First, I will translate the relevant passages in your paper from mathematese.
The argument in your reference
You are studying an X-Y model with the constraint that neighboring spins have to always be within a certain angle of each other. You define the collection of statistical-mechanics Gibbs distributions using a given boundary condition at infinity, as the boundaries get further and further away. Then you note that if the field at the boundary makes the spin turn around from top to bottom the maximum possible amount, then the spins are locked in place--- they can't move, because they need to make a certain winding, and they unless they are at the maximum possible angle, they can't make the winding.
Using these boundary conditions, there is no free energy, there is no thermodynamics, there is no spin-wave limit, and the Mermin Wagner theorem fails.
You also claim that the theorem fails with a translation invariant measure, which is just given by averaging the same thing over different centers. You attempt to make the thing more physical by allowing the boundary condition to fluctuate around the mean by a little bit $\delta$. But in order to keep the boundary winding condition tight, as the size of the box $N$ goes to infinity, $\delta$ must shrink as $1\over N$, and the resulting Free energy of your configuration will always be subextensive in the infinite system limit. If $\delta$ does not shrink, the configurations will always randomize their angles, as the Mermin-Wagner theorem says.
The failures of the Mermin-Wagner theorem are all coming from this physically impossible boundary situation, not really from the singular potentials. By forcing the number of allowed configurations to be exactly 1 for all intents and purposes, you are creating a situation where each different average value of the angle has a completely disjoint representative in the thermodynamic limit. This makes the energy as a function of the average angle discontinuous (actually, the energy is infinite except for near one configuration), and makes it impossible to set up spin waves.
This type of argument has a 1d analog, where the analog to Mermin-Wagner is much easier to prove.
1-dimensional mechanical analogy
To see that this result isn't Mermin-Wagner's fault, consider the much easier one-dimensional theorem--- there can be no 1d solid (long range translational order). If you make a potential between points which is infinite at a certain distance D, you can break this theorem too.
What you do is you impose the condition that there are N particles, and the N-th particle is at a distance ND from the first. Then the particles are forced to be right on the edge of the infinite well, and you get the same violation: you form a 1d crystal only by imposing boundary conditions on a translation invariant potential.
The argument in 1d that there can be no crystal order comes from noting that a local defect will shift the average position arbitrarily far out, so as you add more defects, you will wash out the positional order.
Mermin-Wagner is not affected
The standard arguments for the Mermin-Wagner theorem do not need modification. They are assuming that there is an actual thermoodynamic system, with a nonzero extensive free energy, an entropy proportional to the volume, and this is violated by your example. The case of exactly zero temperature is also somewhat analogous--- it has no extensive entropy, and at exactly zero temperature, you do break the symmetry.
If you have an extensive entropy, there is a marvelous overlap property which is central to how physicists demonstrate the smoothness of the macroscopic free-energy. The Gibbs distribution at two angles infinitesimally separated sum over almost the same exact configurations (in the sense that for a small enough angle, you can't tell locally that it changed, because the local fluctuations swamp the average, so the local configurations don't notice)
The enormous, nearly complete, overlap between the configurations at neighboring angles demonstrates that the thermodynamic average potentials are much much smoother than the possibly singular potentials that enter into the microscopic description. You always get a quadratic spin-wave density, including in the case of the model you mention, whenever you have an extensive free energy.
Once you have a quadratic spin-wave energy, the Mermin Wagner theorem follows.
Quick answer
the Gibbs distributions for orientation $\theta$ and the Gibbs distributions for orientation $\theta'$ always include locally overlapping configurations as $\theta$ approaches $\theta'$. This assumption fails in your example, because even an infinitesimal change in angle for the boundary condition changes the configurations completely, because they do not have extensive entropy, and are locked to within a $\delta$, shrinking with system size, of an unphysically constrained configuration.
Lack of convergence does not mean there is nothing mathematically rigorous one can extract
from perturbation theory. One can use Borel summation. In fact, Borel summability of perturbation theory has been proved for some QFTs:
- by Eckmann-Magnen-Seneor for $P(\phi)$ theories in 2d, see this article.
- by Magnen-Seneor for $\phi^4$ in 3d, see this article.
- by Feldman-Magnen-Rivasseau-Seneor for Gross-Neveu in 2d, see this article.
In fact these articles obtain such results by using an alternative to ordinary perturbation theory called a multiscale (or phase cell or phase space) cluster expansion.
The latter is based on combinatorial structures which mimic Feynman diagrams. However, these expansions converge at small coupling.
Edit as per Timur's comment: Glimm and Jaffe's book is what you want to read in order to understand why one needs cluster expansions. It is excellent at giving the big picture: how axiomatic, Euclidean, constructive QFTs fit together, as well as with scattering theory. But for learning how to do a cluster expansion the book is outdated.
The cluster expansion explained in GJ is the early one invented by Glimm, Jaffe and Spencer in their Annals of Math article. It was the first in the QFT context and as such
quite a mathematical feat. However there has been many improvements and simplifications since then (around 1973). If you want to learn about cluster expansions in 2011, here is a more efficient path:
- Learn about the Mayer expansion for the polymer gas: a quick intro is in the "Additional material" at the bottom of my course webpage.
- Learn about the single scale cluster expansion, i.e., controlling the infinite volume limit when both UV and IR cut-offs are present: look at the article "Clustering bounds on n-point correlations for unbounded spin systems" on the same webpage. For a more cleaned up version see the published version, but this is not freely accessible.
- Finally the real McCoy: the multiscale cluster expansion where one tries to do all of the above and remove the cut-offs. This is somewhat like an infinite volume limit in phase space. Here there is no easy reference. All accounts of the subject are extremely difficult to read. I plan to write a pedagogical article on this in the next
few months. In the meantime you could try the following: the book by Rivasseau "From Perturbative to Constructive Renormalization", the book "Wavelets and Renormalization" by Battle, and also this recent article by Unterberger (in French).
Best Answer
Your statement
is not really correct, and seems to have a common misunderstanding in it. The technical difficulties from QFT do not come from infinities. In fact, ideas basically equivalent to renormalization and regularization have been used since the beginning of math--see, e.g., many papers by Cauchy, Euler, Riemann, etc. In fact, G.H. Hardy has a book published on the topic of divergent series:
http://www.amazon.com/Divergent-AMS-Chelsea-Publishing-Hardy/dp/0821826492
There is even a whole branch of math called "integration theory" (of which things like Lebesgue integration is a subset) that generalizes these types of issues. So having infinities show up is not an issue at all, in a sense, they show up out of convenience.
So the idea that infinities have anything to do with making QFT axiomatic is not correct.
The real issue, from a more formal point of view, is that you "want" to construct QFTs via some kind of path integral. But the path integral, formally (i.e., to mathematicians) is an integral (in the general sense that appears in topics like "integration theory") over a pretty pathological looking infinite dimensional LCSC function space.
Trying to define a reasonable measure on an infinite dimensional function space is problematic (and the general properties of these spaces doesn't seem to be particularly well understood). You run into problems like having all reasonable sets being "too small" to have a measure, worrying about measures of pathological sets, and worrying about what properties your measure should have, worrying if the "$\mathcal{D}\phi$" term is even a measure at all, etc...
At best, trying to fix this problem, you'd run into an issue like you have in the Lebesgue integral's definition, where it defines the integral and you construct some mathematically interesting properties, but most of its utility is in letting you abuse the Riemann integral in the way you wanted to. Actually calculating integrals from the definition of the Lebesgue integral is not generally easy. This isn't really enough to attract the attention of too many physicists, since we already have a definition that works, and knowing all of its formal properties would be nice, and would certainly tell us some surprising things, but it's not clear that it would be all that useful generally.
From an algebraic point of view, I believe you run into trouble with trying to define divergent products of operators that depend on renormalization scheme, so you need to have some family of $C^*$-algebras that respects renormalization group flow in the right way, but it doesn't seem like people have tried to do this in a reasonable way.
From a physics point of view, we don't care about any of this, because we can talk about renormalization, and demand that our answers have "physically reasonable" properties. You can do this mathematically, too, but the mathematicians are not interested in getting a reasonable answer; what they want is a set of "reasonable axioms" that the reasonable answers follow from, so they're doomed to run into technical difficulties like I mentioned above.
Formally, though, one can define non-interacting QFTs, and quantum mechanical path integrals. It's probably the case that formally defining a QFT is within the reach of what we could do if we really wanted, but it's just not a compelling topic to the people who understand how renormalization fixes the solutions to physically reasonable ones (physicists), and the formal aspects aren't well-understood enough that it's something one could get the formalism for "for free."
So my impression is that neither physicists or mathematicians generally care enough to work together to solve this problem, and it won't be solved until it can be done "for free" as a consequence of understanding other stuff.
Edit:
I should also add briefly that CFTs and SCFTs are mathematically much more carefully defined, and so a reasonable alternative to the classic ideas I mentioned above might be to start with a SCFT, and define a general field theory as some kind of "small" modification of it, done in such a way to keep just the right things well-defined.