I would be very grateful if you could help me with some insight into the concepts of time translation and dynamic evolution in QM and QFT.
Any relativistic quantum system must be Poincare covariant. Cf. Wigner, the Poincare transformations are represented by unitary operators in the system's Hilbert space. The infinitesimal generators of the unitary operators are Hermitian operators corresponding to basic observables. The very definition of the way the time translation operator acts in either Schroedinger or Heisenberg picture is formally equivalent with the Schroedinger and Heisenberg equations of motion (in integrated form), respectively. For example, $\psi (t+a) = U(a)\psi (t)$, etc.
Question 1: Is it sufficient to postulate Poincare covariance instead of postulating the Schroedinger or Heisenberg eqs. of motion? That is, are the Schroedinger or Heisenberg eqs. of motion really consequences (byproducts) of postulating Poincare invariance?
Question 2: I've read a bit axiomatic qft (AQFT) and saw that there is no axiom (postulate) for dynamics among Wightman's postulates. Why is it so? Is it not needed due to an yes answer to Question 1?
I understand that it's hard to come up with a nice Hamiltonian $H$ for a realistic system in AQFT, but the equations of motions should exist at least formally, with an yet to be found $H$. Am I wrong? What are the dynamical equations for a general system (field or otherwise)?
Question 3: Is there a difference between time translation and dynamical evolution? If yes, which is that from the math and physics point of view? Are they described by different operators? Could you provide a simple example for illustration?