## EIGENSPINORS OF DIRAC OPERATOR ON THREE-SPHERE: WAVE-PARTICLE DUALITY OF FERMIONS

March 26, 2016 by zulfahmed

Quantum theory posits some strange features of particles such as ‘wave-particle’ duality. Here I note that the eigenspinors of the Dirac opertor on a three-sphere have a wavelet-type localization. In other words, even without a quantum theory, compactness of the space is sufficient to produce localization of Dirac eigenspinors. From eigenspinors to the solution of the Dirac wave equation is just multiplication by a phase factor, i.e.

solves . With a mass term, one has the Dirac equation

Now the eigenspinors have an explicit expression for spheres given by Camporesi-Higuchi. They find the Dirac eigenspinors by separation of variables for which the first factor can be used to show the ‘wave-particle duality’ issue. Approximation using the Darboux formula of this factor gives us

where

Now we can pick a high value say and look at the graph of with . We can see the localization of the eigenspinor clearly. Note that no strange theory of wave-particle duality is needed in this case as this is a static eigenspinor of the Dirac operator. This particular shape of the Dirac eigenspinor is due to the geometry of , the global geometry. I suggest that much of the complex obfuscation of quantum theory would be clarified if one developed physics directly on compact hypersurfaces of a scaled four-sphere which I claim is the actual shape of the universe. Mystical interpretations of physics arises from this key missing link that we’re in a compact universe. We need to have a clearer science of physics. The Big Bang never happened either. God does not play dice with the universe and so on.

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