# A Constructive QFT - current status

#### Auto-Didact

I haven't been up to date on the state of the art of the field for quite some years now; the contemporaneity of my knowledge ends with the review by Rivasseau, 2000. A quick gander at the topic over at the n-Cat Lab shows that practically nothing has changed.

Is anyone working in the field here more up to date on the current state of the art willing to address whether there have been major progress to the problems as listed in Rivasseau's review? More explicitly, has there been a full constructive formulation of QFT in 4 dimensions? And if not, how far away are we projected to be?

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#### A. Neumaier

has there been a full constructive formulation of QFT in 4 dimensions? And if not, how far away are we projected to be?
No. We are as far away as it takes to make an unforeseen breakthrough.

#### Auto-Didact

Thanks... I will try not to cry myself to sleep later

#### DarMM

Gold Member
There has been some progress (more field theories constructed in curved background in lower dimensions) however nothing of major note to somebody not deeply interested in the field.

However it's often the case that the issue is presented as something "deep" about the four dimensional case, where as that is not so much true as it is the difficulties related to being "only" renormalizable and requiring coupling constant renormalization.

Many of the techniques in constructive field theory involve slicing the Path Integral into several subintegrals, each confined to a certain four-momenta band. One then performs perturbation theory to a certain order at each length scale via a form of functional integration by parts with a nonperturbative remainder. We can perform renormalization on the perturbative part to render them finite and then bound the nonperturbative part, thus we are able to analytically bound the approach to the continuum limit and prove that it is finite. Typically we need to go further into perturbation theory as the length scale goes to zero (Energy goes to infinity) in order to bound the nonperturbative part.

However there are three problems.

If the theory is only renormalizable the divergences within perturbation theory itself make the whole expansion difficult to control. The nonperturbative remainder will be divergent and for this reason one has to essentially always obtain optimal estimates on every aspect of the functional integrals, non-optimal bounds will mask the very precise cancellations that permit the existence of a continuum limit. In superrenormalizable theories we can make incredibly non-optimal and crude bounds and still prove convergence.

Secondly most techniques operate via estimates against Gaussian integrals/the free theory. This is very easy to do when the coupling constant is simply a number $\lambda$ with the free case being given by the $\lambda = 0$ case. We can prove estimates bounding things in terms of polynomials in $\lambda$, demonstrate continuity in $\lambda$ of bounds etc. However when $\lambda$ itself has divergences that balance those in the integral all of this goes out the window.
Also coupling constant renormalization introduces overlapping divergences and we in addition have renormalons when summing the perturbative series which affect estimating the perturbative part of these expansions.

Third when the theory has "special features" that need to be preserved by cutoffs. For example the Gross-Neveu model is in a certain sense "as difficult" as $\phi^{4}_{4}$, but the later has a positivity of the interacting term that needs to be preserved or otherwise estimates will be insufficiently tight.

We have very poor control and constructive results even in $d = 3$ for just renormalizable theories or ones which require coupling constant renorms.

In $d = 4$ all theories are like this and they have a very special structure that needs to be preserved, Gauge symmetry. Note though that anything like this in $d = 3$ would be beyond current methods.

#### Demystifier

2018 Award
In my opinion, the whole program of search for a mathematically rigorous continuous field theory is fundamentally misguided. The continuous field theories (such as the Standard Model) that we have are just effective theories that at very small distances must be replaced by completely different theories.

#### DarMM

Gold Member
In my opinion, the whole program of search for a mathematically rigorous continuous field theory is fundamentally misguided. The continuous field theories (such as the Standard Model) that we have are just effective theories that at very small distances must be replaced by completely different theories.
Why would their physical inaccuracy at small length scales imply they have no rigorous formulation?

For example non-relativistic QM and General Relativity are both incorrect in certain regimes but have a mathematically rigorous formulation.

Why can field theories in lower dimensions be constructed?

#### Demystifier

2018 Award
Why would their physical inaccuracy at small length scales imply they have no rigorous formulation?
It wouldn't. It just implies that we don't so strongly need such a rigorous formulation, even if it exists.

For example non-relativistic QM and General Relativity are both incorrect in certain regimes but have a mathematically rigorous formulation.
It's indeed nice when a theory has a rigorous formulation, but if that theory is not fundamental, then it's not such a big problem if it hasn't a rigorous formulation.

Why can field theories in lower dimensions be constructed?
Well, maybe field theories in 4 dimensions also have a rigorous formulation awaiting to be discovered. But if someone desperately searches for it beacuse he thinks that it must exist for otherwise the Nature would be inconsistent, I think it's wrong.

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#### DarMM

Gold Member
Isn't this basically an argument against looking for mathematical rigour for any physical theory?

#### atyy

It's indeed nice when a theory has a rigorous formulation, but if that theory is not fundamental, then it's not such a big problem if it hasn't a rigorous formulation.
Rigor is conceptually important for quantum mechanics, as a relativistic quantum theory would prove the wave function is not real

#### Demystifier

2018 Award
Isn't this basically an argument against looking for mathematical rigour for any physical theory?
No. Rigour is desirable, but not a must. I am against a rigour for its own sake when it destroys some more important properties of the theory. An example is a rigorous result in statistical mechanics that there are no mathematical phase transitions in finite systems. This conflicts with the initial goal to describe the physical phase transitions (such as freezing of water) that obviously exist in finite systems. Another example is the Haag's theorem.

#### DarMM

Gold Member
No. Rigour is desirable, but not a must. I am against a rigour for its own sake when it destroys some more important properties of the theory. An example is a rigorous result in statistical mechanics that there are no mathematical phase transitions in finite systems. This conflicts with the initial goal to describe the physical phase transitions (such as freezing of water) that obviously exist in finite systems. Another example is the Haag's theorem.
I don't understand I have to say. What does Haag's theorem "destroy"?

Same for the statistical mechanical example, to me that just shows that sharp seperation between phases is an infinite volume idealization. I would have found that interesting rather than considering it to "destroy" a goal of statistical mechanics.

#### Demystifier

2018 Award
I don't understand I have to say. What does Haag's theorem "destroy"?

Same for the statistical mechanical example, to me that just shows that sharp seperation between phases is an infinite volume idealization. I would have found that interesting rather than considering it to "destroy" a goal of statistical mechanics.
Well, when a rigorous result is interpreted that way, it's perfectly welcomed. But in my experience, some mathematical physicists tend to make less reasonable conclusions from some rigorous theorems. They just not have a good intuition about which of the assumptions (from which the theorem was derived) should be questioned.

#### DarMM

Gold Member
Well to my mind having a rigorous formulation of the theory means you understand the theory better and can begin to apply more mathematical methods to it. Also you may find counterexamples to folk wisdom that show the theory to be richer and more complex than its naive formulations.
For example the Gross-Neveu model is perturbatively non-renormalizable but is actually completely well-defined and renormalizable non-perturbatively.

#### Demystifier

2018 Award
Well to my mind having a rigorous formulation of the theory means you understand the theory better
That's because your mind has a well developed philosophical component, so you understand what you are doing at a deeper meta-level. Not all mathematical physicists have that.

#### atyy

No. Rigour is desirable, but not a must. I am against a rigour for its own sake when it destroys some more important properties of the theory. An example is a rigorous result in statistical mechanics that there are no mathematical phase transitions in finite systems.
But this is also an intuitive result. Are you against intuitive and rigorous results?

This conflicts with the initial goal to describe the physical phase transitions (such as freezing of water) that obviously exist in finite systems.
So how do you describe physical phase transitions? Do you discard classical statistical mechanics and thermodynamics, since they take the unphysical infinite systen limit?

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#### Demystifier

2018 Award
But this is also an intuitive result. Are you against intuitive and rigorous results?
Well, it's not intuitive to me. Can you explain why is that intuitive to you?

So how do you describe physical phase transitions? Do you discard classical statistical mechanics and thermodynamics, since they take the unphysical infinite systen limit?
One can do it non-rigorously which, in a sense, is an approach in which the system is both infinite and non-infinite. (This is somewhat analogous to intuitive calculus before Cauchy, where $dx$ is both zero and non-zero.) Essentially, one first computes intensive quantities (pressure, temperature, concentration, ...) in the limit $N\rightarrow 0$, but then computes extensive quantities (energy, number of particles, ...) in a finite volume $V$. The goal of mathematical physicists is to explain why such an ill defined procedure gives correct results, but "ordinary" physicists can just use their naive intuition to work that way in practice.

#### atyy

Well, it's not intuitive to me. Can you explain why is that intuitive to you?
A phase boundary is non-analytic behaviour. If one writes the partition function with a finite number of particles, the function is analytic, and it is hard to see how one gets the non-analytic behaviour. If one allows the limit to infinity to be taken, then it seems the non-analytic behaviour could be possible.

Eg. David Tong's notes: http://www.damtp.cam.ac.uk/user/tong/sft.html (lecture 1, p13)

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#### Auto-Didact

Well, it's not intuitive to me. Can you explain why is that intuitive to you?
What is intuitive to someone depends on knowledge of and experience with certain prototypical concepts within a branch of mathematics: this changes with time, focus and exposure.
A phase boundary is non-analytic behaviour.
Actually this is often due to an approximation of some mathematically yet unresolved boundary layer; the approximation then tangentially or asymptotically matches the actual analytic process with an unknown, often non-visible limited range of validity.

In such cases, the mathematical method of approximation itself then is the cause for any defective mathematical properties of the effectively derived function ascribed to the described phenomenon and the general inconsistency thereof with - e.g. non-generalizability to - the actually sought after function.

No amount of rigour or precision can alleviate such a problem because the problem is completely mathematical yet through a premature misinterpretation invalidly gets projected onto the physics leading to endless misunderstandings and paradoxes; sounding familiar yet?

This why constructive mathematical demonstrations are so crucially important: once the mathematical foundations of a physical theory crumbles, all the secondary structures that were built on top of it come crashing down as hopelessly inadequate and deeply misguided; if lucky the theory can survive as a limiting case.

#### Demystifier

2018 Award
A phase boundary is non-analytic behaviour. If one writes the partition function with a finite number of particles, the function is analytic, and it is hard to see how one gets the non-analytic behaviour. If one allows the limit to infinity to be taken, then it seems the non-analytic behaviour could be possible.

Eg. David Tong's notes: http://www.damtp.cam.ac.uk/user/tong/sft.html (lecture 1, p13)
Thanks, now it's intuitive to me too.
That's yet another demonstration that Tong's lectures are really great.

#### microsansfil

An example is a rigorous result in statistical mechanics that there are no mathematical phase transitions in finite systems.
Doesn't this show that there is no isomorphism between the model expressed in the language of mathematics and the physical phenomenology?

A model can lead to over-specification (e.g : Gödel metric/Closed timelike curves) or to underspecification to describe physical phenomenology.

/Patrick

#### DarMM

Gold Member
I think it's just that actual transitions between phases aren't sharp for real systems. Having sharp transitions simplifies many treatments, but is technically an infinite volume limit idealisation. However since we have analytic control over that limit we can bound the errors we are making when we treat real systems in this manner and see that it is virtually irrelevant.

That's another point of rigorous constructions, if we had a rigorius continuum limit for Yang-Mills we could bound the systematic errors in Lattice simulations exactly.

#### atyy

I think it's just that actual transitions between phases aren't sharp for real systems. Having sharp transitions simplifies many treatments, but is technically an infinite volume limit idealisation. However since we have analytic control over that limit we can bound the errors we are making when we treat real systems in this manner and see that it is virtually irrelevant.

That's another point of rigorous constructions, if we had a rigorius continuum limit for Yang-Mills we could bound the systematic errors in Lattice simulations exactly.
Unless we have something bizarre like this ?

"The standard approach of trying to gain insight into such models by solving numerically for larger and larger lattice sizes is doomed to failure; the system could display all the features of a gapless model, with the gap of the finite system decreasing monotonically with increasing size. Then, at some some threshold size, it may suddenly switch to having a large gap."

#### vanhees71

Gold Member
In my opinion, the whole program of search for a mathematically rigorous continuous field theory is fundamentally misguided. The continuous field theories (such as the Standard Model) that we have are just effective theories that at very small distances must be replaced by completely different theories.
Well, I think the physical validity and comprehensibility of theories have little to nothing to do with the possibility to find a mathematically rigorous formulation.

E.g., Newtonian classical mechanics is a mathematically well-defined theory with well-understood and mathematically interesting theorems and proofs. Nature doesn't like it though in a sense: It's only an approximation valid under well-understood circumstances. The limits of applicability are defined by relativity as well as (non-relativistic and of course also relativistic) Q(F)T.

That can also be said about classical electrodynamics as long as only a classical continuum treatment of the (charged) matter is concerned. The interacting theory for point-particles and the electromagnetic field is mathematically not well defined and only approximate descriptions are possible (with the Landau-Lifshitz approximation to the Abraham-Lorentz-Dirac equation the most convincing one, but with not too strong empirical justification though it seems to work well enough for accelerator physicists to construct accurate enough accelerators).

GR is also a mathematically well-defined theory, but it's physically for sure incomplete. Our ignorance is manifest by the unavoidable singularities in the solutions for non-trivial cases (the universe, black holes).

Non-relativistic quantum mechanics seems to be well-undertood and rigorously formulated mathematically but of course it has its limits of applicability as soon as relativistic situations are reached.

Finally, as discussed in this thread, the Standard Model of elementary particle physics is mathematically not fully understood, but this is for a rather unphysical case anyway, namely the infinite-volume limit (where strictly speaking even the perturbative formulation is inconsistent due to Haag's theorem). Treated in the right way as an effective field theory it's the most successful theory ever, including high-precision calculations for some fundamental quantities (g-2 for electrons, Lamb shifts of hydrogen(-like) atoms/ions, quantum-optics experiments concerning the very foundations aka Bell tests; just now also a demonstration of the EPR paradox in its original form about position and momentum: https://doi.org/10.1103/PhysRevLett.123.060403 ; guess, why the "violation of the HUP" is no true contradiction to the HUP here ;-)), to be discussed in a separate thread).

Of course, from a academic perspective a mathematically well-defined interacting QFT in (1+3) dimensions would be desirable, maybe also shedding light on the physics. Usually deep mathematical problems seem to have also interesting meanings for the understanding of the physics described by them.

It's also the other way around: Sloppy physicists' math can contain interesting mathematical context. E.g., Dirac's $\delta$ distribution (however already defined and used by Sommerfeld around 1910) triggered the development of an entire new field of mathematics, functional analysis.

Sommerfeld called this, in reference to Leibniz, the "prestabilized harmony between maths and physics" ;-)).

#### DarMM

Gold Member
Finally, as discussed in this thread, the Standard Model of elementary particle physics is mathematically not fully understood, but this is for a rather unphysical case anyway, namely the infinite-volume limit (where strictly speaking even the perturbative formulation is inconsistent due to Haag's theorem)
A slight correction, I would say "where the normal derivation of the perturbative formalism does not hold". Even in the infinite volume limit the perturbative series is the correct expansion, it just has to be derived differently from how it's done in textbooks.

#### vanhees71

Gold Member
Ok, what different derivation do you have in mind? Are there papers/books understandable to the usual mortal QFT practitioner?

"Constructive QFT - current status"

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