I have expanded, a little, the lead-in paragraph of the Idea-section (here)

and added a graphics schematically illustrating the covariant and the canonical phase space and their relation.

]]>Thanks for adding that missing arXiv link.

Now I also added today’s

- Alejandro Corichi, Juan D. Reyes, Tatjana Vukasinac,
*On covariant and canonical Hamiltonian formalisms for gauge theories*[arXiv:2312.10229]]

Added an arXiv link to Margalef-Bentabol–Villaseñor.

]]>added pointer to:

- Juan Margalef-Bentabol, Eduardo J. S. Villaseñor,
*Geometric formulation of the covariant phase space methods with boundaries*, Phys. Rev. D**103**(2021) 025011 [doi:10.1103/PhysRevD.103.025011]

added pointer to:

- Daniel Harlow, Jie-qiang Wu,
*Covariant phase space with boundaries*, J. High Energ. Phys.**2020**146 (2020). [arXiv:1906.08616, doi:10.1007/JHEP10(2020)146]

added pointer to:

- Jerzy Kijowski, Wlodzimierz M. Tulczyjew,
*A Symplectic Framework for Field Theories*, Lecture Notes in Physics**107**(1979, 2005) [doi:10.1007/3-540-09538-1]

have boosted several bibitems in the entry (adding/fixing hyperlinks, ects.)

]]>I have fixed the hierarchy of the subsections and re-instantiated the table of contents.

(Also I changed the part about the “Application to the inverse problem” from a subsection to a Remark.)

]]>Deleting the table of contents for now to make the article parse.

See also https://nforum.ncatlab.org/discussion/10105/bugs-and-feature-requests/?Focus=105801#Comment_105801 for a description of this bug.

]]>This article no longer parses and produces an XML error.

]]>Back in October 2011, in revision 29, I had added to *covariant phase space* the observation “Via the BV-complex”, originating in discussion with Igor Khavkine, which says that the BV 2-form corresponding to the anti-bracket serves as a homotopy/coboundary between the two canonical symplectic forms that are associated via the covariant phase space method to two cobordant Cauchy surfaces.

Now I see that this is precisely the key compatibility condition as stated in equation (9) of

- Aberto S. Cattaneo, Pavel Mnev, Nicolai Reshetikhin,
*Classical BV theories on manifolds with boundary*(arXiv:1201.0290)

I have added a brief pointer to the entry. This deserves to be expanded on further, eventually.

]]>Igor Khavkine kindly added to covariant phase space a new section Application to the inverse problem of the calculus of variations

]]>have pasted into covariant phase space a bunch of information (eg. long commented list of literature) kindly provided by Igor Khavkine

]]>Not sure if I will be quick at all. Don’t have much of an idea yet.

But it might help to formulate this more in terms that are more common in the literature.

Here is a related exercise, that one should look into:

let’s consider dg-geometry: take the $\infty$-site of formal duals of cochain dg-algebras in non-positive degree and the oo-topos over that, $Sh_\infty(dgAlg_-^{op})$ (choice of topology won’t matter much, as I’ll oly be looking at pullbacks, so I guess you can just as well assume the $\infty$-presheaf topos).

Then the object that I like to write $\mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1$ ought to be given by the cotangent complex functor

$\mathbf{\flat}_{dR} \mathbf{B}\mathbb{R} :A \mapsto \mathbb{L}\Omega^1_K(A)$and the morphism that I like to write

$\theta : \mathbb{A}^1 \to \mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1$ought to be given by

$\theta_A : (a \in Q A) \mapsto d a \in \Omega^1_K(Q A) \,,$where we observe that $\mathbb{A}^1 : A\mapsto A$ and where $Q A$ denotes a cofibrant resolution of $A \in dgAlg_-$.

Now let $\mathfrak{g} \to T X = T Spec B$ be a Lie algebra action on some space and write $X//\mathfrak{g}$ for the corresponding quotient Lie algebroid, given by the Chevalley-Eilenberg dg-algebra: the BRST complex for the action.

Then a morphism

$S : X//\mathfrak{g} \to \mathbb{A}^1$is the same thing as a $\mathfrak{g}$-invariant function on $X$. We want to compute in $Sh_\infty(dgAlg_-)$ the homtopy fiber of

$d S : X//\mathfrak{g} \stackrel{S}{\to} \mathbb{A}^1 \stackrel{\theta}{\to} \mathbb{L} \Omega^1_K(-) \,.$That homotopy fiber ought to be given by the BV-BRST complex for the data $X, \mathfrak{g}, S$, representing the derived critical locus of $S$.

I see people go around and call the BV-BRST complex the “derived critical locus” of $S$. (I know two authors who do this, probably there are more.) But I haven’t seen this little exercise worked out, which would be necessary to fully prove the intuition correct.

]]>Very nice question. I would like to think about it, but I am sure you will be quicker, so hopeless to start…

]]>I am now wondering about the following;

in the abstract language, given an action functional

$S : X \to \mathbb{R}$the covariant phase space is the fiber

$\array{ X|_{d S = 0} \\ \downarrow \\ X &\stackrel{d S}{\to}& \mathbf{\flat}_{dR}\mathbf{B}\mathbb{R} }$of $d S$.

Now that statement going back to Zuckerman, Witten and maybe others is that if $S$ comes from a sensible Lagrangian, then there is canonically a presymplectic structure on $X|_{d S = 0}$. But this just means that there is a closed 2-form $\omega$ on it (not necessarily non-degenerate). Therefore abstractly this means that there is canonically a morphism

$\omega : X|_{d S = 0} \to \mathbf{\flat}_{dR} \mathbf{B}^2 \mathbb{R} \,.$I am wondering how I can see the existence of this morphism abstractly. That would be very interesting to have.

It begins to look a little like a kind of Postnikov tower

$\array{ Y \\ \downarrow \\ X|_{d S = 0} &\stackrel{\omega}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^2\mathbb{R} \\ \downarrow \\ X &\stackrel{d S}{\to} &\mathbf{\flat}_{dR} \mathbf{B}\mathbb{R} } \,,$where $Y$ is the locus on which $\omega$ completely degenerates (is empty for a genuine symplectic structure)

What’s going on here?

]]>This will upgrade you from “Doctor Schreiber” to “Professor Schreiber”, right?

Unfortunately not exactly, otherwise I might have been more enthusiastic about it. Technically a “professor” here is not a degree but a position.

Writing a “Habilitation” used to be the mandatory prerequisite in Germany for applying for professorship, hence for permanent position. But a few years back the government noticed that German professors used to be on average much older than elsewhere in the world. They decreed to change the system, declared the “Habilitation” to be no longer a formal prerequisite and introduced instead something roughly akin to “tenure track” in other countries, which they called “junior professorship”.

But, probably because all the committees consist of all the old professors who don’t see any reason for why what was good for them should not be good for everyone, in practice what happens is that if you apply somewhere in Germany for a permanent position, the committee probably will want to see that you have written a Habilitation. Or thather, if you have written one probably nobody will care much about it, but if you don’t have written one, they might raise their eyebrows.

This implies that what also happens in practice is that the whole process degenerates to a pointless formality: a Habilitation is usually quite literally an Introduction followed by the publication record stapled together.

]]>Typo: http://ncatlab.org/nlab/show/Jerzy+Kyjowski

I changed the name to http://ncatlab.org/nlab/show/Jerzy+Kijowski but the cash bug also keeps the old page.

The diffiety school is partly motivated by similar issues:

A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., Vol. 2, 1984, p. 21.

A. M. Vinogradov, Symmetries and conservation laws of partial differential equations: basic notions and results, Acta Appl. Math., Vol. 15, 1989, p. 3.

A. M. Vinogradov, Scalar differential invariants, diffieties and characteristic classes, in: Mechanics, Analysis and Geometry: 200 Years after

This is supposed to serve as what in Deutschland they call a “habilitation thesis”

This will upgrade you from “Doctor Schreiber” to “Professor Schreiber”, right?

In America, my students would sometimes call me “Professor” while I was still a graduate student. (And today, teaching in a school where most instructors don’t have PhDs, they often mistakenly call me “Mister”.)

]]>By the way, stub Jerzy Kijowski.

This seems to neither exist nor to be linked to from phase space.

]]>Great go on…this great work of yours should be published soon :)

Yes. There is always loose ends (and tying them up produces more) but I need to produce something that serves as “version 1.0” very soon. I was supposed to hand this in already last summer. This is supposed to serve as what in Deutschland they call a “habilitation thesis”, but I am not sure if it is stil of the form they expected me to produce.

]]>By the way, stub Jerzy Kijowski.

There is also lots of recent references (many at arXiv, cf. hep-th list here) of G. Sardanashvily who studies classical phase space and constraints with lots of references to jet bundles.

As far as de Witt, one reference just for the record (I am not sure if this is the correct entry for it)

- Bryce S. DeWitt, Dynamical theory of groups and fields. 1964 Relativité, Groupes et Topologie (Lectures, Les Houches, 1963 Summer School of Theoret. Phys., Univ. Grenoble) pp. 585–820 Gordon and Breach, New York (Russian transl. Dinamičeskaja teorija grupp i poleĭ, Nauka, Moskva 1987)

Great go on…this great work of yours should be published soon :)

I added the above survey M. J. Gotay, J. Isenberg, R. Montgomery, J. E. Marsden to the references at phase space.

]]>of what?

Of differential cohomology in a cohesive topos (schreiber)

Only missing section is the very last one on $\infty$-Chern-Simons theory. Which I’ll fill in now, with the material at infinity-Chern-Simons theory (schreiber)

]]>