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## There cannot be massless fermions in our universe

UPDATE:  The following note is incorrect.  It is true and standard that a manifold with positive scalar curvature does not have spinors annihilated by the Dirac operator.  However, this is when one uses the spin connection.  For the fermions, one does not use the spin connection but a Yang-Mills connection.  For those, we don’t have a simple result but we can learn something about the dimension of the zero sets of the Dirac operator using the Atiyah-Singer index theorem.  The equation for zero-eigenvalue modes in that case is
$i \gamma^\nu ( \partial_\mu + A_\mu) \psi = 0$

and the Atiyah-Singer index theorem tells us the difference

$n_+ - n_- = \frac{1}{32\pi^2}\int d^4x *F_a^{\mu\nu} F_{a\mu\nu}$

The right side is essentially the Pontrjagin number of the bundle.

I have argued elsewhere that the actual universe is a scaled four-sphere.  It has constant positive sectional curvature and hence constant positive scalar curvature.  Now fermions appear essentially as solutions to $D\psi = m \psi$.  Spinors on a four sphere are spanned by Killing spinors.  A Bochner-Weitzenbock formula reads
$D^2 = \nabla^*\nabla + C(K)$

where the additional term is the scalar curvature multiplied by a universal constant.  Then taking inner product and integration by parts shows that nonconstant spinors do not exist.  The argument itself is classical.  What is interesting is that on a four dimensional sphere there can be no massless fermion by this simple argument.