Why is Quantum Field Theory so topological?

Question

I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always felt a bit stupid to ask this in real:

Many parts of physics look mainly analytic to me, i.e. electrodynamics and fluid dynamics look like an application of vector analysis and PDEs, quantum mechanics seems to rely heavily on functional analysis. I also understand that there is a lot of structure hidden in those theories: The best example would be symplectic geometry and classical mechanics or topological phases in quantum mechanics. Despite, it seems to me that this is a more modern development in the physics community and the core of those theories as they would be taught to undergraduates would still be mainly analytic. (Sorry if I am wrong about this assumptions but it is my feeling).

However, when it comes to Quantum Field Theory, I feel that very much revolves (especially from the math-community side) around topological and algebraic questions. There is for example a visible math-community with analysis background working on mathematical quantum mechanics, but I never noticed this community in Quantum Field Theory.

Please consider these thoughts of mine, to justify my question: Is there no analysis present in QFT or why do mathematicians concentrate so heavily on those other aspects?

Answer 1

As Robert Israel indicates, if you really wanted to define a QFT, it would certainly be very analytic, say, to define the path integral. So analytic, in fact, that no one can do it for even semicomplicated theories.

However, one of the big math/physics developments in the last few decades is a class of QFTs where the observables are topological in nature. These are the topological QFTs or TQFTs. In these theories, you can ignore all or most of the (too) hard analysis and deal with much more well-defined spaces. Of course, from the mathematics point of view, these theories still involve a path integral that isn't defined (due to all that hard analysis we're ignoring), but enough structure can be found and has been developed to lead to all sorts of cool mathematics (see the mathematical definition of TQFTs, most recently axiomatized by Lurie building on lots of prior work). And even without this structure, physical intuition about these not mathematically well-defined theories has led to countless conjectures, theorems and the like, for example in mirror symmetry and various invariants like Donaldson and Seiberg-Witten.

Answer 2

The framing of your question is a bit ambiguous and perhaps there are two different questions here depending on the context and interpretation. One could approach your question from the point of view of intrinsic scientific content and ask: why QFT seems to be intrinsically more related to topology than analysis? (Question A). But one can also approach the question from the angle of how this is reflected in human activity (mathematicians doing mathematics) in these subjects. Namely, you could ask the question: why, among mathematicians interested in QFT, there are more topologists than analysts? (Question B).

Here is a stab at answering these two very different questions.

Question A:

This is moot since it is based on a false premise. QFT is not only related to topology but also to analysis and, I even venture to say, to almost all of mathematics. The reason for this is that QFT or the problem of rigorously defining functional integrals is the logical and natural continuation of the development of calculus as I explained in this MO answer. After the usual calculus sequence (I, II, III) concerning the finite-dimensional situation, it is natural to explore differentiation (Calc IV) and integration (Calc V) in infinite dimension. Although Calc IV can be traced back to the early work on the calculus of variations by Maupertuis, Euler and Lagrange, I think its mathematical development started in earnest with the work of Volterra. As for Calc V, Wiener's construction of Brownian motion would come to mind as an important early milestone. The intrinsic programmatic content of Calc V is exactly what the analysis of QFT is about. As Robert mentioned, this area of mathematics already exists and is called constructive quantum field theory (CQFT), although nowadays it is also called rigorous renormalization group theory.

To get an idea of what is going on in the field, have a look at the reports for the two recent Oberwolfach meetings:

• "Recent Mathematical Developments in Quantum Field Theory"

• "The Renormalization Group".

Another recent meeting around the new developments by Hairer and others in the strongly related field of stochastic quantization is:

• "Rough Paths, Regularity Structures and Related Topics".

The problem of defining a QFT functional integral is a well posed mathematical problem (see this MO answer for details). In a nutshell, one starts by putting UV and IR cutoffs as is familiar in the theory of Schwartz distributions and one lets bare couplings vary with these cutoffs. The problem is to find the set of all weak limits for the corresponding probability measures on Schwartz distributions. The main difficulty is to construct such weak limit points that are not Gaussian or free measures. One would also like to parametrize this collection of weak limits by a finite number of parameters called renormalized couplings. The main tool to do this is the renormalization group (RG). In this MO answer I briefly explained what the RG is, but I did not give details about how the RG provides a strategy for solving the above problem about weak limit points. For more explanations about this strategy see my article "QFT, RG, and all that, for mathematicians, in eleven pages" and my answer to the physics.stackexchange question Wilsonian definition of renormalizability.

What Robert said "I think there is a feeling that the "easy" questions have been answered, and much of what remains may be impossibly hard" is not quite correct. There are plenty of doable problems to work on at present in CQFT other than $YM_4$.

For example one has analogous conjectures for the 2d Gross-Neveu model and the 2d $\sigma$-model. These are not impossibly hard like $YM_4$ and they do not really require extraterrestrial "new ideas". As in the millenium problem, what one has to do is a construction of the model without UV cutoffs and in infinite volume together with a proof of mass gap.

Another interesting problem (the one I focused on in the above references) is to make contact with conformal field theory. The good class of examples to study in this regard are three-dimensional $N$ component phi-four models with fractional Laplacian $(-\Delta)^{\alpha}$ in the kinetic term.
The cases $N=1$, $\alpha=\frac{3}{4}+\epsilon$ as well as $N$ large, $\alpha=1$ should be not be impossibly hard.

Other problems of current interest are: proving the operator product expansion and conformal invariance using the RG. As for what I personally think is most important problem in constructive quantum field theory today, it is to develop a rigorous Wilsonian RG formalism for handling space-dependent couplings.

Question B:

This one is not moot. It is a fact that there are more mathematicians working on the topological aspects of QFT rather than its analytical aspects. I think this state of affairs is simply due to the status quo, i.e., it's just the way things are. With regards to the North American situation in particular, I think the main explanation is that if a graduate student would like to work on the analysis of QFT, chances are there would simply be nobody in their department to teach them the subject to the point of being research-ready. I think there is nothing more to it, but this could change in the future.

Answer 3

Perhaps the main "analytic" area in Quantum Field Theory is known as Constructive Quantum Field Theory. This essentially emerged in the 1960's with the Wightman axioms. There is still work going on there, but I think there is a feeling that the "easy" questions have been answered, and much of what remains may be impossibly hard: e.g. the Millenium Prize problem on Yang-Mills theory could be out of reach. At least it will require some very new ideas. You might look at this review article by Arthur Jaffe.

Answer 4

it's not. actually it's very difficult to build a quantum field theory that does not depend on the base manifold. that's how Witten got a Fields medal in 1988, for constructing a QFT version on the Jones polynomial.

Actually the most handwaviest thing ever. I prefer to read Witten directly instead of Reshetihkin and Turaev.

Quantum field theory and the Jones polynomial https://projecteuclid.org/euclid.cmp/1104178138

two good questions I think:

• what are the invariants he predicts?
• how does Witten argue (e.g. index theory) that his model indeed has Jones polynomial as partition function.

There is a lot to disagree with, but perhaps that's the strength of his argument.

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Attempting to address other aspects of your question: Electricity and Magnetism can be thought of as the study of solutions to Maxwell's Equations.

• $\nabla \cdot E = 4\pi \rho$
• $\nabla \cdot B = 0$
• $\nabla \times E = - \frac{1}{c} \frac{\partial B}{\partial t} $
• $\nabla \times B = \frac{1}{c} \big( 4\pi j + \frac{\partial E}{\partial t}\big) $

Then in light of Special Relativity the Electricity and Magnetism got merged into a 2-tensor. And there are only two equations

• $dF \;\;\;= 0$
• $d \ast F = J$

This is already being hinted at in a Freshman E&M textbook like Purcell but certainly in Jackson. This is kind of misleading since I don't think we have understood the geometric content of E and M

• we don't understand the geometry of solutions to Maxwell's equations (e.g. [1])
• I challenge you to find engineering or physics problems where $E$ and $B$ are given and you find $F$ and the solution is "unified".

Yang-Mills Theory is a generalization even of this. It is now expressed in variational language. There is an action: $$ S = \frac{1}{2e^2} \int d^4 x \; \mathrm{Tr} |F|^2 $$ There's no way I can cover all the literature on this topic. If we had trouble solving , we certainly have trouble solving this one. This is a PDE but it is still not a topological invariant.

• Yang Mills Theory over Riemann Surfaces Atiyah, Bott (1982)
• Stable bundles and integrable systems Hitchin (1987)
• TASI Lectures on Solitons Tong (2005)
• Twistor theory at fifty: from contour integrals to twistor strings Atiyah, Dunajski, Mason (2017)

I don't think Freshman physics students are thinking about the Hopf fibration or any serious geometry. Instead they are busy trying to pass to pass their course. And let many opportunities pass by.

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The notion of the instanton (or more generally the soliton) -- these are the things our partition functions are counting. Individual solutions to partial differential equations may be too difficult to evaluate point-wise but we can still get qualitative information about solutions to PDE.

• Physics 253a: Quantum Field Theory Sidney Coleman (1985) [Video, YouTube]
• Aspects of Symmetry Sidney Coleman (1988)