# Phenomenologies in Hypersphere Soliton and String Photon Models
## 3 Intrinsic Frequencies of Baryons  
Now, we investigate the intrinsic frequencies of baryons in the first-class Hamiltonian formalism in the HSM. To do this, in the HSM we construct the equivalent first-class Hamiltonian \(\hat{H}^{\prime}\) as follows  
\[\hat{H}^{\prime}=\hat{H}+\frac{1}{4\mathcal{I}}\pi_{\theta}\hat{\Omega}_{2}, \tag{35}\]  
to yield the corresponding Gauss law constraint algebra  
\[\{\hat{\Omega}_{1},\hat{H}^{\prime}\}=\frac{1}{2\mathcal{I}}\hat{\Omega}_{2},\ \ \ \{\hat{\Omega}_{2},\hat{H}^{\prime}\}=0. \tag{36}\]  
Note that \(\{\hat{\Omega}_{1},\hat{H}\}=0\) and \(\{\hat{\Omega}_{2},\hat{H}\}=0\). We then find the Hamiltonian spectrum for \(\hat{H}^{\prime}\)  
\[\langle\hat{H}^{\prime}\rangle=E_{B}+\frac{1}{2\mathcal{I}}\bigg{[}I(I+1)+ \frac{1}{4}\bigg{]}, \tag{37}\]  
which is equal to that for \(\hat{H}\) in (25), as expected. Next, we consider the equation of motion in Poisson bracket form  
\[\hat{\mathcal{W}}=\{\hat{W},\hat{H}^{\prime}\},\ \ \text{for the first class variable}\ \hat{\text{W}}, \tag{38}\]  
where the over-dot denotes the time derivative. Making use of the equation of motion in (38), we obtain these two equations  
\[\hat{u}^{\mu}=\{\hat{u}^{\mu},\hat{H}^{\prime}\}=\frac{1}{4\mathcal{I}}\hat{ \pi}^{\mu},\ \ \ \hat{\pi}^{\mu}=\{\hat{\pi}^{\mu},\hat{H}^{\prime}\}=-\frac{1}{4\mathcal{I}}\hat {\pi}^{\
u}\hat{\pi}^{\
u}\hat{u}^{\mu}, \tag{39}\]where the first-class fields \(\bar{a}^{\mu}\) and \(\tilde{\pi}^{\mu}\) are given as follows  
\[\tilde{a}^{\mu}=a^{\mu}\bigg{(}\frac{a^{\
u}a^{\
u}+2\theta}{a^{\
u}a^{\
u}} \bigg{)}^{1/2},\quad\tilde{\pi}^{\mu}=(\pi^{\mu}-a^{\mu}\pi_{\theta})\bigg{(} \frac{a^{\
u}a^{\
u}}{a^{\
u}a^{\
u}+2\theta}\bigg{)}^{1/2}. \tag{40}\]  
In order to formulate the equations in (39), we have used the following identities among the physical fields  
\[\{\tilde{a}^{\mu},\tilde{\pi}^{\
u}\} = \delta^{\mu\
u}-\tilde{a}^{\mu}\tilde{a}^{\
u},\qquad\{\tilde{\pi }^{\mu},\tilde{\pi}^{\
u}\}=\tilde{\pi}^{\mu}\tilde{a}^{\
u}-\tilde{a}^{\mu} \tilde{\pi}^{\
u},\] \[\{\tilde{a}^{\mu},\tilde{H}\} = \frac{1}{4\mathcal{I}}(\tilde{\pi}^{\mu}-\tilde{a}^{\mu}\tilde{a} ^{\
u}\tilde{\pi}^{\
u}),\quad\{\tilde{\pi}^{\mu},\tilde{H}\}=\frac{1}{4 \mathcal{I}}(\tilde{\pi}^{\mu}\tilde{a}^{\
u}\tilde{\pi}^{\
u}-\tilde{a}^{\mu }\tilde{\pi}^{\
u}\tilde{\pi}^{\
u}),\] \[\{\tilde{a}^{\mu},\pi_{\theta}\} = \tilde{a}^{\mu},\qquad\qquad\qquad\{\tilde{\pi}^{\mu},\pi_{ \theta}\}=-\tilde{\pi}^{\mu}. \tag{41}\]  
Applying the equation of motion algorithm in (38) to \(\tilde{a}^{\mu}\), we find  
\[\tilde{a}^{\mu}=\{\tilde{a}^{\mu},\tilde{H}^{\prime}\}=\frac{1}{4\mathcal{I}} \hat{\pi}^{\mu}=-\frac{1}{4\mathcal{I}^{2}}\bigg{[}I(I+1)+\frac{1}{4}\bigg{]} \tilde{a}^{\mu}, \tag{42}\]  
to yield the equation of motion for a simple harmonic oscillator  
\[\{\hat{\Omega}_{1},\hat{H}^{\prime}\}=\frac{1}{2\mathcal{I}}\hat{\Omega}_{2},\ \ \ \{\hat{\Omega}_{2},\hat{H}^{\prime}\}=0. \tag{36}\]0  
where \(\omega_{I}\) is the intrinsic frequency of pulsating baryon with isospin quantum number \(I\) given by  
\[\{\hat{\Omega}_{1},\hat{H}^{\prime}\}=\frac{1}{2\mathcal{I}}\hat{\Omega}_{2},\ \ \ \{\hat{\Omega}_{2},\hat{H}^{\prime}\}=0. \tag{36}\]1  
Making use of the formula for \(\omega_{I}\) in (44) for the nucleon \(N\)\((I=\frac{1}{2})\) and the delta baryon \(\Delta\)\((I=\frac{3}{2})\), we obtain predictions of intrinsic frequencies \(\omega_{N}\) and \(\omega_{\Delta}\) of the baryons given in \[\{\hat{\Omega}_{1},\hat{H}^{\prime}\}=\frac{1}{2\mathcal{I}}\hat{\Omega}_{2},\ \ \ \{\hat{\Omega}_{2},\hat{H}^{\prime}\}=0. \tag{36}\]2 Note that we find the identity \(\omega_{\Delta}=2\omega_{N}\). Finally, it seems appropriate to comment on the gauge fixing problem within the first-class constraints of the Dirac Hamiltonian formalism.