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.

]]>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, đź™‚

Marko

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).

]]>“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, đź™‚

Marko

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.

]]>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.

]]>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) ]]>

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, đź™‚

Marko

“The issue is quantization. Are there interesting QFTs that are not in any known sense the â€śquantizationâ€ť 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 theyâ€™re strongly coupled QFTs that donâ€™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, đź™‚

Marko