In the C*-algebraic setting of quantum theory, the physical states S are represented by a large convex subset of the states of a unital C*-algebra, and the bounded observables by the self adjoint part of this C*-algebra A [12, 16, 18, 25, 31]. For a precise formulation of what is meant by a large set of states see [18]. Time development, at least for Markov processes, is given by a continuous semigroup {zl:^ 0} of afline transformations on the physical states. We adopt the position that this flow arises as follows, noting that this assumption can be rigorously deduced in certain circumstances, see [18]. There is a strongly continuous semigroup of necessarily positive unital maps {Tt: t^ 0} on the C* algebra A such that in the Schrodinger picture states evolve as f-* vJ= foTt feS, t&0 whilst in the Heisenberg picture, observables evolve as a-> T,(a) aeAh, t> 0.

We denote the infinitesimal generator of the semigroup {Tt: t^ 0} by L. If the dynamics is reversible then the flow xt extends to a group of alTine transformations on the state space S. In this case Kadison [18] showed further that the corresponding strongly continuous group of positive unital maps Tt on A is automatically a group of*-automorphisms, and so the generator L is a derivation. If the group is norm continuous, then whenever n is a representation of A on a Hilbert space H, there exists [19, 28] a self-adjoint operator h in the weak closure of n (A) such that L is implemented by ih in the representation n. That is to say