Introduction to Perturbative Quantum Field Theory - Comments

[A. Neumaier]

for smooth sets which are not manifolds there appear different inequivalent concept of tangent spaces. But for applications to field theory, it turns out all one needs is differential forms on the spaces of field histories.

I don't believe this. Already for classical existence proofs one needs to extend the infinitesimal description in terms of differential equations to a global description in terms of histories, and they are related as Lie algebras and Lie groups. In this relation the notion of tangent space is essential.

To be more concrete, can you show me a paper where Lie groups are studied from a smooth set point of view, and the typical difficulties in infinite dimensions are absent?

 

[Urs Schreiber]

I don't believe this. Already for classical existence proofs one needs to extend the infinitesimal description in terms of differential equations to a global description in terms of histories, and they are related as Lie algebras and Lie groups. In this relation the notion of tangent space is essential.

But not the tangent space to the space of histories.

To be more concrete, can you show me a paper where Lie groups are studied from a smooth set point of view, and the typical difficulties in infinite dimensions are absent?

The term to look for is "diffeological groups". For instance here

 

[Urs Schreiber]

Maybe I should re-amplify the point about diffeology:

The concept of "smooth sets" subsumes that of diffeological spaces , and essentially all examples of relevance in field theory fall in the class of diffeological space.

The diffeological spaces are the "concrete smooth sets". So the concept of diffeological spaces is a generalization of that of smooth manifolds, and the concept of smooth sets is yet a further generalization of diffeological spaces.

As far as the formalism is concerned, it is no harder to work in the generality of smooth sets than it is to work in the intermediate generality of diffeological spaces. But essentially all examples of smooth sets that appear in the context of field theory are actually diffeological spaces, and so if you are looking for literature on the subject, you should look for the keywords "diffeological spaces".

In particular, to highlight this once more, there is a down-to-earth non-categorical completely introductory and detailed textbook introducing all the standard material of differential geometry in terms of diffeological spaces.This is

• Patrick Iglesias-Zemmour:
  Diffeology
  Mathematical Surveys and Monographs
  Volume: 185; AMS 2013;

I don't think that studying this is necessary for following my notes, since the basic idea is really simple and really close to how physicists think anyway, but to all readers who do want to dig deeper into this differential geometric background to the theory I recommend looking at this textbook.

 

[dextercioby]

Urs, can you write in no more than 2 lines the connection (if any) between diffeologic spaces and topological spaces?

 

[Urs Schreiber]

Urs, can you write in no more than 2 lines the connection (if any) between diffeologic spaces and topological spaces?

Good question.

1. A topological space is a set $X$ equipped with information which functions $\mathbb{R}^n \longrightarrow X$ are continuous.
2. A diffeological space is a set $X$ equipped with information which functions $\mathbb{R}^n \longrightarrow X$ are smooth.

A leisurely exposition of the grand idea behind this is at motivation for sheaves, cohomology and higher stacks.

 

[DrDu]

It is since the number of loops counts the powers of $hbar$. This is clear from the path-integral formalism since you can understand the Dyson series also as saddle-point approximation of the path integral. See, e.g., Sect. 4.6.6 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

Yes, but the free theory which forms the starting point of the perturbation expansion contains already quantized electrons, photons, etc. How comes we consider this to be a classical theory?

 

[A. Neumaier]

Yes, but the free theory which forms the starting point of the perturbation expansion contains already quantized electrons, photons, etc. How comes we consider this to be a classical theory?

In an $\hbar$ expansion, the theory is expanded around the interacting classical limit, not around a free quantum field theory.

 

[DrDu]

In an $\hbar$ expansion, the theroy is expanded around the interacting classical limit, not around a free quantum field theory.

I know what you mean but I don't see how this fits with my naive view of e.g. QED where, when describing the scattering of, say, an electron from a photon. I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.

 

[vanhees71]

The point is that you can get the Klein-Nishina formula for Compton scattering, i.e., in the modern way by just evaluating the tree-level Feynman diagrams, by investigating scattering of a classical electromagnetic wave on an electron using the Dirac equation of the electron. The same holds for the photoeffect. You come quite far with the semiclassical approximation in QED, i.e., quantizing only the electron and keep the em. field as classical. The most simple argument for the necessity to quantize also the em. field is the existence of spontaneous emission, which afaik cannot be derived from the semiclassical theory.

 

[DrDu]

But you need a quantized electron, or is it sufficient to use a classical Grassmann valued field for the electron?

 

[vanhees71]

It needs not even be Grassmann. The original paper dealt with the Dirac equation as if you could use it in the same way as the non-relativistic Schrödinger wave function.

Klein, O. & Nishina, Y. Z. Physik (1929) 52: 853. https://doi.org/10.1007/BF01366453

English translation

O. Klein and Y. Nishina, "On the Scattering of Radiation by Free Electrons According to Dirac's New Relativistic Quantum Dynamics", The Oskar Klein Memorial Lectures, Vol. 2: Lectures by Hans A. Bethe and Alan H. Guth with Translated Reprints by Oskar Klein, Ed. Gösta Ekspong, World Scientific Publishing Co. Pte. Ltd., Singapore, 1994, pp. 113–139.

 

[DrDu]

Wow, you never stop learning! So all this Feynman stuff in tree order is basically only first quantization?
So if I want count photons and electrons, I have to go beyond tree level. Can you show me how to see this?

 

[A. Neumaier]

I know what you mean but I don't see how this fits with my naive view of e.g. QED where, when describing the scattering of, say, an electron from a photon. I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.

The collection of all tree diagrams really describes perturbation theory of a classical field theory in terms of powers of the coupling constant (one power per vertex)! This shows that Feynman diagrams have nothing to do with particles, except as a suggestive way of talking!

 

[vanhees71]

One should also note that in relativistic QFT particle number is only well defined for asymptotic free states. That's why cross sections and related quantities are defined via the S-matrix which gives transition rates between asymptotic free states.

 

[Urs Schreiber]

An account that makes explicit how the tree level perturbation series is just the perturbation series for the classical field equations is in

• Robert Helling,
  Solving classical field equations
  (pdf)

 

[DrDu]

• Solving classical field equations
  (pdf)

Wonderful! Does not even require a master in category theory! :-)

 

[A. Neumaier]

I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.

See also https://physics.stackexchange.com/questions/348942

 

[Urs Schreiber]

Wonderful!

Glad you like it

Does not even require a master in category theory! :-)

That makes me sad. I carefully avoid all category theory in the notes and give down-to-earth discussion of everything. If you find any statement that requires mastery of category theory, I'll remove it.

 

[DrDu]

That makes me sad. I carefully avoid all category theory in the notes and give down-to-earth discussion of everything. If you find any statement that requires mastery of category theory, I'll remove it.

I wasn't referring to your explanations but rather to the other articles you were citing.

 

[A. Neumaier]

1. A topological space is a set $X$ equipped with information which functions $\mathbb{R}^n \longrightarrow X$ are continuous.
2. A diffeological space is a set $X$ equipped with information which functions $\mathbb{R}^n \longrightarrow X$ are smooth.

A leisurely exposition of the grand idea behind this is at motivation for sheaves, cohomology and higher stacks.

The link gives an error.

I think that giving in your proposed next insight article a category-free definition of a diffeological space such as that given in Wikipedia (rather than just saying ''information with the natural properties expected'') would be helpful, together with saying how a smooth set extends that notion.

There is also a useful book, Patrick Iglesias-Zemmour, Diffeology, 2013, and the author's diffeology blog.

 

[Urs Schreiber]

The link gives an error.

Only as of a few minutes back, sorry for that. Our admin is fiddling with the installation right now.

There is also a useful book, Patrick Iglesias-Zemmour, Diffeology, 2013

Yup, I have pointed that out before, last time in #63 .

 

[Urs Schreiber]

Our admin is fiddling with the installation right now.

He brought it back now.

I think that giving in your proposed next insight article a category-free definition of a diffeological space such as that given in Wikipedia (rather than just saying ''information with the natural properties expected'') would be helpful, together with saying how a smooth set extends that notion.

To repeat, there is a detailed pedestrian introduction at geometry of physics -- smooth sets . I am getting the impression you have not actually looked at that yet. If you get distracted by the top level "Idea"-section you should jump right to where the discussion starts at "Model layer" (meaning: the down-to-earth version of the stroy) which starts out with the inviting sentence:

In this Model Layer we discuss concretely the definition of smooth spaces and of homomorphisms between them, together with basic examples and properties.

 

[A. Neumaier]

To repeat, there is a detailed pedestrian introduction at geometry of physics -- smooth sets . I am getting the impression you have not actually looked at that yet. If you get distracted by the top level "Idea"-section you should jump right to where the discussion starts at "Model layer" (meaning: the down-to-earth version of the stroy) which starts out with the inviting sentence:

In this Model Layer we discuss concretely the definition of smooth spaces and of homomorphisms between them, together with basic examples and properties.

Well, I didn't know this page. In pAQFT 1: A first idea of quantum fields, you referred at the first mention to smooth sets, which is quite abstract. You should have referred instead to the page you just mentioned, and you should add your present comment there at the top.

In geometry of physics -- smooth sets, Definition 2.1 is still unmathematical and hence empty. It doesn't tell what sort of formal object a plot is, and it is not explained afterwards either. I guess you mean ''The elements of $X(R^n)$ are referred to as plots of $X$'? This should then be part of Definition 2.2.1.

In Definition 2.2.2 it is clearer to write ''for each smooth function $f$ (called in the present context an abstract coordinate transformation)'' in place of ''for each abstract coordinate transformation, hence for each smooth function $f$...'' and property 2.2.2 would read clearer if you wouldn't talk informally about change but only about composition. The informal interpretation (''to be thought of'') should not be part of the definition (which should be pure mathematics, introducing concepts, names, notation and properties) but a comment afterwards that adds intuition to the stuff introduced.

''But there is one more consistency condition'' - Is this still part of the definition, or is this a preamble to the definition of a smooth space in Definition 2.6?

And at that point (or later) I still don't know what a smooth set is! Is it just another word for a smooth space? Then why have two very similar names for it?

Nowhere the connection is made to diffeological spaces and to manifolds (except in a introductory sentence superficially justified very late in Remark 2.29, which is again quite abstract and does not make the connection transparent). But these should be the prime examples and hence figure prominently directly after Definition 2.2, to connect the general abstract concept to traditional objects more likely to be familiar to the reader. The example of the irrational torus as a diffeological space which is not a manifold would be instructive.

 

[Urs Schreiber]

Well, I didn't know this page ,

Thanks for the feedback. I have edited a little in reaction, bringing up the examples further up front.

And I fixed the terminology overloading: "smooth space" is synonymous to "smooth set" (At some point I thought it would better match with "diffeological space", but then changed my mind). I made it read "smooth set" throughout now. Thanks for catching this.

 

[David Neves]

What do you think of the following paper about QED?

https://journals.aps.org/prd/pdf/10.1103/PhysRevD.96.085002

Here is the ansatz.

Infrared divergences in QED revisited
Daniel Kapec, Malcolm Perry, Ana-Maria Raclariu, and Andrew Strominger
Phys. Rev. D 96, 085002 (2017) – Published 10 October 2017

It has been found recently that the vacuum state of quantum electrodynamics (QED) is infinitely degenerate. The authors exploit this fact and show that any non-trivial scattering process in QED is necessarily accompanied by a transition among the degenerate vacua, making the scattering amplitude finite at low energy scales (infrared finite).

Recently, it has been shown that the vacuum state in QED is infinitely degenerate. Moreover, a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here, we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of Faddeev and Kulish.

Also, I am hoping that towards the end of your series that you will also talk about conformal field theory (CFT).

 

[A. Neumaier]

Thanks for the feedback. I have edited a little in reaction, bringing up the examples further up front.

And I fixed the terminology overloading: "smooth space" is synonymous to "smooth set" (At some point I thought it would better match with "diffeological space", but then changed my mind). I made it read "smooth set" throughout now. Thanks for catching this.

Thanks. A misprint: diffetrential

 

[A. Neumaier]

What do you think of the following paper about QED?

You should open a new thread fro discussing this!

 

[Urs Schreiber]

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...

 

[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...

Thanks!

In the yet missing discussion of QED, you might also want to discuss the Lamb shift. Then you'll see that the perturbative approach (algebraic or not) is still severely deficient in the infrared and cures nothing...

 

[vanhees71]

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