Higher Prequantum Geometry III: The global action functional - cohomologically - Comments

[Urs Schreiber]

Urs Schreiber submitted a new PF Insights post

Higher Prequantum Geometry III: The Global Action Functional - Cohomologically

Continue reading the Original PF Insights Post.

 

[David Corfield]

One day I mean to take another look at http://arxiv.org/abs/physics/0210081 to see whether the modality of classical mechanics there matches your ideas.

 

[Urs Schreiber]

One day I mean to take another look at http://arxiv.org/abs/physics/0210081 to see whether the modality of classical mechanics there matches your ideas.

It seems to me that the use of the word "modality" there is more vague than what is used in Modern physics formalized in Modal homotopy type theory . In any case, on pages 12-13 of physics/0210081 there is the following, in brief summary and in my (re-)formulation:

• his 1st modality is about the question: to which extent does the whole field bundle $E$ matters beyond the critical locus (the physical shell) $\mathcal{E}$ inside it
• his 2nd modality refers to field bundles $E$ which may not describe anything seen in nature;
• his 3rd modality refers to Lagrangians $L$ which may not describe anything seen in nature.

The second and third issue is the one of concrete particulars of general abstract theories. The question is: once we have the fundamental theory of nature (such as Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory) are there models ("vacua") that are somehow singled out as being really realized. See also our discussion re "universal exceptionalism", the brane bouquet, etc.

The first relates more to a concrete technical question. In the next article in the series I describe how one needs to lift the "Euler-Lagrange$p$-gerbe" to a "Lepage $p$-gerbe" and then restrict that to the shell. The lift needs to happen off-shell, and so it's indeed a good question how much of a neighbourhood of the classical shell one needs to take into account for the lift.

The same kind of issue appears in Noether's theorem, where one considers symmetries that must be defineable off-shell but in the end are only considered in their restriction on shell.

In the end this has to do with the question to which extent the path integral may be substituted by (or rather: be made rigorous by) constructions that happen just on the covariant phase space: the path integral by definition integrates over all sections of $E$, not just those of the shell. The covariant phase space however is just the shell $\mathcal{E}$ equipped with some prequantum information $\mathbf{\Theta}$ about how it deforms into $E$. There is a somewhat subtle question here as to how much of the original $E$ is seen by $(\mathcal{E},\mathbf{\Theta})$.

 

[David Corfield]

So we could do with an nLab entry on 'shell'. At the moment we just have "off-shell Poisson bracket' and ''on-shell recursion'. You're saying the shell is the critical locus of the EL-functional.

Hmm, I see there are some basics I haven't sorted out. Why doesn't the process of finding the critical locus deliver a subspace of $[\Sigma, E]$? Instead we have solutions as $[\Sigma, \mathcal{E}]$ where $\mathcal{E}$ is a subspace of $E$. And this $\mathcal{E}$ is said to be an equation. Presumably there's the employment of equivalences, such as when a differential equation is identified with its space of solutions, but I'm still struggling with what $\mathcal{E}$ is if it's also a subspace of $E$. Is there a simplest example of a classical field to work through?

 

[Urs Schreiber]

Good, point, I have started an nLab entry shell. Does the text there now answer your question?

 

[Urs Schreiber]

Regarding worked examples: there is standard literature, Anderson's book is really good

 

[David Corfield]

That's helpful, thanks.

Returning to Butterfield's text, regarding his modality type 2, " it considers a counterfactual number of degrees of freedom, or a counterfactual potential function," wouldn't that include varying the Lagrangian, which you have above as type 3?

His 3rd type, as I read it, sounds like looking off-shell. " the counterfactual histories share the initial and final conditions, but do not obey the given deterministic laws of motion, with the given forces."

Since we may be able to change the Lagrangian to make one of these non-legal trajectories into a legal trajectory for the new Lagrangian, he says " I also agree that my distinction between (Modality;3rd) and (Modality;2nd), between varying the laws and varying the forces, is not as hard-and-fast as it first seems"

His type 1 is just shifting through the solution space by varying initial conditions.

So what you call modality type 1 (which I think is Butterfield's type 3) might be construed in terms of infinitesimal invariance around the shell, so link to the kind of jet comonad modality we've discussed?

 

[Urs Schreiber]

Regarding Butterfield's points, ah, maybe I numbered them in the wrong order, true. In either case, I am not sure if I recognize the "necessary in nearby possible worlds"-modality in his text, but I still think and agree that this may be a fun way to think of the jet comonad as being about infintiesimal necessity and as the shell as being a modal type for that modality.

(Now of course nobody following us here will know what we are talking about, so maybe I should close with adding that this is about the analogy of the jet comonad to the necessity modality.)

 

[David Corfield]

I see the Anderson you mentioned above has moved into computer techniques for differential geometry. That might provide a point of comparison if you succeed in encouraging people to implement modal homotopy type theory.

 

[Urs Schreiber]

I see the Anderson you mentioned above has moved into computer techniques for differential geometry.

Yes, apparently he has said that his interest in this computer algebra is what prevented him from fully finalizing his book on the variational bicomplex.

That might provide a point of comparison if you succeed in encouraging people to implement modal homotopy type theory.

I feel that eventually there will be a considerable gain in having computer support not only for the tensor calculus representation of variational calculus (and ultimately of prequantum field theory) but also of its conceptual structure. For instance, as the next article in the series will discuss, Noether's variational theorem has a more fundamental representation than the usual component-derivation, and algebra software that could handle differential cohesion would be able to reason about this. But then, presently this is apparently a topic too far in the future. We'll see.

On a different note, I have added to the entry above a paragraph which makes explicit that homotopies between maps into the Deligne moduli $\mathbf{B}_H^{p+1}(\mathbb{R}/_{\!\hbar}\mathbb{Z})_{\mathrm{conn}}$ are equivalently coboundaries between the corresponding Cech-Deligne cocycles. I'll be using this a lot in article number V in the series, and so I went back and made it more explicit here.