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I have expanded the previous stub-entry phase space
I emphasized the notion of covariant phase space = space of solution to the EL equations as the fundamental notion and so far have some vague remarks on how reduced phase spaces are obtained from this (see the references that I list)
gave a formal definition of covariant phase space using cohesive $\infty$-topos technology;
indicated the beginning of a discussion of examples in derived geometry: Koszul-Tate resolutions as derived critical loci of action functionals = derived covariant phase spaces.
This needs further discussion. But I think it’s a start. Have also created a quick note at configuration space in a similar spirit
also edited action functional in a similar spirit. Much more should be said here, but I am running out of steam for the moment.
Crnković was a student of Witten, now unfortunately in finance. Please distinguish ć and č, č is in Russian names, Croatians have both č and ć which carry a weight of a distinct phonems. In my memory, K. Gawędzki wrote an article or review in which he said that the infinite-dimensional symplectic formalism from Crnković-Witten papers is, unlike the impression from the paper, in essence not new, and listed several papers from diverse authors preceding it which tackled the subject.
Maybe these are some of the relevant references
J. Kijowski, W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys. 46, 1976, 183–206.
J. Kijowski, W. Tulczyjew , A symplectic framework for field theories, Lecture Notes in Physics 107, Springer 1979.
See recent survey
Quite a bibliography on Hamiltonian approach and phase space in classical field theory compiled by Sam at http://www.physicsforums.com/showthread.php?t=388700
[1] I.V. Kanatchikov: On Field Theoretic Generalizations of a Poisson Algebra, Rep. Math. Phys. 40 (1997) 225-234, hep-th/9710069.
[2] I.V. Kanatchikov: Canonical Structure of Classical Field Theory in the Polymomentum Phase Space, Rep. Math. Phys. 41 (1998) 49-90, hep-th/9709229.
[3] J.F. Cari˜nena, M. Crampin & L.A. Ibort: On the Multisymplectic Formalism for First Order Field Theories, Diff. Geom. Appl. 1 (1991) 345-374.
[4] M.J. Gotay, J. Isenberg & J.E. Marsden: momentum Maps and Classical Relativistic Fields I: Covariant Field Theory, physics/9801019.
[5] M. Forger & H. R¨omer: A Poisson Bracket on Multisymplectic Phase Space, Rep. Math. Phys. 48 (2001) 211-218; math-ph/0009037.
[6] H. Goldschmidt & S. Sternberg: The Hamilton-Cartan Formalism in the Calculus of Variations, Ann. Inst. Four. 23 (1973) 203-267.
[7] V. Guillemin & S. Sternberg: Geometric Asymptotics, Mathematical Surveys, Vol. 14, American Mathematical Society, Providence 1977.
[8] J. Kijowski: A Finite-dimensional Canonical Formalism in the Classical Field Theory, Commun. Math. Phys. 30 (1973) 99-128; Multiphase Spaces and Gauge in Calculus of Variations, Bull. Acad. Pol. Sci. SMAP 22 (1974) 1219-1225.
[9] J. Kijowski & W. Szczyrba: Multisymplectic Manifolds and the Geometrical Construction of the Poisson Brackets in the Classical Field Theory, in: “G´eometrie Symplectique et Physique Math´ematique”, pp. 347-379, ed.: J.-M. Souriau, C.N.R.S., Paris 1975.
[10] J. Kijowski & W. Szczyrba: Canonical Structure for Classical Field Theories, Commun. Math. Phys. 46 (1976) 183-206.
[11] J. Kijowski & W. Tulczyjew: A Symplectic Framework for Field Theories, Lecture Notes in Physics, Vol. 107, Springer-Verlag, Berlin 1979.
[12] M. Forger, C. Paufler & H. R¨omer: More about Poisson Brackets and Poisson Forms in Multisymplectic Field Theory.
[13] C. Crnkovi´c & E. Witten: Covariant Description of Canonical Formalism in Geometrical Theories, in: “Three Hundred Years of Gravitation”, pp. 676-684, eds: W. Israel & S. Hawking, Cambridge University Press, Cambridge 1987.
[14] C. Crnkovi´c: Symplectic Geometry of Covariant Phase Space, Class. Quant. Grav. 5 (1988) 1557-1575.
[15] G. Zuckerman: Action Principles and Global Geometry, in: “Mathematical Aspects of String Theory”, pp. 259-288, ed.: S.-T. Yau, World Scientific, Singapore 1987.
[16] R.E. Peierls: The Commutation Laws of Relativistic Field Theory, Proc. Roy. Soc. Lond. A 214 (1952) 143-157.
[17] B. de Witt: Dynamical Theory of Groups and Fields, in: “Relativity, Groups and Topology, 1963 Les Houches Lectures”, pp. 585-820, eds: B. de Witt & C. de Witt, Gordon and Breach, New York 1964.
[18] B. de Witt: The Spacetime Approach to Quantum Field Theory, in: “Relativity, Groups and Topology II, 1983 Les Houches Lectures”, pp. 382-738, eds.: B. de Witt & R. Stora, Elsevier, Amsterdam 1984.
[19] S.V. Romero: Colchete de Poisson Covariante na Teoria Geom´etrica dos Campos, PhD thesis, Institute for Mathematics and Statistics, University of S˜ao Paulo, June 2001.
[20] M. Forger & S.V. Romero: Covariant Poisson Brackets in Geometric Field Theory.
[21] R. Abraham & J.E. Marsden: Foundations of Mechanics, 2nd edition, Benjamin/ Cummings, Reading 1978.
[22] V. Arnold: Mathematical Methods of Classical Mechanics, 2nd edition, Springer, Berlin 1989.
[23] H.A. Kastrup: Canonical Theories of Lagrangian Dynamical Systems in Physics Phys. Rep. 101 (1983) 3-167.
[24] G. Martin: A Darboux Theorem for Multisymplectic Manifolds, Lett. Math. Phys. 16 (1988) 133-138.
[25] G. Martin: Dynamical Structures for k-Vector Fields, Int. J. Theor. Phys. 41 (1988) 571-585.
[26] F. Cantrijn, A. Ibort & M. de Le´on: On the Geometry of Multisymplectic Manifolds, J. Austral. Math. Soc. (Series A) 66 (1999) 303-330.
[27] C. Paufler & H. R¨omer: Geometry of Hamiltonian n-Vector Fields in Multisymplectic Field Theory, math-ph/0102008, to appear in J. Geom. Phys.
[28] C. Paufler: A Vertical Exterior Derivative in Multisymplectic Geometry and a Graded Poisson Bracket for Nontrivial Geometries, Rep. Math. Phys. 47 (2001) 101-119; math-ph/0002032.
[29] L. Faddeev, in Quantum Fields and Strings: A Course for Mathematicians, Vol 1, P513-550. American Mathematical Society.
elsewhere I have seen the idea of “covariant phase space” attributed to deWitt. I am not sure what the history is.
But let’s add that list of references.
By the way, Tamarkin has presented in Paris 2004, a derived version of Marsden-Weinstein symplectic reduction. I think he never puboished that nice result. For example the topological A-model and topological B-model are obtained in quite a simple way using a derived symplectic reduction from the Poisson sigma model (of Schaller-Strobl and advanced later by Cattaneo-Felder and Kontsevich). Unfortunately I did not keep full notes and those which I retained are burried somewhere in my old stack of papers…
Urs: surely, we can not resolve things now and we should include more references (though we should make some useful choices). On the other hand, there are two things: one is about making the formalisation of the constrained phase space and the other about passing from finite dimensional to the infinite dimensional formalism. Crnković-Witten, as far as I recall in reading them, emphasised on the fact that they moved into the infinite-dimensional world, what was definitely studied many times before. It may be that the specific of the covariant phase space in field theory is due de Witt. I have his book about it.
Thanks for the hint. I need to collect everything I can get on this matter. We will be running our group’s seminar on something along these lines this semester.
But not right now. I need to finish my writeup. My goal is to have a version 1.0 anounced by… tonght. Famous last words.
Sound nice. 1.0 of what ?
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)
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.
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)
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.
This seems to neither exist nor to be linked to from phase space.
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”.)
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 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.
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?
Very nice question. I would like to think about it, but I am sure you will be quicker, so hopeless to start…
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.
have pasted into covariant phase space a bunch of information (eg. long commented list of literature) kindly provided by Igor Khavkine
Igor Khavkine kindly added to covariant phase space a new section Application to the inverse problem of the calculus of variations
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
I have added a brief pointer to the entry. This deserves to be expanded on further, eventually.
This article no longer parses and produces an XML error.
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.
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