Quanta magazine has an interesting new piece up, an interview of Witten by Natalie Wolchover.
One topic covered in the interview is the question discussed in a recent posting, that of whether a different formulation of QFT exists, one not based on a choice of Lagrangian. Here Witten is non-committal, leaning to the idea such a thing might exist only in special cases:
Now, Nati Seiberg [a theoretical physicist who works down the hall] would possibly tell you that he has faith that there’s a better formulation of quantum field theory that we don’t know about that would make everything clearer. I’m not sure how much you should expect that to exist. That would be a dream, but it might be too much to hope for; I really don’t know…
I find it hard to believe there’s a new formulation that’s universal. I think it’s too much to hope for. I could point to theories where the standard approach really seems inadequate, so at least for those classes of quantum field theories, you could hope for a new formulation. But I really can’t imagine what it would be.
The standard example of where such a formulation might be needed is the 6d superconformal (2,0) theory, about which Witten says:
From the (2,0) theory’s existence and main properties, you can deduce an incredible amount about what happens in lower dimensions. An awful lot of important dualities in four and fewer dimensions follow from this six-dimensional theory and its properties. However, whereas what we know about quantum field theory is normally from quantizing a classical field theory, there’s no reasonable classical starting point of the (2,0) theory.
About the current state of M-theory, there’s this exchange:
You proposed M-theory 22 years ago. What are its prospects today?
Personally, I thought it was extremely clear it existed 22 years ago, but the level of confidence has got to be much higher today because AdS/CFT has given us precise definitions, at least in AdS space-time geometries. I think our understanding of what it is, though, is still very hazy. AdS/CFT and whatever’s come from it is the main new perspective compared to 22 years ago, but I think it’s perfectly possible that AdS/CFT is only one side of a multifaceted story. There might be other equally important facets.
What’s an example of something else we might need?
Maybe a bulk description of the quantum properties of space-time itself, rather than a holographic boundary description. There hasn’t been much progress in a long time in getting a better bulk description. And I think that might be because the answer is of a different kind than anything we’re used to. That would be my guess.
Are you willing to speculate about how it would be different?
I really doubt I can say anything useful. I guess I suspect that there’s an extra layer of abstractness compared to what we’re used to. I tend to think that there isn’t a precise quantum description of space-time — except in the types of situations where we know that there is, such as in AdS space. I tend to think, otherwise, things are a little bit murkier than an exact quantum description. But I can’t say anything useful.
The hope of 22 years ago was that it was non-perturbative string theory which would provide the desired “description of the quantum properties of space-time itself”. Over the years though studies of gauge-gravity duality have moved away from the use of string theory to provide this bulk description. Witten’s take on the current situation: “There hasn’t been much progress in a long time in getting a better bulk description. And I think that might be because the answer is of a different kind than anything we’re used to.” seems reasonable.
It’s interesting to hear that Witten was going back to Wheeler to see if he had any inspiration to offer the current “It from Qubit” program. This requires a patience for the “vague but inspirational” that Witten has more of these days than he used to:
Why do you have more patience for such things now?
I think when I was younger I always thought the next thing I did might be the best thing in my life. But at this point in life I’m less persuaded of that. If I waste a little time reading somebody’s essay, it doesn’t seem that bad.
This patience is not infinite though: among Witten’s many admirable qualities are the way he responds to:
Do you have any ideas about the meaning of existence?
No. [Laughs.]
What does it mean for a theory to not have a Lagrangian? That its solutions do not satisfy a variational principle?
Davide,
From one point of view a QFT, like any quantum theory, is some algebra of operators acting on some state space, perhaps with some interesting group acting by automorphisms on the algebra. The usual way to produce such a thing is to start with classical fields, a Lagrangian and a group acting leaving the Lagrangian invariant. Then, produce matrix elements of the operators by either
1. Computing a path integral using the Lagrangian to get the action and thus a “measure”
2. Using a variational principle and the Lagrangian to produce an equation of motion, taking as phase space the space of solutions, quantizing by “canonical quantization”, or maybe geometric quantization.
It seems that for the 6d (2,0) superconformal theory there is an appropriate operator algebra, but no known Lagrangian that would give this algebra using 1. or 2.
Unlike Witten, I’m more willing to speculate, and very fond of the philosophy that interesting quantum systems are sometimes understandable as representations of some group or algebra, accessible via the methods of representation theory. I’d guess that there are various cases where such methods can produce representations that are not of a sort that the standard methods 1. or 2. can produce (or, that they can’t produce in a conventional way: physicists may be able to expand their usual ways of doing 1 or 2 to accommodate more possibilities, and end up with something they still might call a “Lagrangian”.)
Can anyone please explain this response to me:
“Personally, I thought it was extremely clear it existed 22 years ago, but the level of confidence has got to be much higher today because AdS/CFT has given us precise definitions, at least in AdS space-time geometries. I think our understanding of what it is, though, is still very hazy. ”
It seems he is claiming M-Theory has an even higher level of confidence today than 22 years ago but then says his understanding of what M-Theory even is has become very hazy. How can you have even more confidence in something that you understand even less?
John,
“M-theory” is basically a conjecture that there’s an unknown theory that behaves in a certain way in certain limits. I think what he’s saying is that we understand the behavior in some of these limits much better, in a way that agrees with the conjecture. Away from the limits, at a generic point, nothing people have tried has really worked, thus the “still very hazy”.
One thing that I’ve never really understood is the often-expressed claim that M-theory is “unique”, even though one doesn’t know what it is.
Gregory Moore’s Felix Klein lectures for 2012 ( PDF file: http://www.physics.rutgers.edu/~gmoore/FelixKleinLectureNotes.pdf ) has the following, has the situation changed in the last five years?
Quote:
In what follows we try to explain the reasons many string theorists believe in the existence of the remarkable (2, 0) superconformal field theories. In §??? we attempt to write down with some precision the ground rules physicists use when speculating about these theories.
Mathematicians will find this discussion extremely frustrating. A mathematician could well ask: “Is this mathematics?” The answer is “No.” It is not even physical mathematics. The relevant question is “Can it be turned into mathematics?”
End quote
Anonyrat,
I don’t think much has changed concerning what is known about the underlying theory since those lectures. There may very well be important developments I’m unaware of though. The Moore lectures give an excellent overview about what is known about this still not understood theory. See sections 6.6 and 6.7 for this.
Peter,
In discussing QFT without Lagrangian, why do you neglect the axiomatic QFT, such as the Wightman theory, the algebraic QFT of Haag et al., etc. ? As a typical example, I take the 2-dimensional massless scalar field thoery: both scalar field and its dual field satisfy the 2-dim. d’Alembert equation. While if the existence of the Lagrangian is assumed, both fields cannot have independent degree of freedom so that the duality is necessarily broken, if one admits to formulate the theory in the Wightman framework, it is possible to realize the complete duality. But I, personally, prefer the Lagrangian formalism.
Nakanishi,
I’m not trying to discuss what the right general framework for QFT should be (because I don’t know…). I’m just noting the fact that Tachikawa, Seiberg, and Witten, very much mainstream theorists, have each been publicly discussing the idea that the standard textbook way theorists are trained to produce QFTs, by starting with a Lagrangian, doesn’t produce all the QFTs they are interested in. This kind of public acknowledgement that maybe one should be rethinking the foundations of QFT, looking at different fundamental formulations than the Lagrangian one, seems to me a positive development.
Davide #1, check out Yuji Tachikawa’s awesome talk slides http://indico.ipmu.jp/indico/event/134/contribution/17/material/slides/0.pdf (this was linked to a few posts back on this blog)
Peter
I am just pointing out that the axiomatic approach provides us a broader framework for QFT, which is independent of Lagrangian. It is irrelevant to whether or not the three persons you quoted are “mainstream” theorists. Probably, they forget to mention the axiomatic QFT.
Interesting title. There are many ways I would describe Witten, but ‘physicist’s physicist’ would never be one of them.
piscator,
I think what was meant was “theoretical physicist’s theoretical physicist”….
Reportedly Sheldon Glashow already bestowed the “physicist’s physicist”title on Sidney Coleman.
The way I see Witten often described, he sounds more like some near-miraculous alien intelligence inhabiting a human body.
The aforementioned slides of the talk by Yuji Tachikawa is about the unsatisfactory state of QFT, and says that axiomatic QFT is too broad and generic to carry out the calculation of the anomalous magnetic moment of the electron.
Peter,
Are you saying that there exist QFT’s that can be proven to have no Lagrangian formulation, or simply that there are some QFT’s whose Lagranian is not currently known?
(By ‘proven’ I mean at a physics level of rigor, Coleman-Mandula rigor, say.)
ronab,
The conjectured 6d superconformal theory discussed here comes with a physics argument why it shouldn’t have a Lagrangian. Lacking a real definition of that theory though one has no way of knowing if, once defined, there won’t be something you could interpret as its Lagrangian.
In principle, if you know how to write down a partition function (with sources) for your QFT, you can take the logarithm of that to get the energy functional, and then do a Legendre transformation to obtain an effective action functional. Then, assuming that $\hbar$ exists as a deformation parameter in your QFT, it will sit somewhere in the effective action — then you can study the limit $\hbar\to 0$ for the effective action, and the result can be provisionally called “classical action”. Then you define a Lagrangian as the kernel of that action.
So in cases where your QFT is defined in the language where the above procedure can be performed, eventually you end up with *a* Lagrangian. The question whether this Lagrangian is *the* Lagrangian of your QFT (in the sense that the initial partition function can be reconstructed from that Lagrangian using some quantization scheme) is of course nontrivial, and the answer usually needs to be studied on a case-by-case basis.
That’s how I’d like to understand this whole story.
HTH, 🙂
Marko
Witten is of course an exceptional researcher, but really a Physicists’s Mathematician.
Fermi… who hand built capacitors and small rolling tables for the Chicago Cyclotron, and who estimated the Trinity yield from casting torn up paper in the air just before the blast wave hit, and used the displacement in the fall to the ground.. of who Feynman said, “he was like me, only ten times better at it”… now he was a Physicist’s Physicist.
Of course, these names and qualifiers don’t really matter. But Fermi’s collected works, and his family of mentorees, dwarf in breadth and depth any other physicist I can think of, including Einstein. Good these days to remember… he mentored the only woman other than Marie Curie to win a Nobel Prize in Physics.
Resorting to saying there may be no Lagrangian is akin to saying instead of tinkering with the recipe let’s blame the pot, in my view.
Is the only justification for the “no lagrangian” business the 6d superconformal theory? Because that seems to me to be far from sufficient.
sdf,
The 6d theory is one of the reasons Witten and others have started publicly asking the question of whether our usual Lagrangian-based understanding of QFT is too limited, but I think there are other reasons to be asking this question. For instance, one thing to notice about a lot of studies of 2d conformal field theory is that the way you get detailed information about these theories is by studying representations of certain infinite dimensional groups (Virasoro and loop groups). The theory may also have a Lagrangian, but this isn’t anything particularly useful for studying the theory. It may be that there’s always some way to assign a “Lagrangian” to a theory (as vmarko does above for instance), but that this in some cases won’t uniquely characterize the theory or tell you anything useful about it.
Probably you’ve already seen this, but if not: Since the interaction of cutting-edge mathematics and physics is of particular interest, the latest Quanta article about Minhyong Kim seems worth a look.
My limited knowledge of the subject would lead me to believe that something like the specific “action” principle he is searching for might relate somehow to this discussion of Lagrangians.
Dear Peter,
I would like to understand a bit more about ‘the philosophy that interesting quantum systems are sometimes understandable as representations of some group or algebra, accessible via the methods of representation theory’.
I’ve read part of your book and wanted to ask whether you have ever done something with ‘dualities’ between different quantum field theories in the context of representation theory. For 1+1 dimensional bosonization, this seems to be well understood in some cases. The free and massless fermions and bosons are for example related to the unique irreducible representation of the Virasoro and Kac-Moody algebra’s that specify these theories. I was wondering whether it is known how one can find the exact relation between the fermionic and bosonic degrees of freedom just from representation theoretic arguments, and whether it is known how this works generalizes to the massive interacting cases.
In the case of 3+1 dimensional (supersymmetric) Yang-Mills theories, the partition function can be written as a theta function. S-duality (when we look at the partition function as a path integral) then seems to be related to a ‘functional Fourier-transformation’ being performed on this partition function. I don’t know enough about the way modular forms transform under Fourier transformations and how this generalizes in this case… I tried to read part of the notes by Edward Frenkel on the Langlands programme (though it’s a bit too mathematical for me) hoping this would explain a bit more about how the representations of the original Lagrangian are transformed under this ‘functional Fourier transform’ into representations of the Langlands dual group that describes the S-dual theory.
I am very interested whether you ever wrote something about dualities in the context of representation theory, or if you could give me a few hints or articles in the right direction?
Many thanks.
Jeroen,
What’s interesting to mathematicians about these QFT dualities is that many of them don’t correspond to anything well-understood in mathematics (in particular in representation theory). For T dualities there is a clear understanding of their origin (both in math and physics), which is the usual story of Fourier series and the Abelian Fourier transform.
For S dualities, you can often interpret them as part of a modular group action and thus the story of modular forms, but this is exactly where the representation theory issues get very deep and poorly understood. It’s important to realize that there is a lot of mystery at the heart of the Langlands program and thus these issues. It’s not that mathematicians have a simple idea about why such things should be true, and are just having trouble making things rigorous.
The geometric Langlands program and its Langlands duality is about very non-trivial structure of spaces of bundles and connections over Riemann surfaces, it was formulated partly by looking for the geometric analog of the number theory Langlands program. Witten related this to S-duality of 4d QFTs, but this just relates two different mysteries. The only conceptual explanation of the S-duality I know of is based on starting with the existence of a superconformal 6d theory, compactifying on a torus, and getting the modular group action using the torus. This explanation is yet a different kind of mystery, the existence of this 6d QFT.
So, sorry this isn’t much of an answer, just an indication of how pursuing the connections between math and physics based on S-duality leads to relations between deep mysteries, but no truly satisfactory explanation of these mysteries.
Jeroen,
Also, forgot to make this clear: I haven’t written anything about this. In the book I wrote I decided to not try and say anything about conformal field theory. That would require a whole other volume, and it’s a subject where there’s already an extensive literature from the representation theory point of view, in which you likely could find discussion of cases of dualities that are well understood.
The quick way to see that there is an issue with the (2,0)-superconformal field theory in 6d to be Lagrangian is to observe that it involves a self-dual 2-form higher gauge field B, meaning that the 3-form field strength H = dB is proportional to its own Hodge dual star H, which means that the would-be kinetic Lagrangian H /\ star H vanishes identically.
This is one example in a hierarchy of self-dual higher gauge theories which in 2d starts with the chiral boson and in 10d continues with the self-dual RR-fields. All these share the same issue with being Lagrangian, and all of them are either known or conjectured to be defined instead as the holographic boundary theory in one dimension higher which is Lagrangian. For the chiral boson this is abelian 3d Chern-Simons theory, for the 6d theory it is AdS7/CFT6 duality, and specifically for the self-dual higher gauge sector inside the 6d theory it is 7d Chern-Simons theory, and for the 10d self-dual RR fields there is a proposal in terms of an 11-dimension Chern-Simons theory whose fields are differential K-cocycles.
But generally, that not every field theory is Lagrangian is an old hat, this was fully understood for free field theories by Helmholtz way back in 1887. In his honor the corresponding obstruction for the general case is known today as the cohomology of the Helmholtz operator, which is one of the differentials in the Euler-Lagrange complex that controls all things variational. A good introductory discussion to these matters is in Anderson’s book The variational bicomplex, see p. viii of the introduction.