, in \(A\) and the subscript \(r\) and \(l\) denotes right and left derivatives, respectively. The BRST charge for the first-class constraint algebra related with \(\tilde{H}^{\prime}\) is then given by  
\[\{\hat{\Omega}_{1},\hat{H}^{\prime}\}=\frac{1}{2\mathcal{I}}\hat{\Omega}_{2},\ \ \ \{\hat{\Omega}_{2},\hat{H}^{\prime}\}=0. \tag{36}\]3  
We choose the unitary gauge  
\[\{\hat{\Omega}_{1},\hat{H}^{\prime}\}=\frac{1}{2\mathcal{I}}\hat{\Omega}_{2},\ \ \ \{\hat{\Omega}_{2},\hat{H}^{\prime}\}=0. \tag{36}\]4 \begin{table}
\begin{tabular}{c c c} \hline
**Particle Type** & **Notation** & **Intrinsic Frequency** \\ \hline Nucleon & \(\omega_{N}\) & \(0.87\times 10^{23}\) s\({}^{-1}\) \\ Delta baryon & \(\omega_{\Delta}\) & \(1.74\times 10^{23}\) s\({}^{-1}\) \\ Photon & \(\omega_{\gamma}\) & \(9.00\times 10^{23}\) s\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 2: The intrinsic frequencies of particles [9; 13]. by selecting the fermionic gauge fixing function \(\Psi\): \(\Psi=\mathcal{C}_{i}\chi^{i}+\bar{\mathcal{P}}_{i}\mathcal{N}^{i}\). Exploiting the BRST charge \(Q\) in (45), we find the BRST transformation rule defined as \(\delta_{Q}F=\{Q,F\}\) for a physical field \(F\)  
\[\delta_{Q}a^{\mu}=-\mathcal{C}^{2}a^{\mu}, \delta_{Q}\pi^{\mu}=2\mathcal{C}^{1}a^{\mu}+\mathcal{C}^{2}(\pi^{ \mu}-2a^{\mu}\pi_{\theta}), \delta_{Q}\theta=\mathcal{C}^{2}a^{\mu}a^{\mu}, \delta_{Q}\pi_{\theta}=2\mathcal{C}^{1}, \delta_{Q}\mathcal{C}^{i}=0, \tag{47}\] \[\delta_{Q}\bar{\mathcal{P}}_{i}=\bar{\Omega}_{i}, \delta_{Q}\mathcal{P}^{i}=0, \delta_{Q}\mathcal{N}^{i}=-\mathcal{P}^{i}, \delta_{Q}\bar{\mathcal{B}}_{i}=0.\]  
Note that \(\bar{\mathcal{H}}^{\prime}\) is not BRST invariant, which implies that \(\delta_{Q}\bar{\mathcal{H}}^{\prime}\
eq 0\). Next, we obtain the gauge-fixed Hamiltonian  
\[\bar{H}^{\prime\prime}=\bar{H}^{\prime}-\frac{1}{2\mathcal{I}}\mathcal{C}^{1 }\bar{\mathcal{P}}_{2}, \tag{48}\]  
which is now invariant under the BRST transformation rule in (47), namely \(\delta_{Q}\bar{H}^{\prime\prime}=0\). Note that the BRST charge \(Q\) in (45) is nilpotent so that we can have \(\delta_{Q^{2}}F=\{Q,\{Q,F\}\}=0\) for a physical field \(F\). Note also that \(H_{\mathit{eff}}\equiv\bar{H}^{\prime\prime}-\{Q,\Psi\}\) is the BRST invariant Hamiltonian including the fermionic gauge-fixing function \(\Psi\).