Zeta-Functions and Star-Products

We use the definition of a star (or Moyal or twisted) product to give a phasespace definition of the $\zeta$-function. This allows us to derive new closed expressions for the coefficients of the heat kernel in an asymptotic expansion for operators of the form $\alpha p^2+v(q)$. For the particular case of the harmonic oscillator we furthermore find a closed form for the Green's function. We also find a relationship between star exponentials, path integrals and Wigner functions, which in a simple example gives a relation between the star exponential of the Chern-Simons action and knot invariants.