What Does As a Sheaf of Functions on Phase Space Mean?

[Urs Schreiber]

Urs Schreiber submitted a new PF Insights post

Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively

Continue reading the Original PF Insights Post.

 

[ShayanJ]

I don't know whether this is a proper question to ask here or not, but I can't resist any more.
This whole series is way above my head but I somehow understand its importance and beauty, so I'm eager to learn it, actually adding it to my list of to-learn things!
So what branches of Mathematics do I need to learn?
I think they are algebraic topology and algebraic geometry (and category theory?), are they?
Thanks

 

[David Corfield]

There are some (three?) occasions where the Latex didn't compile.

And some typos: bunle; invariace; codimenion; depening

This post will need some re-reading on my part. I wonder how much these constructions are arising through abstract general considerations. E.g. where one to want such a thing, would they go through in the complex analytic realm? And how much (presumably less) in the realm of arithmetic jet spaces?

 

[dextercioby]

I don't know whether this is a proper question to ask here or not, but I can't resist any more.
This whole series is way above my head but I somehow understand its importance and beauty, so I'm eager to learn it, actually adding it to my list of to-learn things!
So what branches of Mathematics do I need to learn?
I think they are algebraic topology and algebraic geometry (and category theory?), are they?
Thanks

This is the highest level tutorial ever posted on PF. High level differential geometry, diff. and algebraic topology. To reach this level of maths knowledge, you need to start early. Did you read Arnold's mechanics text ?

 

[ShayanJ]

This is the highest level tutorial ever posted on PF. High level differential geometry, diff. and algebraic topology. To reach this level of maths knowledge, you need to start early. Did you read Arnold's mechanics text ?

I know Arnold's text, but the book I plan to read about more theoretical aspects of classical mechanics is "Mechanics: From Newton's Laws to Deterministic Chaos" by Florian Scheck. Is it more or less equivalent to Arnold's?

 

[Urs Schreiber]

There are some (three?) occasions where the Latex didn't compile. And some typos: ...

Thanks! Fixed now.

I wonder how much these constructions are arising through abstract general considerations.

Everything! For the present series I am downplaying the general abstract perspective, clearly, but everything I am saying here flows naturally out of differentially cohesive homotopy theory. The key point is that the variational bicomplex is locally equivalent to the de Rham complex. This means that as we start with ordinary differential cohomology in the base differentially cohesive infinity-topos and then send that to the topos of PDEs, there a "finer Poincare lemma" appears which allows to resolve the constant real coefficients by a chain complex adapted to the horizontal stratification. This way the variational Euler-differential appears all by itself as the curvature of those "Euler-Lagrange p-gerbes". Anyway, I should talk about this in more detail later.

And how much (presumably less) in the realm of arithmetic jet spaces?

One has to beware that the arithmetic jet spaces of Buium do not capture the general concept of jets. I think the right way to put it is that for $X$ an arithmetic scheme, then Buium's arithmetic jet space is to be thought of as the jets of the bundle $X \times \mathrm{Spec}(\mathbb{Z}) \to \mathrm{Spec}(\mathbb{Z})$. More generally one need jet bundles of more general arithmetic bundles.

 

[Urs Schreiber]

So what branches of Mathematics do I need to learn? I think they are algebraic topology and algebraic geometry (and category theory?), are they?
Thanks

I am rather freely speaking category theory here, that's true. But for the moment all the series is really referring to is the concept of presheaf and the Yoneda lemma. Under the hood the Beck monadicity theorem is playing a key role, but just to follow the story you don't need that.

And then of algebraic topology the series presently needs mainly the concept of sheaf hypercohomology in its presentation by Cech cohomology. I have been writing lecture notes that gradually introduce precisely the material that I am using here at

• nLab: geometry of physics

These notes consist of a list of sub-pages that have detailed exposition and introduction to the various subjects needed, complete with further pointers to the literature.

Essentially this same lecture series is also available in pdf format, as section 1.2 of

• https://dl.dropboxusercontent.com/u/12630719/dcct.pdf

(just focus on section 1).

 

[David Corfield]

Thanks. I see Anderson has a G-equivariant version for some group of symmetries of E, which delivers a G-invariant Euler-Lagrange complex.

He paints a picture of an ambitious research program. Would it be fair to say this work hasn't been exploited by the community as much as it might have been?

 

[Urs Schreiber]

Would it be fair to say this work hasn't been exploited by the community as much as it might have been?

A point that Igor Khavkine has been making for a while (for instance in this talk) is that the full power of available mathematical machinery for handling local field theories has rarely been fully applied, notably so in the case of effective gravity.

A striking example is the construction of local and gauge-invariant observables in gravity. Folk-lore held that these simply do not exist, a claim that leads to a wealth of apparent problems that are often claimed to necessitate speculative modifications of established physical principles. But a careful mathematical analysis shows that these observables do indeed exist, not as global functions, but as a sheaf of functions on phase space:

• Igor Khavkine, Local and gauge invariant observables in gravity, Class. Quantum Grav. 32 185019, 2015 (arXiv:1503.03754, CQG+, http://www.science.unitn.it/~moretti/convegno/khavkine.pdf for talk at http://www.science.unitn.it/~moretti/convegno/convegno.html Levico Terme, Italy, September 2014)

 

[dextercioby]

I know Arnold's text, but the book I plan to read about more theoretical aspects of classical mechanics is "Mechanics: From Newton's Laws to Deterministic Chaos" by Florian Scheck. Is it more or less equivalent to Arnold's?

His ambition is there, to be at the same level, but his presentation is not something I like. He starts at the average level of Marion/Thornton for about 330 pages, then turns the mathematics on what he already wrote about.

 

[David Corfield]

There are some (three?) occasions where the Latex didn't compile. And some typos: ...

Thanks! Fixed now.

I wonder how much these constructions are arising through abstract general considerations.

Everything! For the present series I am downplaying the general abstract perspective, clearly, but everything I am saying here flows naturally out of differentially cohesive homotopy theory. The key point is that the variational bicomplex is locally equivalent to the de Rham complex. This means that as we start with ordinary differential cohomology in the base differentially cohesive infinity-topos and then send that to the topos of PDEs, there a "finer Poincare lemma" appears which allows to resolve the constant real coefficients by a chain complex adapted to the horizontal stratification. This way the variational Euler-differential appears all by itself as the curvature of those "Euler-Lagrange p-gerbes". Anyway, I should talk about this in more detail later.

And how much (presumably less) in the realm of arithmetic jet spaces?

One has to beware that the arithmetic jet spaces of Buium do not capture the general concept of jets. I think the right way to put it is that for $X$ an arithmetic scheme, then Buium's arithmetic jet space is to be thought of as the jets of the bundle $X times mathrm{Spec}(mathbb{Z}) to mathrm{Spec}(mathbb{Z})$. More generally one need jet bundles of more general arithmetic bundles.

"More generally one need jet bundles of more general arithmetic bundles."

Is he as limited as this? In 'Differential calculus with integers' (http://arxiv.org/abs/1308.5194) he's talking about taking jet spaces of spec(W(R)) (p-typical Witt vectors of R) in section 1.2.12.

And problem 3.1 states:
"Study the arithmetic jet spaces Jn(X) of curves X (and more general varieties) with bad reduction."

Many of the constructions appearing in these posts appear there, e.g., variational complex, Euler-Lagrange total differential form and Noether's theorem appear in 1.2.9. But maybe just as exposition.

It seems it's his 'Arithmetic Differential Equations' where one sees his "arithmetic" Euler-Lagrange operators and Noether's theorem (p. 98 of https://books.google.co.uk/books?id=aqwYKFjW5nwC). It's not so easy to find out the breadth of jet spaces treated there.

 

[Jimster41]

A point that Igor Khavkine has been making for a while (for instance in this talk) is that the full power of available mathematical machinery for handling local field theories has rarely been fully applied, notably so in the case of effective gravity.

A striking example is the construction of local and gauge-invariant observables in gravity. Folk-lore held that these simply do not exist, a claim that leads to a wealth of apparent problems that are often claimed to necessitate speculative modifications of established physical principles. But a careful mathematical analysis shows that these observables do indeed exist, not as global functions, but as a sheaf of functions on phase space:

• Igor Khavkine, Local and gauge invariant observables in gravity, Class. Quantum Grav. 32 185019, 2015 (arXiv:1503.03754, CQG+, http://www.science.unitn.it/~moretti/convegno/khavkine.pdf for talk at http://www.science.unitn.it/~moretti/convegno/convegno.html Levico Terme, Italy, September 2014)

By "as a sheaf of functions on phase space" does that mean an observable would be spread out in space-time?
If so does that mean distributed discretely, where some parts of the phase space don't support those functions?I found another paper that seems to be saying something like what Khavkine is saying.

http://arxiv.org/pdf/hep-th/0512200v4.pdf
Observables in effective gravity
Steven B. Giddings, Donald Marolf, James B. Hartle
(Submitted on 16 Dec 2005 (v1), last revised 8 Sep 2006 (this version, v4))
We address the construction and interpretation of diffeomorphism-invariant observables in a low-energy effective theory of quantum gravity. The observables we consider are constructed as integrals over the space of coordinates, in analogy to the construction of gauge-invariant observables in Yang-Mills theory via traces. As such, they are explicitly non-local. Nevertheless we describe how, in suitable quantum states and in a suitable limit, the familiar physics of local quantum field theory can be recovered from appropriate such observables, which we term `pseudo-local.' We consider measurement of pseudo-local observables, and describe how such measurements are limited by both quantum effects and gravitational interactions. These limitations support suggestions that theories of quantum gravity associated with finite regions of spacetime contain far fewer degrees of freedom than do local field theories.

 

[Urs Schreiber]

By "as a sheaf of functions on phase space" does that mean an observable would be spread out in space-time?

No, that means that on sufficiently small neighbourhoods of field configurations there is a good supply of spacetime local and gauge invariant observables that are functions just of those field configurations in that small neighbourhood. For larger neighbourhoods no such observables will exist anymore, but the small neighbourhoods for which they do exist cover all of the (phase) space of fields.