# Introduction to Perturbative Quantum Field Theory - Comments

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Gold Member
But do you get the same strong existence results as one gets for infinite-dimensional Lie groups, say?

The statement about the full embedding that I quoted means precisely that all standard theory embeds.

What I meant is: To understand the concept of a smooth set you apparently need the whole category.

Not in any non-trivial sense, no. A smooth sets is, as I mentioned, a declaration for each Cartesian space (abstract coordinate system) of what counts as a smooth function out of that space into the smooth set, subject to the evident condition that this is compatible with smooth functions between Cartesian space (abstract coordinate changes).

That''s it.

Or can you define a smooth set without reference to a category, just as you can define a group without reference to a category?

Sure. But this is a triviality unless you read some superficial scariness into the innocent word "category". When you define what a group is, you also want to know what counts as a homomorphism between two groups. The high-brow term for this is "the category of groups" but it means nothing else but "groups and maps between them".

Leo1233783 and dextercioby
A. Neumaier
A smooth sets is, as I mentioned, a declaration for each Cartesian space (abstract coordinate system) of what counts as a smooth function out of that space into the smooth set, subject to the evident condition that this is compatible with smooth functions between Cartesian space (abstract coordinate changes).
Well, I'd like to have a mathematically precise specification. Can I replace Cartesian space by ##R^n##? Are there other significantly different Cartesian spaces that need to be catered for? Can one use fields like the p-adic numbers in place of the reals? (Some people are interested in p-adic physics!) Do the smooth functions have to be defined on all of ##R^n## or only on open subsets? What is the precise compatibility condition?

I wonder whether after all these things have been spelled out, the definition is really simpler than that of a manifold over a convenient vector space (in the sense of Kriegl and Michor), say.

I'd also like to be able to decide on the basis of the information provided questions such as whether any algebraic variety is a smooth set in an appropriate sense, or what restrictions apply.
When you define what a group is, you also want to know what counts as a homomorphism between two groups. The high-brow term for this is "the category of groups" but it means nothing else but "groups and maps between them".
But one can fist define what a group is, and then define what a homomorphism between groups is, and one does not the second concept to understand the first and to play with examples. I'd like to have a definition of smooth sets phrased in the same spirit. The categorial interpretation should be a second step that allows one to make certain universal constructions available, and not something already integrated into the definition.

dextercioby
A. Neumaier
Another quite popular non-perturbative approach is the renormalization-group approach ("Wetterich equation").

I guess, these more or less handwaving methods are not subject to the mathematically more rigorous approach discussed here, or can the here discussed approaches like pAQFT provide deeper insight to understand, why such methods are sometimes amazingly successful?
I believe that the Wetterich equation can be described on a reasonably rigorous level, though still with some uncontrolled approximations. But I haven't seen any concrete work in this direction by mathematical physicists.

vanhees71
A. Neumaier
By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here
In Section 2 of your nLab draft, you are unnecessarily onomatopoetic: ''expressions with repeated indicices''

A. Neumaier
The statement about the full embedding that I quoted means precisely that all standard theory embeds.
Apparently it means only that the standard examples are examples of the generalized concepts, not that the standard theorems also hold for the generalized concepts. I was asking for the latter. To get strong results for infinite-dimensional Lie groups, the latter probably need a full manifold structure, and not only a smooth set structure.

A. Neumaier
By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here
The title is somewhat misleading: 98% of the text is about classical field theory and only a few paragraphs at the end hint at quantum field theory through a sequence of remarks, without giving significant substance or interpretation. More appropriate would be something like ''The classical background needed for quantum field theory''.

vanhees71 and dextercioby
Gold Member
Well, I'd like to have a mathematically precise specification.

A detailed introduction is here: geometry of physics -- smooth sets . The quick way to state the definition is to say that a smooth set is a sheaf on the site whose objects are Cartesian spaces, whose morphisms are smooth functions between them, and whoe Grothendieck pre-topology is that coming from good open covers. But the introduction at geometry of physics -- smooth sets spells this out in elementary terms, not assuming any sheaf-theoretic background (or any other background except the concept of smooth functions between Cartesian spaces).

Can I replace Cartesian space by ##R^n##?

Here "Cartesian space" means precisely : ##\mathbb{R}^n##s.

Can one use fields like the p-adic numbers in place of the reals? (Some people are interested in p-adic physics!)

The analous definition work for any choice of test spaces with a concept of covering defined. If you take something like affinoid domains as in rigid analytic geometry you get somethng that deserves to be called "p-adic analytic sets" or the like. More relevant for physics is for instance the Choice of Stein spaces, in order to get "complex analytic sets". If you take affine schemes, you get ordinary algebraic spaces (among which ordinary schemes).

Do the smooth functions have to be defined on all of ##R^n## or only on open subsets? What is the precise compatibility condition?

One may equivalently take the site of open subsets of Cartesian spaces. Some authors do that. It does't change the resulting concept, though. The compatibility condition is gluing: The choice of what counts as a smooth function into your smooth set must be so that if you cover one Cartesian space by a set of other Cartesian spaces, then the smooth functions out of the former must be uniquely fixed by their restriction to those patches of the cover.

I'd also like to be able to decide on the basis of the information provided questions such as whether any algebraic variety is a smooth set in an appropriate sense, or what restrictions apply.

There is once you decide on what should count as a smooth function from a Cartesian space to the algebraic variety. In general there will not be a useful such choice, but if your algebraic variety happens to be complex-analytic, then of course there is, and you recover the underlying smooth manifold.

But one can fist define what a group is, and then define what a homomorphism between groups is, and one does not the second concept to understand the first and to play with examples.

Same for smooth sets. To recall, a smooth set is defined to be a choice, for each ##n \in \mathbb{R}^n## of a set, regarded as the set of smooth functions from ##\mathbb{R}^n## to the smooth set (called "plots"), such that this choice is compatible with smooth functions ##\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}## and respects gluing, as above.

That's the definition. Next, a homomorphism between smooth sets is a map that takes these plots to each other, again respectiing the evident compositions.

Gold Member
In Section 2 of your nLab draft, you are unnecessarily onomatopoetic: ''expressions with repeated indicices''

Thanks, fixed now.

Gold Member
It keeps growing. In a few weeks the quantization will be covered, please have a little patience.

You see, this is my source from which I was going to successively produce Insights-articles here, one at a time. You are only seeing my master source only because the conversion to Insights-articles is running into technical difficulties.

Gold Member
Apparently it means only that the standard examples are examples of the generalized concepts, not that the standard theorems also hold for the generalized concepts. I was asking for the latter. To get strong results for infinite-dimensional Lie groups, the latter probably need a full manifold structure, and not only a smooth set structure.

One needs full manifold structure for surprisingly few things. Everything that involves only differential forms instead of vector fields generalizes to all smooth sets. If vector fields get involved one needs to be careful, as 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.

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?

Gold Member
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

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

Urs Schreiber
Gold Member
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.

vanhees71
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.

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vanhees71
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
Gold Member
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
Gold Member
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
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