  We consider an elliptic self-adjoint first order pseudodifferential operator
acting on columns of m complex-valued half-densities over a connected compact
n-dimensional manifold without boundary. The eigenvalues of the principal
symbol are assumed to be simple but no assumptions are made on their sign, so
the operator is not necessarily semi-bounded. We study the spectral function,
i.e. the sum of squares of Euclidean norms of eigenfunctions evaluated at a
given point of the manifold, with summation carried out over all eigenvalues
between zero and a positive lambda. We derive a two-term asymptotic formula for
the spectral function as lambda tends to plus infinity. We then restrict our
study to the case when m=2, n=3, the operator is differential and has
trace-free principal symbol, and address the question: is our operator a
massless Dirac operator? We prove that it is a massless Dirac operator if and
only if the following two conditions are satisfied at every point of the
manifold: a) the subprincipal symbol is proportional to the identity matrix and
b) the second asymptotic coefficient of the spectral function is zero.
