Fermions in orthodox physics are described not simply as solutions of Dirac’s equation on spinors. If this were the definition of particles, then with additional assumption that the geometry of the universe is a scaled 4-sphere with physical universe a hypersurface (determined by the Einstein-Hilbert functional) and mass is identified with mean curvature, then we can safely conclude non-existence of zero mass fermions by showing that nontrivial harmonic eigenspinors cannot exist because they can be extended to as harmonic eigenspinors which is impossible because the scalar curvature there is positive.

The subtleties begin here on the question of mass gap for fermions with a non-Abelian gauge group such as of the strong force. Looking at the paper of Vafa and Witten where they consider uniform upper bounds for Dirac eigenvalues, fermions in gauge theories are eigenvectors of a twisted or extended Dirac operator. Suppose is a riemannian spin manifold with spin bundle . Then let be a Hermitian vector bundle which carries the representation of some gauge group which could be in the case of particular interest. For any connection on one extends the Dirac operator of by adding on sections of and refers to it as .

Vafa and Witten (VafaWittenEigenvalueInequalities and also the wonderful mathematical exposition AtiyahDiracEigenvalues) observe and use the fact that the Atiyah-Patodi-Singer index theorem tells us that the index of a Dirac operator is a topological invariant computable in terms of characteristic numbers of and . They show that the index can be forced to be positive forcing a zero eigenvalue for a twisted Dirac operator.

This means that regardless of whether has no zero solutions (corresponding to ‘zero mass fermion’) in on one bundle, there will be a nontrivial zero for a sufficiently twisted bundle. They produce twisted bundles for by taking a map to a sphere and pulling back bundles with nontrivial ‘winding numbers’.

The above constructions lead one to ask for some clarity on when we should consider a spinor or a section of for to be a ‘massless fermion’, i.e. the interpretation. The Vafa-Witten analysis suggests that this twisting essentially moves the eigenvalues ‘down’ forcing zero eigenvalue by the index theorem.

In any case, showing that a model of physical universe cannot have harmonic spinors in is not sufficient to show nonexistence of massless fermions; one has to show somehow that the the actual bundle for the gauge theory has sufficiently low complexity topology as measured by Chern classes or winding numbers to ensure that the index of does not become nontrivial.

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