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S6d. Is there a rigorous interacting QFT in 4 dimensions?
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The Wightman axioms and the Osterwalder-Schrader axioms
[see, e.g., math-ph/0001010 or the book by Glimm and Jaffe] 
are currently the basis on which rigorous quantum field theory 
(at least for massive particles) is discussed.

In spite of many attempts (and though numerous uncontrolled 
approximations are routinely computed), no one has so far succeeded 
in rigorously constructing a single QFT in 4D which 
has nontrivial scattering. Not even QED is a mathematical object, 
although it is the theory that was able to reproduce experiments 
(anomalous magnetic moment of the electron; see the entry 
''Is QED consistent"" in this FAQ) with an accuracy of 1 in 10^12. 
But till today no one knows how to formulate the
theory in such a way that the relevant objects whose approximations 
are calculated and compared with experiment are logically well-defined.
See, e.g., the S.P.R. threads
   http://groups.google.com/groups?q=Unsolved+problems+in+QED
   http://groups.google.com/groups?q=What+is+well-defined+in+QED

This probably explains the high prize tag of 1.000.000 US dollars,
promised for a solution to one of the Clay millenium problems,
that asks to find a valid 
construction for d=4 quantum Yang-Mills theories that is strong 
enough to prove correlation inequalities corresponding to the 
existence of a mass gap. The problem is to explain rigorously
why the mass spectrum for compact Yang Mills QFT begins at a positive 
mass, while the classical version has a continuous spectrum
beginning at 0.

The mass gap is a property of the theory, not of a wave function.
Intuitively, it means that, in the rest frame of the total system,
the ground state (=vacuum) is an isolated eigenstate of the 
Hamiltonian H, i.e., that the spectrum of H is a subset of 
{0} union [E_1,inf]. The largest E_1 with this property defines 
the mass gap m_1=E_1/c^2.

This would make proper sense for a nonrelativistic theory.
For a relativistic theory one has to read between the lines and 
interpret everything in terms of suitable analogies, 
for lack of a consistent mathematical theory. 
The millenium problem essentially asks for a rigorous mathematical 
setting in which the above can be made precise and proved.

The real problem is the rigorous construction of a Hilbert space with 
a unitary representation of the Poincare group, such that a 
perturbation argument recovers the traditional renormalized order by 
order approximation of quantum field theory.


The state of the art at the time the problem was crowned by 
a prize is given in
   www.claymath.org/Millennium_Prize_Problems/Yang-Mills_Theory/_objects/Official_Problem_Description.pdf
and the references quoted there. See also
   http://www.claymath.org/millennium/Yang-Mills_Theory/ym2.pdf
I don't think significant progress has been published since then.
(The paper hep-th/0511173 which claims to have solved the problem 
only consists of a bunch of heuristic arguments. That the author calls 
it a proof doesn't turn it into a mathematical proof.)

Yang-Mills theories are (perhaps erroneously) believed 
to be the simplest (hopefully) tractable case, 
being asymptotically complete while not having the 
extra difficulties associated with matter fields.
(There are only gluons, no quarks or leptons.)

Of course, one would like to show rigorously that QED is consistent.
But QED has certain problems (the Landau pole, see below) that are
absent in so-called asymptotically free theories, of which 
Yang-Mills is the simplest.


Note that rigorous interacting relativistic theories in 2D and 3D exist;
see, e.g., 
   J. Glimm and A Jaffe,
   Quantum Physics: A Functional Integral Point of View,
   Springer, Berlin 1987. 
This book is quite difficult on first reading.
Volume 3 of Thirring's Course in Mathematical Physics
(which only deals with nonrelativistic QM but in a reasonably 
rigorous way) might be a good preparation to the functional analysis
needed. A more leisurely introduction of the physical side of the
matter is in
   Elcio Abdalla, M. Christina Abdalla, Klaus D. Rothe 
   Non-Perturbative Methods in 2 Dimensional Quantum Field Theory
   World Scientific, 1991, revised 2nd. ed. 2001.
   http://www.wspc.com/books/physics/4678.html
The book is about rigorous results, with a focus on solvable models.
Note that 'solvable' means in this context 'being able to
find a closed analytic expression for all S-matrix elements'.
These solvable models are to QFT what the hydrogen atom is to
quantum mechanics. The helium atom is no longer 'solvable' in the
present sense, though of course very accurate approximate calculations
are possible.

Unfortunately, solvable models appear to be restricted to 2 dimensions. 
The deeper reason for the observation that dimension d=2 is special 
seems to be that in 2D the line cone is just a pair of lines.
Thus space and time look completely alike, and by a change of variables 
2g. (light front quantization), one can disentangle things nicely 
and find a good Hamiltonian description.

This is no longer the case in higher dimensions. (But 4D light front 
quantization, using a tangent plane to the light cone, is well alive
as an approximate technique, e.g., to get numerical results from QCD.)

Thus, while 2D solvable models pave the way to get some rigorous 
understanding of the concepts, they are no substitute for the
functional analytic techniques needed to handle the non-solvable
models such as Phi^4 theory.