Feeds:
Posts

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

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 $D_{S^3} \psi_0(x) = \lambda \psi_0(x)$ to the solution of the Dirac wave equation is just multiplication by a phase factor, i.e.

$\psi(t,x) = e^{i\lambda t/\hbar} \psi_0(x)$

solves $D\psi = i\hbar \partial_t \psi$.  With a mass term, one has the Dirac equation

$(mc + D)\psi = i\hbar\partial_t \psi$

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

$\phi_{nl}(\theta) = 1/\sqrt{n\pi} \cos(N\theta + \gamma)$

where $a=3/2+l-1$

$b=3/2+l$

$N=n+(a+b+1)/2$

$\gamma = -(a+1/2)\pi/2$

Now we can pick a high value say $n=10^{12}$ and look at the graph of $\phi_{n0}(\theta)$ with $\theta \in (-\epsilon_0, \epsilon_0)$.  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 $S^3$, 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.