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Last Build Date: Wed, 22 Nov 2017 23:21:10 +0000
Wed, 22 Nov 2017 23:21:10 +0000
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, :-) MarkoWed, 22 Nov 2017 23:08:36 +0000
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, :-) MarkoWed, 22 Nov 2017 22:21:22 +0000
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.Wed, 22 Nov 2017 17:08:41 +0000
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.Wed, 22 Nov 2017 16:16:26 +0000
Can you elaborate what you can do in QFT without Lagrangian or Hamiltonian? Do you calculate S matrix directly?Wed, 22 Nov 2017 16:12:06 +0000
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.Wed, 22 Nov 2017 14:52:34 +0000
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.Wed, 22 Nov 2017 14:45:53 +0000
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, :-) MarkoWed, 22 Nov 2017 01:06:12 +0000
The paragraph describing Sci Con #13 reads like it could have been written in 1994.