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## IN S4 UNIVERSE ALL MASSIVE FERMIONS WILL BE POINT PARTICLES

Consider the exact formula for Dirac eigenspinors on spheres computed by Camporesi-Higuchi (Camporesi-Higuchi-EigenfunctionDiracOnSpheres).  Use the Darboux approximation for Jacobi polynomials.  The major issue is that using the Darboux approximation (see this)and separation of variables, we have the approximation of one of the factors (as $n\rightarrow \infty$)

$\phi_{nl} = \frac{2}{\sqrt{\pi n}} \frac{\cos(Kx+\gamma)}{\sin(x)} + O(n^{-3/2})$

The $\sin(x)$ in the denominator produces a sharp localization.

One can check this directly even for small $n$.

import matplotlib.pyplot as plt
import numpy as np

x=np.linspace(-5e-300,5e-300,1000)
def f(N,l):
a=3./2+float(l)-1
b=3./2+float(l)
gamma=-(a+0.5)*np.pi/2.
K=N+(a+b+1.)/2.
z1=np.sign(np.sin(x))*np.exp(np.log(np.abs(2./np.sin(x)))-0.5*np.log(np.pi*N))
z=z1*np.cos( K*x + gamma)
return(z)

plt.plot(x,f(5,1))
plt.show()

CarrMadan_OptionValuationUsingtheFastFourierTransform_1999