Insights Introduction to Perturbative Quantum Field Theory - Comments

[A. Neumaier]

Mh? The lambshift is among the great successes of perturbative QED. What's deficient there?

Well, the mathematical basis is deficient, as in most discussion of anything involving infrared problems. (Note that this is a thread about rigorous QFT!)

The usual discussions (e.g., Weinberg, Vol. 1, Section 14.3) involve a significant amount of handwaving that is hard to make rigorous, even from a perturbative point of view.

Even the Faddeev-Kulish procedure for treating dressed electrons (the simplest infrared problem) is at present not really rigorous; see https://www.physicsforums.com/posts/5863748 .

 

[A. Neumaier]

Prodded by the feedback here, especially that by Arnold Neumaier, I have been restructuring the geometry background: Now it's being developed incrementally with the rest of the theory all in the one text here. I also brought in a skeleton of the remaining material including quantization. But not done typing yet...

So you require spacetime to be a manifold but the bigger objects (history spaces etc) only to be diffeological spaces?

 

[Urs Schreiber]

So you require spacetime to be a manifold but the bigger objects (history spaces etc) only to be diffeological spaces?

Yes. The only point in the development that requires a diffeological space (or more generally a smooth set or super smooth set) to actually be a smooth manifold is if we want to integrate differential forms over it. For spacetime this is what we want to do in order to define local observables, and therefore it is required to be a smooth manifold.

From the broader perspective of algebraic topology this is a familiar phenomenon: The theory lives on very general kinds of spaces, but as soon as one requires fiber integration to exist one gets that the fibers need to be manifolds equipped with suitable tangential structure.

 

[Urs Schreiber]

By the way, another use of diffeological spaces is that in terms of these distribution theory becomes a topic native to differential geometry: see at
distributions are the smooth linear functionals .

This is very natural in pAQFT, as it makes the entire theory exist in differential geometry, with no fundamental recourse to functional analysis (except for convenience.)

 

[A. Neumaier]

By the way, another use of diffeological spaces is that in terms of these distribution theory becomes a topic native to differential geometry: see at
distributions are the smooth linear functionals .

The link is blank.

This is very natural in pAQFT, as it makes the entire theory exist in differential geometry, with no fundamental recourse to functional analysis (except for convenience.)

But surely functional analysis must enter once you have to show that solutions to differential equations exist!
It is also needed for defining the spectrum of the Hamiltonian!

 

[A. Neumaier]

Yes. The only point in the development that requires a diffeological space (or more generally a smooth set or super smooth set) to actually be a smooth manifold is if we want to integrate differential forms over it. For spacetime this is what we want to do in order to define local observables, and therefore it is required to be a smooth manifold..

How then do you define partition functions, which require infinite-dimensional integration!?

 

[Urs Schreiber]

The link is blank.

Sorry, here: ncatlab.org/nlab/show/distributions+are+the+smooth+linear+functionals

But surely functional analysis must enter

Yes, that's why I said "except for convenience": You want the traditional tools to reason about distributions, but the concept of distribution as such does not come externally onto the differential geometry of the space of field histories, but is part of it.

The "microcausal local observables" out of which the Wick algebra and then the interacting field algebra are built want to be multilinear smooth functions on the diffeological space of field histories. Traditionally one ignores this and instead declares that they are given by distributions in the functional-analytic sense. But the statement is: it's the same! What looks like a functional-analytic definition of observables in fact is secretly the definition of smooth functionals on the diffeological space of field histories.

 

[Urs Schreiber]

How then do you define partition functions, which require infinite-dimensional integration!?

I suppose you are really thinking of taking the trace of a trace class operator?

 

[Urs Schreiber]

The "microcausal local observables" out of which the Wick algebra and then the interacting field algebra are built want to be multilinear smooth functions on the diffeological space of field histories. Traditionally one ignores this and instead declares that they are given by distributions in the functional-analytic sense. But the statement is: it's the same! What looks like a functional-analytic definition of observables in fact is secretly the definition of smooth functionals on the diffeological space of field histories.

I should add: To appreciate the usefulness, compare to the major trouble that Collini 16 has to go through with establishing the relevant smooth structure on observables (def. 15 and downwards).

 

[A. Neumaier]

I suppose you are really thinking of taking the trace of a trace class operator?

Actually limits of traces (e.g. over denser and denser lattice discretizations), and these limits are infinite-dimensional (i.e., functional) integrals.

 

[Urs Schreiber]

Actually limits of traces (e.g. over denser and denser lattice discretizations), and these limits are infinite-dimensional (i.e., functional) integrals.

Not sure what you want me to say. I won't be considering explicit path integrals. Maybe you could point to some concrete article and say something like: "How would you phrase that construction in terms of diffeological spaces!"?

 

[A. Neumaier]

Not sure what you want me to say. I won't be considering explicit path integrals. Maybe you could point to some concrete article and say something like: "How would you phrase that construction in terms of diffeological spaces!"?

You refer to the path integral in Remarks 15.4 and 16.2 of
https://ncatlab.org/nlab/show/geometry+of+physics+--+A+first+idea+of+quantum+field+theory
So these are considered only as loose heuristics, not with a diffeological interpretation?

 

[Urs Schreiber]

You refer to the path integral in Remarks 15.4 and 16.2 of
https://ncatlab.org/nlab/show/geometry+of+physics+--+A+first+idea+of+quantum+field+theory
So these are considered only as loose heuristics, not with a diffeological interpretation?

Absolutely. I don't consider explicit path integrals. These remarks are meant for the reader who will have been exposed to the usual informal path integral lore and are meant to explain how the axiomatic construction of the S-matrix and of the interacting observables in causal perturbation theory correspond to that informal lore.

I'll try to rephrase these remarks a little to make sure that this becomes clear.

 

[Urs Schreiber]

I'll try to rephrase these remarks a little to make sure that this becomes clear.

Okay, I have touched the wording of these two remarks:

• Intuitive interpretation of the perturbative S-matrix as a "path integral"
• Intuitive interpretation of Bogoliubov's formula in terms of a "path integral"

 

[A. Neumaier]

Okay, I have touched the wording of these two remarks:

• Intuitive interpretation of the perturbative S-matrix as a "path integral"
• Intuitive interpretation of Bogoliubov's formula in terms of a "path integral"

misprint: simimlarly

 

[Leo1233783]

With Greg we are looking for a solution now. A technically simple solution would be to simply include that webpage inside an "iframe" within the PF-Insights article. But maybe this won't be well received with the readership here? If anyone with experience in such matters has a suggestion, please let me know.

Did you found a solution ? if not, I can try. What or where are exactly a typical input and a typical output ?

 

[Urs Schreiber]

Did you find a solution ?

No, I didn't.

if not, I can try.

That would be great!

What or where are exactly a typical input and a typical output ?

Okay, my input source is here .

The output format that is needed for PF-Insights is mainly standard HTML, except for the maths formulas.

For the maths formulas

1. single dollar sign delimiters in my source need to be turned into double hash delimiters
2. backslash followed by "hookrightarrow" in my source needs to be turned into plain "rightarrow" (because the hookrightarrows come out as strange graphics otherwise - alternatively if you know how to generate a decent hookrightarrow here, that would be welcome).

For the text decoration:

1. underscore delimiter needs to be turned into the HTLML "em"-tag environment
2. star delimiter needs to be turned into the HTML "strong"-tag environment

For the hyperlinks:

1. In my source a string "some text" inside double square brackets wants to become a hyperlink to "https://ncatlab.org/nlab/show/some+text".
   
2. Alternatively I have single square brackets around "some text" followed by "page#anchor" in round brackets, and this wants to be turned into "some text" equipped with a hyperlink to "https://ncatlab.org/nlab/show/page#anchor".

Other markup I use could just be stripped off. For instance

1. I use environments for automatically numbered Definitions/Propositions/Proofs. Of course it would be great to automatically turn them into something saying "Definition" or "Proposition" etc. but to first approximation just discarding that environment code would work.
2. Similarly I have a handful of tables in the code, in some probably self-explanatory encoding. In principle these could be turned into HTML-tables, but to first approximation we could just discard them.

Please let me know if this is information that you can use. Otherwise I'll be happy to try to say more.

Thanks again for looking into this!

 

[Leo1233783]

Data received. A simple line transformer will be probably enough and extensible. Else I have other solutions. I'll come back to you in a private message :)

 

[Urs Schreiber]

To everyone:

The terrific Leo1233783 above (who seems to like to remain anonymous) has now provided tremendous help with converting my source to PF-Insights format. A million thanks for that!

This way we are finally set for the series on QFT to begin:

A First Idea of Quantum Field Theory – 20 Part Series

So far that page has just a few lead-in words and then a table of contents. In the next days and weeks, each chapter in that table of contents will appear as a separate PF-Insights article, and will be hyperlinked from that page.