An origin of spins of fields

Abstract

Spins of fields are investigated in terms of the zero-energy eigenstates of 2-dimensional Schr$\ddot {\rm o}$dinger equations with central potentials $V_a(\rho)=-a^2g_a\rho^{2(a-1)}$ ($a\not=0$, $g_a>0$ and $\rho=\sqrt{x^2+y^2}$). We see that for $a=N/2$ ($N=$positive odd integers) one half spin states can naturally be understood as states with the angular momentum $l=1$ in the $\zeta_a$ plane which is obtained by mapping the $xy$ plane in terms of conformal transformations $\zeta_a=z^a$ with $z=x+iy$. It is shown that the scalar and the 1/2-spin fields can obtain masses. Vortex structures and a supersymmetry for the zero-energy states are also pointed out.