  The principal object in noncommutatve geometry is the spectral triple
consisting of an algebra A, a Hilbert space H, and a Dirac operator D. Field
theories are incorporated in this approach by the spectral action principle,
that sets the field theory action to Tr f(D^2/\Lambda^2), where f is a real
function such that the trace exists, and \Lambda is a cutoff scale. In the
low-energy (weak-field) limit the spectral action reproduces reasonably well
the known physics including the standard model. However, not much is known
about the spectral action beyond the low-energy approximation. In this paper,
after an extensive introduction to spectral triples and spectral actions, we
study various expansions of the spectral actions (exemplified by the heat
kernel). We derive the convergence criteria. For a commutative spectral triple,
we compute the heat kernel on the torus up the second order in gauge connection
and consider limiting cases.
