  We consider the problem of realizing hyperbolicity cones as spectrahedra,
i.e. as linear slices of cones of positive semidefinite matrices. The
generalized Lax conjecture states that this is always possible. We use
generalized Clifford algebras for a new approach to the problem. Our main
result is that if -1 is not a sum of hermitian squares in the Clifford algebra
of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our
result also has computational applications, since this sufficient condition can
be checked with a single semidefinite program.
