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Hilbert space and Hamiltonian in relativistic quantum field theory
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Most of current quantum field theory (i.e., everything with exception
of 2D and 3D constructive field theory - which doesn't even cover QED)
does not have a well-defined Hilbert space at all, in which a
time operator would be defined.
Well-defined are only the asymptotic Hilbert spaces of in and out
states for scattering experiments. These are Fock spaces of
free particles, and hence defined on a mass shell.
There is a basic result called Haag's theorem which states that
these asymptotic Fock spaces cannot carry a nontrivial local dynamics,
as would be required for a field theory.
The full dynamics can in general be defined only indirectly, via the
Wightman functions; see the entry ''Time evolution in quantum field
theories'' in this FAQ. These can be computed approximately using
CTP (closed time path) integration. But from a rigorous point of view,
they are subject to the unsolved problem of rigorous, nonperturbative
renormalization procedures.
Rigorously constructing for a relativistic field theory a physical
Hamiltonian which is bounded below is really difficult, and has
been achieved only in less than 4D theories.
The construction is usually based on a preferred time coordinate
which is needed in all cases I am familiar with;
- in the Foldy-Wouthuysen transformation (for the Dirac equation,
where p_0 also fails to have the right properties),
- in the Newton-Wigner construction (for single particles in
an arbitrary massive irreducible representation of the
Poincare group) and
- in the Osterwalder-Schrader reconstruction theorem (for
Lorentz-invariant field theories from Euclidean field theories).
While the Hilbert space and the Hamiltonian depend on the choice of
the time coordinate, the physics is independent of it since all these
Hilbert spaces are isomorphic via isomorphisms that maps the
Hamiltonians into each other.