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is Fock space Poincaré invariant? As far as I can see, the scalar product in Fock space involves the scalar products in its N-particle subspaces, which, in turn, are the integrals of the properly (anti-)symmetrized wave functions over space.

This works well in a Galilei-invariant theory because a Galilei transformation just tilts the time axis and leaves the space axis untouched, so space is the same in all inertial systems. But a Lorentz transformation also tilts the space axis (in 2D for simplicity). This means that the spatial integrals in one inertial system extend over points of space-time with different time coordinates in another inertial system. So the dynamics seems to play an important role in guaranteeing the normalization of the state. Is this accounted for in the construction of Hamiltonians in QFT?

What's more, the state has to be normalized for any inertial observer: for any possible space axis the sum over particle number of the spatial integrals has to be one. This seems to be an extremely restrictive condition to me. Can this be guaranteed? Can Fock space be a state space for interacting QFT at all?

Thanks for your answers and comments!

Cheers,

Gedankenspiel