First, two local events, involving well-known physics bloggers:
- Last Thursday I had the pleasure of attending an event at NYU featuring Sabine Hossenfelder and Natalie Wolchover in conversation. You can watch this for yourself here. If you’re not following Hossenfelder on her blog and at Twitter (and planning to read her forthcoming book), as well as reading Wolchover’s reporting at Quanta magazine, you should be.
- Next week there will be an event out in Brooklyn advertised as covering the Scientific Controversy over string theory. The idea seems to be to address this controversy by bringing to the public two well-known and very vocal proponents of one side of it.
For a Q and A with another well-known physics blogger, there’s Tommaso Dorigo at Physics Today.
For a couple of encouraging indications that the theoretical physics community may finally be taking seriously the need to give up on failed thinking and try something new, there’s
- A conference next month in Italy on Weird Theoretical Ideas (Thinking outside the box).
- An interesting talk at a recent IPMU conference by Yuji Tachikawa. I like his conclusion:
Basically, all the textbooks on quantum field theories out there use an old framework that is simply too narrow, in that it assumes the existence of a Lagrangian.
This is a serious issue, because when you try to come up e.g. with a theory beyond the Standard Model, people habitually start by writing a Lagrangian … but that might be putting too strong an assumption.
We need to do something
In General Relativity related news, there’s a new edition out of Misner, Thorne and Wheeler, the book from which many of us learned both geometry and GR. It comes with new prefaces from David Kaiser as well as Misner and Thorne (which an appropriate search on the Amazon preview might show you…). In other Wheeler-related news, Paul Halpern has a new book out, The Quantum Labyrinth, which tells the entangled stories of Feynman and Wheeler.
Finally, also GR related, the Perimeter Institute has announced the formation of a new cosmology-focused “Centre for the Universe”, funded by an anonymous 10-year $25 million donation. It will be led by cosmologist Neil Turok, who is soon to step down as director of Perimeter.
Thanks for the mention. It was good to finally meet you in person 🙂
Has David Gross paid off his bet?
I’m sorry if I’m missing it, and a joke explained is not a joke, but is the “encouraging indication” re Tachikawa a joke? Because his argument and conclusion sound like trying to give a little extra breathing room to susy and strings. (Although I’m not sure that string theorists have by now written down a Lagrangian, I stopped following this some time ago.)
tulpoeid,
Not a joke at all, although my sympathy with the point being made may come from a different origin than Tachikawa’s. I do think it’s a serious and important point: maybe a lot of our problem is that we’re wedded to a too narrow conception of how to produce QFTs (choose a Lagrangian function, turn path integral or canonical quantization crank). Maybe the SM is the best one can do within this framework, to do better you need a wider notion of what a QFT is. One reason I stayed away from the Lagrangian formalism in my recent book was precisely because for the analysis of quantum theories in terms of representation theory, the Hamiltonian formalism is much more appropriate. Perhaps there is a representation-theoretic framework for understanding QFTs where some things become clear which are hard to impossible to see in the usual Lagrangian formalism.
This is a question about QFTs and how to formulate them, pretty much irrelevant to string theory, which Tachikawa doesn’t even mention.
I am just a layman and picked the Tachikawa link as the one I was going to read through, and I thought it was sort of a win for string theory and was wondered too what you would think. On slide 29 they say “where genuinely N=3 theories were found, using string theory.” I took that to mean this was one of the first theories found to break the lagrangian view and so maybe it was a nice, as you sometimes allow, mathematical use of strings.
boop,
All sorts of interesting QFT phenomena have turned up when people have been looking into questions coming out of string theory. Categorizing new ideas as to whether they’re “a win for string theory” or not is just reducing everything to uninteresting ideology and sloganeering (and to his credit, Tachikawa isn’t doing this).
The actual example Tachikawa gives here (an N=3 4d SUSY QFT) is a rather complicated one and doesn’t look in and of itself very interesting. More interesting is the question of finding new ways to identify and study such QFTs. Sure, you may be able to find them by string theory methods, but people have been trying that for a long time. If you could find other, more insightful, methods that would be more promising.
The paragraph describing Sci Con #13 reads like it could have been written in 1994.
I’m slightly confused by all the noise regarding Lagrangian formalism. I don’t see anything really fundamental in using (or not using) a Lagrangian — it’s just a piece of mathematical formalism, convenient for some purposes, and less so for other purposes.
The question whether a Lagrangian for a given theory exists can be answered in a pretty trivial way. Given any set of partial differential equations (that describe your classical field theory), one can always construct a Lagrangian to reproduce those equations by extremizing the action. Just write the Lagrangian as the LHS of your differential equation times a Lagrange multiplier, and you’re done. Of course, introducing Lagrange multipliers as auxiliary fields into the theory is the price one pays for having a Lagrangian, but this can always be done if you want to work in a Lagrangian formalism.
So, if one can rewrite any classical field theory as a Lagrangian theory, what’s all the fuss about?
Best, 🙂
Marko
I watched the video of the NYU presentation. It was very enjoyable but it was almost spoiled by Robert Lee Hotz, the white-haired interviewer. He commits the cardinal sin of interviewing, which is being more interested in showing his own cleverness than interacting with the people being interviewed. He interviews as if he’s being paid by the word, and there are several times when he rudely interrupts what the interviewees are saying. Sad.
On the other hand, both interviewees had some very good responses, and showed their way of thinking about scientific writing.
vmarko,
The issue is quantization. Are there interesting QFTs that are not in any known sense the “quantization” of a classical field field theory? The standard way of thinking about such things is that they’re strongly coupled QFTs that don’t have parameters that can be taken to some weakly-coupled limit where you do expect a usual relation to a classical field theory.
Can you elaborate what you can do in QFT without Lagrangian or Hamiltonian? Do you calculate S matrix directly?
kashyap vasavada,
That’s the problem, we don’t have much in the way of methods to produce such non-Lagrangian theories. One way to characterize them would be in terms of an S-matrix. I believe this is one motivation for some of the work on amplitudes. S-matrix theory has a long history of pursuing the idea of trying to go even further, getting rid not just of Lagrangians, but also quantum fields.
vmarko, not every differential equation is the EL-equation of a Lagrangian, not even locally. The obstruction is measured by the cohomology of the Euler-Lagrange complex in degree “spacetime dimension +1” (an argument that for linear PDEs was made way back by Helmholtz). Examples of non-Lagrangian QFTs are the chiral WZW model (which is “one chiral half” of a Lagrangian theory) and generally self-dual higher gauge theories. (However, these non-Lagrangian theories are thought to be holographic boundary theories of Lagrangian theories.)
The beauty of Lagrangian field theory is that it comes with its own covariant phase space. This is really what makes rigorous pQFT tick. We are running a series on this over at PhysicsForums Insights here.
Hi Peter,
“The issue is quantization. Are there interesting QFTs that are not in any known sense the “quantization” of a classical field field theory?”
This seems to be a completely separate issue, having nothing to do with Lagrangian formalism. You can also ask the same question for QFT’s which do not have a well-defined Hamiltonian. For example, a QFT which lives on a spacetime manifold which does not have $\Sigma\times \mathbb{R}$ topology, so that you cannot introduce the foliation into space and time, and consequently no Hamiltonian.
I don’t see the existence of such QFT’s to be an argument against the Largrangian or Hamiltonian formalisms.
“The standard way of thinking about such things is that they’re strongly coupled QFTs that don’t have parameters that can be taken to some weakly-coupled limit where you do expect a usual relation to a classical field theory.”
You mean a QFT without a well-defined classical limit, i.e. when $\hbar$ is not allowed to go to zero for some reason? While I agree that this would be an interesting object to study in itself, I don’t really see how would such a QFT be relevant to realistic physics?
Best, 🙂
Marko
Hi Urs,
Thanks for the links, I’ll take a look at the PF in detail.
I don’t think I understand your argument. Say I have a differential equation $D(f)=0$, and I define a Lagrangian as $L=\lambda D(f)$, where $\lambda$ is the Lagrange multiplier. One of the Euler-Lagrange equations of motion will always be the above differential equation, obtained by the variation of the Lagrangian in $\lambda$. The other equations (obtained as variations in $f$) will give other equations involving $\lambda$, to complete the set of EL-equations. In the end the set of solutions to the system of EL-equations should be equivalent to the set of solutions of the original differential equation.
Granted, the construction above actually extends the number of fields you have in the theory, and with it the phase space structure etc., but I don’t see any choice of $D(f)$ where such a construction would be impossible. What am I missing?
Note, the existence of the action is another matter, I agree that integrating the Lagrangian over some manifold may depend on the nontrivial topology of the manifold etc. so that the action may fail to be well defined in general, or may be always equal to zero or whatever. But for the Lagrangian itself I don’t really see what can go wrong?
Best, 🙂
Marko
The complete S matrix of quarks and gluons within QCD will teach you very little about QCD as it is really observed. An S-matrix does not make a theory, an S-matrix and a vacuum state do.
If a string theory solution low energy limit gives “a theory without a lagrangian, but with a hamiltonian and an S-matrix“, I would be skeptical that configuration is “physically consistent“ and that thing is a real field theory (than again, Tachikawa cannot give a definition of this)
Re the NYU conversation, I thought the moderator did a fine job of accomodating a mostly taciturn Dr H. There would have been precious little conversation without him.
Marko: The variational functional you propose for the diffusion equation is not bounded from below, hence pathological. Since the Lagrange multiplier and the diffusion field are independent, you can imagine that the time derivative of one is large and positive in the same region where the other field is negative. This pertains to any action of real fields, first order in derivatives.
To all: Bootstrap methods, which assume no Lagrangian have proven their value, but many of their applications (conformal field theory, integrable bootstrap) are to models for which an action or Hamiltonian is known. What isn’t understood in most cases (a major exception is the Ising model’s spin correlation functions in 2D) is how to relate one formalism from the other. A real problem in stat. mech. is making this connection. In a sense this means trying to reconstruct the Lagrangian/Hamiltonian field theory from the axiomatic field theory.
For example, in some bootstrap theories, we know some or even all of the form factors, hence something about correlation functions. Getting the equations of motion, much less the Lagrangian from this is hard (one of my own research goals is to reconstruct the Lagrangian from exact form factors for the large-N principal chiral model. This doesn’t make my life easy).
If we were lucky enough to get a nice bootstrap/axiomatic theory of QCD, there would remain the question of whether it is REALLY QCD. On the other hand, it is not clear we should care, at least as far as phenomenology is concerned.
“For a couple of encouraging indications that the theoretical physics community may finally be taking seriously the need to give up on failed thinking and try something new”
The titles of the more formal talks at the Italy conference look to me like just more of the same old failed thinking, only not of the mainstream variety.
Peter Orland,
“The variational functional you propose for the diffusion equation is not bounded from below, hence pathological.”
I never mentioned the diffusion equation, but regardless… The requirement that the Lagrangian be bounded from below is something one may or may not care about, but there is nothing pathological about it. The scalar curvature in the Lagrangian of GR is not bounded from below either, but I wouldn’t call the Einstein-Hilbert action “pathological” in any sense. Another example would be a BF theory, which is also not bounded from below. I could probably dig up more examples.
On a more general note, I think we need to separate the issue of the existence of the Lagrangian from issues related to the wishlist of properties we want a corresponding QFT to satisfy. The Lagrangian is a *classical* quantity, and its existence or nonexistence has nothing a priori to do with quantization. And even if one is predominantly interested in quantization itself, I don’t see why the non-existence of a classical Lagrangian would be of any benefit to the construction and analysis of QFT’s.
The only statement in this thread so far that makes sense to me is from Peter Woit, arguing that a QFT with no well-defined classical limit may have no Lagrangian associated to it. But such QFTs are hardly relevant for physics, I cannot think of a situation where such a QFT would be in any way connected to the real world. So interesting mathematics aside, why study those in the first place?
Best, 🙂
Marko
“The scalar curvature in the Lagrangian of GR is not bounded from below either, but I wouldn’t call the Einstein-Hilbert action “pathological” in any sense. ”
Quantum gravity (even with a UV cut-off) IS pathological with signature ++++. Not with signature -+++. There were fights over this very issue by quantum gravity people (which I watched from the sidelines) in the eighties. Hawking was arguing that the right analytic continuation could be used to make sense of ++++, but other quantum gravity types appeared unconvinced.
If an action is unbounded from below, there is generally no ground state in the quantum theory. Exceptions are situations where the phase space volume around minus infinity is small enough (like the hydrogen atom).
Peter Orland,
I’d say that the “pathology” of the Lagrangian which is not bounded from below is actually a shortcoming of the possible QFT description, rather than the shortcoming of the Lagrangian itself. In other words, the quantization of such a Lagrangian cannot be done within the framework of QFT — which is a problem of QFT, not of the Lagrangian. There are other quantization frameworks out there…
In particular, the existence of the ground state is one of the “wishlist” things that one may or may not care about when one discusses QFTs, in particular perturbative QFTs. Another example would be the lack of a unique ground state in QFTs in curved spacetime. If anything, these are shortcomings of the perturbative QFT formalism — if you require minimum energy in the theory, or a global Poincare symmetry of the background spacetime, you simply limit yourself to a certain subset of theories where these properties can be satisfied, while the theories where these properties cannot be satisfied (such as GR) become out-of-scope for your QFT description. That’s why nobody really expects quantum gravity to be a QFT, nor (as a consequence of QG) does anybody expect that QFT should be a fundamental description of nature. Today people are slowly getting disenchanted by QFTs and start talking more and more about *effective* QFTs, with the understanding that a QFT is just an approximate description of reality, while at a fundamental level there should be some non-QFT type of theory.
But all this has nothing to do with the existence of the classical Lagrangian, given some classical differential equations of motion. It is a completely separate issue, and should be kept separate, IMO.
Best, 🙂
Marko
Well, I don’t understand why unstable Lagrangians are useful, even classically. The problem goes beyond quantum mechanics. Maybe there are some weird examples where you never worry the instability, but you are messing with foundational stuff here.
If you allow actions to be unbounded from below, you are inviting all sorts of trouble, unless you have a parameter to squelch the instability (like a small value of 1/N, some Sobolev inequality removing the instability, etc.).
Invoking effective QFT’s seems like a red herring. No matter what the more fundamental theory is, it had better be consistent. Theories without ground states aren’t. Even serious axiomatisists (Streater and Wightman, for example) demand a ground state. S matrix theorists (if there are any still out there) do too; there is a vacuum associated with no particles.
vmarko,
the proposal you make (if I follow what you mean) does not just add more fields and field equations, but it leads to a different phase space structure and hence to inequivalent field theories. Consider this for the simple case of the free real scalar field. Your prescription does not just yield a second field (your would-be “Lagrange multiplier”) which has EOMs of a second scalar field, but the canonical momentum of the original field now becomes the derivative of the “Lagrange multiplier” field and vice versa. Hence the resulting field theory is not the original field theory plus extra stuff.
Hi Urs,
Ok, in general, if your original equation cannot be derived from a Lagrangian on its own, I don’t know how can you construct a Hamiltonian, nor the phase space. In that case, the phase space structure coming from my proposed Lagrangian is as good as any, since it has nothing to compare to.
On the other hand, in the case of the real scalar field that you mention, there do exist “traditional” Lagrangian, Hamiltonian and phase space, and can be compared to the Lagrangian I proposed. In that case, the EL-EoMs give rise to the original EoM for the scalar field plus another EoM for the multiplier, and the corresponding phase space structure indeed appears different than the original one. However, I suspect that there is a canonical transformation that will restore the usual canonical variables describing two noninteracting scalar fields. Therefore, the “traditional” classical theory is a subset of the new theory, since I merely added one more scalar field which does not interact with the original one. After that, quantization gives you the standard quantum theory for the old scalar field plus the new auxiliary scalar field. So the procedure merely extends the theory, without changing any physics of the old theory.
But the main advantage of the approach I propose comes about when the original differential equation does *not* have a well-defined Lagrangian (and consequently the Hamiltonian and the phase space, constructed from the Lagrangian in the standard way). In that case, my point is that one can still manage to define a Lagrangian, at the expense of introducing an auxiliary degree of freedom. In my eyes, having a Lagrangian is an advantage of that approach.
Anyway, these days it is common practice for physicists to introduce new auxiliary fields whenever they please, it’s not a controversial step to make, IMO.
Best, 🙂
Marko
I too thought the interviewer did a great job in getting a conversation going, I didn’t notice that he was particularly hogging the conversation or being overtly ‘clever’, and as he points out right at the beginning Dr H is jet-lagged, hence taciturnity.
I’m a little disappointed that the conference calling for ‘new thinking’ doesn’t mention causal nets which I find interesting, because (apparently) it admits Presentism as well as general covariance, which on the face of it, seem to be at odds.
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Dear vmarko,
I do see your point. But I still want to caution that a field theory is more than its equations of motion, and that in this respect the sectors of field theories that you propose are peculiar, to say the least.
The point about phase space structure that I made above in the example of the free scalar field applies more generally: In your proposal to consider Lagrangians of the form L = lambda P[phi] for P any given differential operator on a field phi, the canonical momenta of phi end up involving the would-be “Lagrange parameter” lambda, and conversely. Specifically in the case that P[phi] is linear, then the canonical momenta for phi are proportional to (derivatives) of lambda and independent of phi.
This in turn implies that the propagators between the phi-s are trivial, because these come from the Poisson bracket pairing.
So if you consider, for linear PDE P, the Lagrangian L = lambda P[phi] and then restrict attention to observables involving only phi, not lambda, then the theory looks entirely like the classical field theory defined by P[phi] = 0.
While I agree that one can consider this situation, its peculiarity makes it be not a counter argument to the claim that it is a special property of a field theory to be Lagrangian, i think.
Notice the similarity between your proposal and the very issue of the non-Lagrangian chiral WZW model (“self-dual boson” in the abelian case, the 2d toy version of that (2,0) superconformal 6d self-dual higher gauge theory that is discussed elsewhere):
Here one may start out with the Lagrangian field theory in 2d that turns out to split into two chiral halfs and then observe that each chiral part is a viable quantum field theory in itself (current algebra). Now even though there is a Lagrangian around which gives the two chiral halfs together, possibly suggesting to regard the other half as an “auxiliary field”, the point is still that either one by itself is not Lagrangian; and this is what makes self-dual higher gauge theories and their holographic/boundary relation to Chern-Simons theories interesting.
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