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I have added to variational calculus a definition of critical loci of functionals, hence a definition of Euler-Lagrange equations, in terms of diffeological spaces. It’s a very natural definition which is almost explicit in Patrick Iglesias-Zemmour’s book, only that he cannot make it fully explicit since the natural formulation involves the sheaf of forms $\Omega^1_{cl}(-)$ which is not concrete and hence not considered in that book.
I was hoping I would find in his book the proof that the critical locus of a function on a diffeological space defined this was coincides with the “EL-locus” – it certainly contains it, but maybe there is some discussion necessary to show that it is not any larger – but on second reading it seems to me that the book also only observes the inclusion.
ah, exactly this point of view is made pretty much explicit in section 2.4 of the recent
from last december. Interesting, I hadn’t seen this article before. It’s a big advertizement for the use of sheaves in general and diffeological spaces in particular for applications in physics.
apparently Paugam’s article draws from his book/lecture notes Towards the mathematics of quantum field theory.
Enjoyably ambitious.
wait, that pullback defintion that I gave is not quite right. It produces too many low-dimensional plots. I need to think.
Enjoyably ambitious.
Is there anything in Paugam’s work you find unexpected?
Is there anything in Paugam’s work you find unexpected?
I haven’t seen anything unexpected yet, but I think he does a good job at trying to bring together modern developments into a coherent story on mathematical physics. For instance his description of variational calculus in the Beilinson-Drinfeld language of D-modules is laudable, I think.
I wrote:
wait, that pullback defintion that I gave is not quite right. It produces too many low-dimensional plots. I need to think.
Sorry, one has to do this in the synthetic differential topos.
Let $\mathbb{R}^*$ be the object whose plots on $U \times D$ for $D$ an infinitesmal thickening are the $U$-plots of $\mathbb{R}$. Then the pullback in question is that of $S$ along $\mathbb{R}^* \to \mathbb{R}$:
its plots are those $U \times D \to C$ such that the postcomposition $U \times D \to C \stackrel{S}{\to} \mathbb{R}$ factors through the projection $U \times D \to U$ (is constant in the infinitesimal directions).
I’ll write this out properly now.
For instance his description of variational calculus in the Beilinson-Drinfeld language of D-modules is laudable, I think.
Tamarkin was showing some bits of this in 2004, but, as I said my notes are very incomplete (at the moment I do not know where they are but most of those written down are preserved somewhere). We asked him why he did not write this into intro of his abandoned article on renormalization and he responded that Drinfeld-Beilinson do the job better. But the wisdom is too much hidden in the latter which is very dense. Paugam’s notes are very useful exposition.
Paugam also tries to connect to the language of Vinogradov’s “diffiety school” (which was entirely very much motivated by variational calculus and conservation laws). This is more or less the nonlinear analogue of D-modules, namely D-schemes (roughly the difference between crystals of quasicoherent modules and crystals of schemes, cf. Lurie’s notes here) from Gaitsgory’s seminar. BD say A-D-modules for the nonlinear/global version.
I updated diffiety (links and text formulation). I choped out the old discussion. It is here:
(Zoran: I object. There may be one idea, but as variety and scheme are not the same, the level of generality should be in mathematics precisely distinguished. D-schemes for example are not necessarily in characteristics zero. Michael: I agree that I was sloppy and should spell out the precise relation between the different definitions, but your remark about characteristic zero is not an objection. If you read Vinogradov he emphazises that everything should be expressed algebraically, and there is no problem in defining diffieties over characteristic p Zoran: but it is still a field: when we work with general rings and schemes, than unlike for varieties over a field, the residue fields vary from a point to a point, this is what I meant as a complication in taking equation approach, but I hope you will clear this out later.)
http://gdeq.org has various related materials, from the diffiety school. Yet more references and links at diffiety.
What is eventually of interest is the connection between the geometry of variational calculus and path integral quantization. Nontrivial corrections to naive Feynman’s picture are of geometric nature. My Ph.D. advisor Joel Robbin has also written two articles on the subject with Dietmar:
Joel W. Robbin, Dietmar A. Salamon, Feynman path integrals and the metaplectic representation, Math. Z. 221 (1996), no. 2, 307–-335, MR98f:58051, doi
Joel W. Robbin, Dietmar A. Salamon, Phase functions and path integrals, Symplectic geometry (Proc., ed. D. Salamon), 203–-226, London Math. Soc. Lecture Note Ser. 192, Cambridge Univ. Press 1993, RobbinSalamonPhaseFunctionsPathIntegrals.djvu:file.
The link to the file works from Joel Robbin but not from the $n$Forum. How to change the syntax for file call to $n$Lab from $n$Forum ?
Thanks, Zoran, for the the useful links.
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