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I am starting an entry locally covariant perturbative quantum field theory.
[edit: renamed to perturbative algebraic quantum field theory]
So far it contains just an Idea-section and some references to go with it. The same idea I also added as a pointer to the entry quantum field theory.
By this winter I hope to expand the entry to contain a detailed introduction.
In the course of writing that Idea-section, I also created a stub entry causal perturbation theory, and a References-entry The Role of locality in perturbation theory.
I am in the process of organizing the material which I am planning to write out in detail on the $n$Lab. After some discussion behind the scenes, I believe I have now extracted the nice global picture. The key insight, which turned what was effectively an ad-hoc recipe (albeit highly successful in applications) into a conceptual theory proceeding from first principles is surprisingly recent, it dates from last year:
Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)
Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)
These authors finally showed that the method of constructing perturbative interacting quantum field theories via “Bogoliubov’s formula” applied to the “causal S-matrix” constructed via “time-ordered fields” is actually nothing but the formal algebraic deformation quantization of the interacting Lagrangian density. This is exactly what one would hope it should be given that it is called “quantization”. But this was in fact open until last year, remarkably.
This has some neat conceptual consequences. For instance it shows that the freedom in normalizing (“re-normalizing”) the field theory is nothing but the ambiguity in choosing a deformation quantization (in contrast maybe to the popular picture of “counter-terms” which rather suggests that it is a freedom in specifying the classical Lagrangian). This could become interesting if one ever understands how to pass from the perturbative formal deformation quantization to a more non-perturbative formulation, since that may be expected to constrain the quantization ambiguities.
In any case, before I start implementing on the $n$Lab the large amount of technical detail that is involved, I thought I needed to give a global picture outline of what’s going on, not to get lost. I have now tried my hands on such a section
So it’s intentional that this has a few gray links. I’ll create entries for these soon.
So the idea is that formal power series star-algebras of observables are being used instead of C*-algebras because the latter are too difficult to find?
not a single relevant example (interacting QFT in spacetime dimensions 4 or greater) is known
So there are some/many relevant examples existing for power series star algebras in spacetime dimensions 4 or greater?
But even with success here, at some stage the full C*-algebra version will be needed?
while a non-perturbative quantization would be required to yield a C*-algebra of quantum observables, or something similar
And then this greater demand will persist even when making the move to homotopical algebras?
So the idea is that formal power series star-algebras of observables are being used instead of C*-algebras because the latter are too difficult to find?
It’s right that these formal power series are all that are in reach of mathematical technology at the moment. So yes, in this sense it is right that we care about them because the full thing is too difficult. But that’s not how the story started: Remarkably, people started caring about these power series because they happened to match what is seen in experiment to high precision!
The formal power series algebra is to be thought of as the infinitesimal neighbourhood around 1 in the $C^\ast$-algebra that is the real answer, hence as its “infinitesimal approximation”. (For that to make strict mathematical sense the $C^\ast$-algebra needs some auxiliary smooth information, maybe the way one sees it in spectral triples, where a sub-algebra of the $C^\ast$-algebra is specified which behaves like an algebra of smooth functions on a smooth manifold. )
That sounds like an immensely coarse approximation to the real thing. And it is. A remarkable fact about nature is that this immensely coarse approximation to reality already explains/predicts almost all phenomena seen in high energy physics, to high numerical precision. There is no a priori reason why we should be that lucky, but we are. In jargon that some people like it means that “the vacuum that we inhabit is very weakly coupled”. It’s an amazing fact of nature.
not a single relevant example (interacting QFT in spacetime dimensions 4 or greater) is known
So there are some/many relevant examples existing for power series star algebras in spacetime dimensions 4 or greater?
Yes, everything that the traditional QFT textbooks and research articles consider is subsumed. The traditional name of these power series is “Feynman perturbation series”, a formal sum of contributions labeled by Feynman diagrams.
But even with success here, at some stage the full C*-algebra version will be needed?
The full $C^\ast$-algebras ought to describe the full QFT, without approximations made, i.e. the “non-perturbative QFT”.
Notice that figuring out just one small aspect of just one type of non-perturbative QFT is among the list of “Millenium Problems” of the Clay Institute: This is the open problem of constructing Yang-Mills theory as a full non-perturbative QFT, or at least enough of it to be able to prove that it has a “mass gap” (which is believed to be a purely non-perturbative effect).
while a non-perturbative quantization would be required to yield a C*-algebra of quantum observables, or something similar
And then this greater demand will persist even when making the move to homotopical algebras?
Yes, the move to homotopical algebras is to lift yet another overly strict approximation: Namely on top of being perturbative in Planck’s constant and in the coupling constant, standard perturbative QFT also disregards the global “topological” nature of gauge fields. Homotopical AQFT aims to capture these.
Which points to a wide open problem: Combining non-perturbation in coupling and Planck’s constant with admitting non-trivial global gauge sectors seems to demand something like homotopical $C^\ast$-algebras. While there are some evident guesses what these could be, it is completely unclear what the right answer is.
This needs much more thinking. As you will know, I think that there is a good chance that the right answer is not phrased in terms of algebras at all, but in something more general which may be presented by algebras is nice cases. But maybe the best way to make progress here is to study the known theory phrased in terms of algebras is more detail.
Thanks, as ever!
I am presently preparing an article titled
pAQFT Part I. Introduction and References
for a series on pAQFT for PhysicsForums – Insights.
I need to polish and finalize this tomorrow. But in case you like to take a pre-look (and I’d be grateful for your comments!) I have pasted my PFI source into the Sandbox for the moment. That loses the PFI formatting, of course, but maybe it’s still readable.
Some typos:
comological; folk lore (one word); the theory due (missing next word ’to’); such a reasoning (delete ’a’); these are the known (delete ’the’); qantum
Also
exist as mathematically well-defined concepts: geometric quantization and algebraic deformation quantization. Remarkably, pAQFT does follow as a special case of formal deformation quantization
better to use ’algebraic’ again than introduce a new word ’formal’.
I imagine the article will be heavily hyperlinked to the nLab. That should make the trickier passages accessible (e.g., for me, around “adiabatic switching”).
Regarding
Presently not a single example of an interacting non-perturbative Lagrangian quantum field theory is understood in spacetime dimension ##\geq 4## (besides numerical simulation).
does that mean the same as the claim I quoted in #3,
not a single relevant example (interacting QFT in spacetime dimensions 4 or greater) is known?
’Understood’ is more demanding than ’known’ (at least ’known of’). What is there numerical simulation of, if not one of these theories?
Thanks! Implementing this now.
I imagine the article will be heavily hyperlinked to the nLab. That should make the trickier passages accessible (e.g., for me, around “adiabatic switching”).
I’ll have hyperlinks, yes, but if it sounds too cryptical I should modify it.
What is there numerical simulation of, if not one of these theories?
Right, this needs to be phrased carefully.
These simulations, called lattice gauge theory, discretize Minkowski spacetime and then simply evaluated the Euclidean, i.e. Wick rotated path integral approximately as a big numerical sum. This gives results that are non-perturbative in the coupling constant but still approximate due to the finite lattice spacing and numerical error.
Despite the word “theory” in “lattice gauge theory”, this is more like computer-simulated experiment. It is due to lattice gauge theory that people are pretty sure that non-perturbative QCD does have a mass gap, because one does see its consequences in the simulation. But since this is just numerical computer output, this provides no real understanding.
Since I came across mass gap with an empty link, I created it and copied over what was relevant at quantization of Yang-Mills theory. Is it still fair to say there has been essentially no progress?
Since I came across mass gap with an empty link, I created it and copied over what was relevant at quantization of Yang-Mills theory.
Thanks. Hopefully an expert sees this and is sufficiently bothered by the stubbiness of the entry to provide some help.
Is it still fair to say there has been essentially no progress?
I believe so, from what I hear (or don’t hear). But it would be good to see what an expert on this matter has to say, as there is a huge literature with a variety of partial attempts and partial aspects, which I am not an expert on.
The article is now publically available on PF-Insights, here.
That book you refer to – Quantum Gauge Theories – A True Ghost Story – is apparently the first edition of a book renamed as
Gauge Field Theories: Spin One and Spin Two: 100 Years After General Relativity, 2016.
The second edition had yet another name.
Thanks for the pointer, I had not been aware of this renaming. We should make a note in the entry here
I made a note. The main difference seems to be a new chapter, 6. Non-geometric general relativity.
- Non-geometric general relativity.
Interesting, I should take a look. What is this about?
Perhaps you can see pp. 247-8, here. The metric tensor is taken as just another field on Minkowski space. Plenty of fun comments there.
Now I got hold of a copy.
By “non-geometric” gravity he just means the school of thought which says that one should ignore the geometric interpretation of GR. As long as we restrict attention to trival spacetime topology, this just comes down to a matter of choosing words, either way, and so, while I am not enthusiastic about it, I can live with it. Moreover, as that section recalls, various very famous physicists did go down this road, so we are silenced by appeal to authority, even if we have the advantage of hindsight, with new insights like AdS/CFT making the “it’s not really geometry”-picture increasingly awkward.
What then actually happens in the chapter is an analysis of isotropic but in-homogeneous solutions to the Einstein equations (as opposed to isotropic and homogeneous, as for the usual FRW models). Many theoretical physicists these days like to chat about such “pocket universes” when they discuss the “multiverse”, but that chapter 6 (even though written 6 years back) is the first time that I see this kind of idea actually worked out in detail in formulas (but that me be just me not following the GR literature closely enough).
Interesting to see the ambition of the big picture of the whole book here: At the beginning it starts with pQFT, then derives the Einstein equations from perturbative quantum gravity, and thus the book ends with classical gravity, reversing the traditional order. Here the classical theory is truly regarded as an output of the more fundamental quantum theory.
It’s rare these days to see independent thinkers go for the big picture in physics while supplying all their arguments with detailed computation. Various famous big-picture seekers these days have turned to informal story-telling instead (e.g Susskind once more in his latest arXiv:1708.03040)
With all of your recent activity, I was trying to get some sense of the range of approaches to AQFT. Would it be straightforward to say what marks the difference between the approaches mentioned above (causal-covariant-perturbative) and the factorization algebra approach?
At factorization algebra of observables, it says
This formalization of algebraic quantum field theory is similar to, but a bit different from, the notion of local net of observables.
At factorization algebra, it says
There seems to be a close relation between the description of quantum field theory by factorization algebras and the proposal presented in
Hollands is someone you associate at perturbative algebraic quantum field theory with the causal approach. In Quantum fields in curved spacetime he seems to think that his operator-product expansions approach will work for curved spacetimes. But then
In summary, and as discussed further in [57], we believe that OPEs provide a promising approach toward the formulation of QFTCS that is independent of, and different from, perturbative methods.
Back to factorization algebras, where you just claimed
While there are other equivalent rigorous formulations of pQFT on Minkowski spacetime, causal perturbation theory is singled out as being the one that generalizes well to QFT on curved spacetimes (Brunetti-Fredenhagen 99), hence to quantum field theory in the presence of a background field of gravity,
does this suggest factorizations algebras won’t generalise well to curved spacetimes, or that they’re similar enough to the causal approach?
The concept of factorization algebra is modeled, via Beilinson-Drinfeld’s perspective of Chiral Algebras, on the concept of vertex operator algebras (VOAs) in their guise as algebras over the holomorphic punctured sphere operad (due to Huang here). VOAs, by design, express the operator product expansion of (2-dimensional) Euclidean (as opposed to Lorentzian) conformal field theories.
In contrast, in causal perturbation theory (as the name indicates) and in (perturbative) AQFT the Lorentzian causal structure is at the heart of the theory (ever since the Haag-Kastler axioms from the 60s). On Minkowski spacetime the Osterwalder-Schrader theorem makes precise the Wick rotation that translates between the Lorentzian and the Euclidean picture, but this fails on more general spacetimes.
(Even then, factorization/vertex operator algebras are not quite the same as Wick rotated nets of algebras of quantum observables, since the latter assigns algebras to spacetime regions, while the former assigns vector spaces/chain complexes which inherit algebraic operation only from the operadic inclusion of spacetime regions.)
Hollands, after having pioneered the development of rigorous perturbative QFT (via causal perturbation theory) on curved spacetimes (construction of renormalized Yang-Mills theory on curved spacetimes) went on to try to make progress on the non-perturbative theory but turning attention to the operator product expansion. I am not an expert on this, but my impression is that this program got stuck at some point, I’d have to check.
I should expand on this comment I made:
vertex operator algebras are not quite the same as Wick rotated nets of algebras of quantum observables, since the latter assigns algebras to spacetime regions, while the former assigns vector spaces/chain complexes which inherit algebraic operation only from the operadic inclusion of spacetime regions.
But the relation between the two had been vaguely known for a long time, and was finally turned into a theorem in Carpi-Kawahigahshi-Longo-Weiner 15
Thanks. That sounds quite like quite a limitation, and Hollands agrees:
Euclidean methods can be generalized so as to apply to static, curved spacetimes, where the transformation “$t \to i t$” takes one from a static Lorentzian spacetime to a Riemannian space. However, a general curved spacetime will not be a real section of a complex manifold that also contains a real section on which the metric is Riemannian. Thus, although it should be possible to define “Euclidean quantum field theory” on curved Riemannian spaces [65], there is no obvious way to connect such a theory with quantum field theory on Lorentzian spacetimes. Thus, if one’s goal is to define quantum field theory on general Lorentzian spacetimes, it does not appear fruitful to attempt to formulate the theory via a Euclidean approach. (Quantum fields in curved spacetime)
I imagine that weaving in the Lorentzian structure as emerging from the extension of the superpoint is still a distant prospect.
That sounds quite like quite a limitation,
It is, if one is interested in QFT as practiced in physics that is eventually concerned with nature. These days, with mathematical axiomatization of QFT attracting more attention, many concepts termed “field theory” are considered that are only vaguely related to field theory in the original sense.
I imagine that weaving in the Lorentzian structure as emerging from the extension of the superpoint is still a distant prospect.
It does however seem noteworthy that the one key structure on top of the quantum principle which governs QFT, namely Lorentzian causality, does emerge from the superpoint. Indeed, it is noteworthy that what is really needed for causal perturbation theory are not just Lorentzian spacetimes, but time-oriented Lorentzian spacetimes, and that the super-Minkowski spacetimes which emerge from the superpoint are canonically not just Lorentzian, but also time-oriented: The super-Lie bracket pairing from spinors to vectors happens to land in just one of the two causal cones, and hence it breaks time-reflection symmetry and picks a time-orientation.
Presently I don’t know yet in detail how to connect the emergence of causality from the superpoint with the quantum principle emerging from dependent linear homotopy types in order to obtain Lorentzian QFT from first principles – but on a rough conceptual level the required ingredients do precisely match.
In case it’s of interest, I see Kasia Rejzner has constructed a table, a “dictionary”, comparing side by side the Fredenhagen-Rejzner and the Costello-Gwilliam approaches on pp. 18-19 of this OWR report.
Thanks for the pointer, this is a useful table. I am just a bit puzzled that I don’t recall that Kasia was talking about this in her talk at that Oberwolfach meeting.
(But then, I remember I was distracted. During the previous talks on “BV-algebras” I got a little irritated that they glossed over the subtlety in that term, and it may have been during Kasia’s talk that I started typing the entry relation between BV and BD.)
Morning speculation: re #22, as we emerge from the superpoint, how much of the dynamics present at a stage is inherited from the previous stage?
If
Spacetime itself emerges from the abstract dynamics of 0-branes propagating on the super-point,
is it that any dynamics of branes propagating in spacetime is determined solely by higher extensions, or does its emergence from the super-point place constraints here?
Perhaps
We observe that the super-$L_\infty$-algebra extension of super-Minkowski space which is classified by a given cocycle is that space such that maps into it are plain maps to spacetime together with the field that is “sourced” by the boundaries of the open $p$-branes whose dynamics is encoded by that cocycle
may be read as supporting the former.
Are there choices as to the “abstract dynamics of 0-branes propagating on the super-point” which could ripple up through the extensions, or does the process just generate a series of superspaces?
There is a textbook in preparation on pAQFT:
It’s not published yet, but I already gave it an entry in category:reference in order to be able to cite it now, the publication details to be included later.
In the course of this I also gave the only other existing textbook on pAQFT its own category:reference entry:
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