We introduce a continuous tensor product system in the sense of W. Arveson for a general completely positive semigroup of B(H) (H separable). This product system is canonically isomorphic to the product system of the minimal dilation E0-semigroup. We use this construction to show that contrary to previous speculation the minimal dilations of all quantized convolution semigroups are completely spatial. The analysis of these examples involves additionally Levy processes and their stochastic area processes.