Stone's theorem tells us when a one-parameter unitary group has a self-adjoint generator. If U(t) is a group --- i.e., it satisfies U(t1+t2)=U(t1)U(t2) --- then we can write U(t)=eiHt for some unbounded H if and only if we have limt→t0U(t)|ψ⟩=U(t0)|ψ⟩ for every |ψ⟩ in Hilbert space. This is the condition that U(t) is strongly continuous . What if we have a unitary group with multiple parameters? Within any one-parameter subgroup we should be able to find a generator. But is there a connection between the generators of different subgroups? Concretely, imagine we have a two-parameter subgroup that forms a representation of R2. I.e., assume we have U(t1+t2,s1+s2)=U(t1,s1)U(t2,s2). Assume further that the map from R2 to unitary operators is strongly continuous. Then for any one-parameter subgroup we have a generator, in particular, we have U(t,0)=eiH1t and...