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Why does QFT look so different from QM?
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This is only because of technical reasons and the power of tradition.
In ordinary quantum mechanics, pure states are described by
wave functions (more precisely by rays) in a Hilbert space,
there is a Hamiltonian H and an associated Schroedinger equations
i hbar psidot = H psi, the time evolution is described by a unitary
operator, the bound states are normalized eigenstates of the
Hamiltonian, etc.
This is also done in traditional quantum field theory, though it
is not directly apparent. But the Schroedinger equation is still the
thing that makes quantum field theory consistent (at least on a formal
level). As one can read in any QFT textbook, it is needed to derive the
form of the S-matrix and its unitarity. Therefore everyone uses the
Schroedinger equation to derive the basics. But it is no longer very
practical to use it in computations since its use breaks manifest
Lorentz covariance, and therefore soon recedes in the background.
But one can see the importance of the Schroedinger equation when
studying constructive field theory.
All relativistic QFTs that have been constructed rigorously have a
Hamiltonian generating both the time evolution and relating to the
S-matrix in the same way as in simple scattering at external potentials.
Therefore, if one can't construct the Hamiltonian as part of a
nontrivial representation of the Poincare group, one hasn't constructed
a relativistic quantum field theory according to today's standards.
Constructive field theory gives everything in case of 2D quantum
fields. There is a well-defined Hilbert space, a well-defined
Hamiltonian constructed without any use of perturbation theory,
a well-defined unitary dynamics, well-defined bound states that
are eigenstates of the Hamiltonian, and everything is invariant under
the 2D Poincare group ISO(1,1). See the book
J. Glimm and A Jaffe,
Quantum Physics: A Functional Integral Point of View,
Springer, Berlin 1987.
The only thing wanting is an explicit formula for H in the traditional
nonrelativistic form H=H_0+V. Instead, H is constructed in a more
abstract way, as analytic continuation of an operator in Euclidean
field theory.
Note that there is nothing pathological for 2D theories. Once one has
prove the Wightman axioms (and they are proved in 2D), one can apply
all the nice results that can be derived from these axioms, including
the existence of a good scattering theory with a covariant S-matrix.
That the 4D case is more difficult has to do with obstacles in getting
tight enough bounds for the analytic estimates needed. These are
mathematical difficulties, but not inconsistencies - no one proved that
there are contradictions, and the practice of QFT suggests that there
are indeed none (at least for asymptotically free theories).
Therefore, QED is currently still too hard for mathematical physicists,
though they can construct various reasonable approximations to QED,
but not one satisfying the Wightman axioms.
The problem is still open. (On the other hand, they can prove that
Phi^4 theory exists in dimensions 2 and 3, and is trivial in
dimensions >4. The 4D case is borderline and therefore hardest.)
On the perturbative level, there is no difficulty at all - see, e.g.
the book
M Salmhofer,
Renormalization: An Introduction,
Texts and Monographs in Physics,
Springer, Berlin 1999.
which constructs the Euclidean theory for Phi^4 theory in 4 dimensions
perturbatively, i.e., in the formal power series topology, with full
mathematical rigor. If this construction would work nonperturbatively
(i.e., give functions instead of formal power series),
analytic continuation using Osterwalder-Schrader theory would do
the rest. The latter is described, e.g., in Chapter 6 of the above
book by Glimm and Jaffe.