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Here's what really bothers me: take QFT, and run a scattering experiment where at t = -∞ the particles going in are asymptotic free states with well-defined energies. Likewise, at t = +∞ one ends up with particles again with well defined energies, and energy is conserved. So there is

*no*possibility here for arbitrarily large momentums - i.e., exactly zero probability that at any point the energy can be larger than the input energy, the energy is bounded.

Now, in some sense (and please correct me if I'm wrong), the entire universe is one large scattering experiment. Then in no case should there ever be a situation in which the momentum wave function for a particle (or whatever this translates to in terms of field excitations in QFT) can have an arbitrarily large momentum (e.g., one should never use a Gaussian with non-zero tails if one truly want real answers... granted the Gaussian makes a great approximation if you basically ignore the tails, which is the case if I understand correctly).

Am I correct with the above?