1

Noether's theorem states that symmetries are accompanied by a conservation law. So for example, if a system is symmetric under translation in space momentum is conserved; or if a system is symmetric under rotation, angular momentum is conserved.

Now a translation in space can be achieved by applying an infinite number of times an infinitesimal generator (which contains the momentum operator) from the initial point to the point you want to reach.

The same can be said in the case of finite a rotation: it is achieved by applying the infinitesimal generator (which contains the angular momentum operator) an infinite number of times.

Or if the system is symmetric in time, energy is conserved (what is the infinitesimal generator in this case? I guess the energy is part it).

Concerning the symmetries of space and time (which form a four-vector) does the inverse also hold? So if a system is symmetric under a translation in energy-momentum space does that imply that time and space are conserved, which is obviously the case, but can we express that in this way?

Deschele Schilder
  • 1
  • 1
  • The title question about an inverse of Noether's theorem is a duplicate of https://physics.stackexchange.com/q/24596/50583. 2. Your actual question seems to be not about an inverse to Noether's theorem, but about "translations in energy-momentum space". What is "energy-momentum space"? The setting of Noether's theorem is either Lagrangian or Hamiltonian mechanics, and neither of these has such a space, so what are you thinking of here? If you are simply thinking of a covariant phase space, then that would just be an application of the usual Hamiltonian Noether's theorem, no?
    • – ACuriousMind May 09 '17 at 23:02
  • To reopen this question (v1), formulate the actually and title question more carefully than just Does it work the other way around?, which is likely to be misunderstood as a duplicate of https://physics.stackexchange.com/q/24596/2451 – Qmechanic May 10 '17 at 02:43