Insights Introduction to Perturbative Quantum Field Theory - Comments

[Urs Schreiber]

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".

 

[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.

 

[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.

 

[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

• pAQFT 1: A first idea of quantum fields

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

• pAQFT 1: A first idea of quantum fields

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''.

 

[Urs Schreiber]

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.

 

[Urs Schreiber]

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

Thanks, fixed now.

 

[Urs Schreiber]

The title is somewhat misleading.

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.

 

[Urs Schreiber]

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?

 

[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?

 

[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.