Introduction to Perturbative Quantum Field Theory - Comments

[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?

 

[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