Insights Introduction to Perturbative Quantum Field Theory - Comments

[Urs Schreiber]

Urs Schreiber submitted a new PF Insights post

Introduction to Perturbative Quantum Field Theory

Continue reading the Original PF Insights Post.

 

[protonsarecool]

As someone who has only learned QFT via the path integral approach so far (and mostly with applications in condensed matter theory in mind), all this is crazily interesting, and i look forward to the rest of this series!

 

[vanhees71]

That's a great article, I've to study in closer detail later. I only wonder, why you only quoate QED, QCD, and quantum gravity but not the full Standard Model, including weak interactions, i.e., quantum flavor dynamics (aka Glashow-Salam-Weinberg model of the electroweak interaction) $\otimes$ QCD.

 

[dextercioby]

Great work (to come), Urs!
Well, once you're done writing it (all articles), I will go print it and store it in my physical library.

 

[Urs Schreiber]

I only wonder, why you only quote QED, QCD, and quantum gravity but not the full Standard Model, including weak interactions, i.e., quantum flavor dynamics (aka Glashow-Salam-Weinberg model of the electroweak interaction) $\otimes$ QCD.

True, I should mention electroweak theory, too, have edited the entry a little to reflect this. (It will take a bit until I get to these applications, I will first consider laying some groundwork.)

 

[Greg Bernhardt]

Great job Urs, looking forward to the next one!

 

[Haelfix]

Nice article. How much of causal perturbation theory has been shown to match SM physics/ordinary qft? When I looked at this (admittedly many years ago), people had successfully constructed the scalar field and there was work being done on spin 1/2, and some sketchy and complicated proof of concepts, but has it really been shown to be completely isomorphic? One step further, do all the successes of causal perturbation theory match on to the new covariant algebraic qft?

My impressions at the time was that this was a little bit like the theory of distributions vs Dirac's delta function formalism. The former is rigorous and nice, but just clutters up notation when you sit down to calculate things.

 

[Urs Schreiber]

How much of causal perturbation theory has been shown to match SM physics/ordinary qft?

It's true that the literature on this topic is still comparatively small, but everything comes out:

Scharf's two books cover much standard basic material of QED, EW, QCD and pQG.

Feynman diagrammatics and dimensional regularization was realized in in Keller 10, Dütsch-Fredenhagen-Keller-Rejzner 14. (These authors speak in terms of scalar fields, but, as with Epstein-Glaser's original article, this is a notational convenience, the generalization is immediate.)

BV-BRST methods were realized in Fredenhagen-Rejzner 11b.

One step further, do all the successes of causal perturbation theory match on to the new covariant algebraic qft?

Yes, that starts with Brunetti-Fredenhagen 00, Hollands-Wald 01 and culminates in the construction of renormalized Yang-Mills on curved spacetimes in Hollands 07.

My impressions at the time was that this was a little bit like the theory of distributions vs Dirac's delta function formalism. The former is rigorous and nice, but just clutters up notation when you sit down to calculate things.

Sure, once the dust of the theory has settled we want to compute leisurely, but we do want to understand what it is our computations are doing. Distribution theory is a good example for how it pays to spend a moment on sorting out the theoretical underpinning before doing computation. Causal perturbation theory shows that all that used to be mysterious about divergencies in pQFT is clarified by microlocal analysis of distributions: Properly treating the product of distributions with attention to their wave front set is what defines the normal-order product of free fields, and then properly treating the extension of distributions to coinciding interaction points is what defines the renormalized time-ordered products. That gives a solid background explaining what's actually going on in the theory. Not every kind of computation will be affected by this, but given that there remain open theoretical questions in pQFT, it will help to have the foundations sorted out.

 

[A. Neumaier]

Note that there is already an insight article on [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] complementing the present one.

Your point 2 for causal perturabtion theory (finitely many free constants) holds of course only for renormalizable theories. Scharf also constructs (low order) perturbative gravity in the causal framework, but there the number of free constants proliferates with the order. (Mathematically, this is not a problem since the same happens for multivariate power series, but physicists used to think of this as non-renormalizability.)

 

[A. Neumaier]

You stated in the article, ''perturbative AQFT in addition elegantly deals with the would-be “IR-divergencies” in pQFT by organizing the system of spacetime localized quantum observables into a local net of observables.''

I don't agree. The infrared problem remains unsolved in perturbative AQFT. You haven't even given a link to a reference where your claim would be addressed.

 

[A. Neumaier]

Could you please make a printable version of your slides https://ncatlab.org/schreiber/files/SchreiberTrento14.pdf, with the repetitions removed? (This is just an additional line in the latex before compilation.)

The link (web) to Schenkel in the nlab article https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields
is not working.

 

[bhobba]

As I said elsewhere, stunning, simply stunning and I have never said that about an insights article before.

I look forward to the whole series.

Note to those that like me do not understand all the mathematical detail. Of course try to correct that, but still read it and try to get a feel for the issues at the frontiers of current physics.

Feel free if interested to start a thread on, what for example, an instanton sea is, don't know that one myself - much food for thought here.

Thanks
Bill

 

[atyy]

How are pQFT and pAQFT related to lattice gauge theory? Would you accept lattice gauge theory, at least Hamiltonian lattice gauge theory, as a non-perturbative and physically relevant version of QED?

Also, why do you say the path integral doesn't exist? At least in 2D and 3D, doesn't constructive field theory, which you mention at the end, show that the path integral exists?

 

[Urs Schreiber]

Sorry for the slow replies, I am seeing the further comments only now for some reason.

Note that there is already an insight article on [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] complementing the present one.

Right, sorry, I should have pointed to that. I do have pointers to your FAQ on the nLab here .

Your point 2 for causal perturabtion theory (finitely many free constants) holds of course only for renormalizable theories.

I was careful to write "at each order there is a finite-dimensional space of choices" (emphasis added).

As you hint at, this is an important subtlety that is usually glossed over in public discussion: At each order of perturbation theory, there is a finite dimensional space of counter-terms to be fixed. As the order increases, the total number of counterterms may grow without bound, and then people say the theory is "non-renormalizable". But this is misleading terminology: The theory is still renormalizable in the sense that one may choose all counterterms consistently, even f there are infinitely many. What the traditional use of "non-renormalizable" really means to convey is some idea of predictivity: The usual argument is that if there is an infinite number of terms to be chosen, then the theory is not predictive. Of course a moment of reflection shows that it is not quite that black-and-white. The true answer is popular under the term "effective field theory": If we specify the counterterms up to a fixed oder (and there are only finitely many of these for any order) then the remaining observables of the theory are its predictions up to that order . As more fine-grained experimental input comes in, we can possibly determine counterterms to the next order by experiment, and then again the remaining observables of the theory are its predictions up to that next higher order. And so ever on.

 

[Urs Schreiber]

You stated in the article, ''perturbative AQFT in addition elegantly deals with the would-be “IR-divergencies” in pQFT by organizing the system of spacetime localized quantum observables into a local net of observables.''

I don't agree. The infrared problem remains unsolved in perturbative AQFT. You haven't even given a link to a reference where your claim would be addressed.

I believe I did provide a pointer, to the section here , but I could have emphasized this further. This will be the topic of the next (or next to next) installment.

What you are referring to, and what remains unsolved in generality, is taking the adiabatic limit of the coupling constant. But the insight of pAQFT is that this limit need not even be taken in order to obtain a well defined (perturbative) quantum field theory!

Namely the observation is that

1. the algebra of quantum observables localized in any spacetime region may be computed, up to canonical isomorphism, already with any adiabatic switching function that is constant on a neighbourhood of that region of support
   
2. as the region of support varies arbitrarily, the system of algebras of localized quantum observables obtained this way do form a causally local net in the sense of the Haag-Kastler axioms (this prop, the only difference to the original axioms being that here they are formal power series algebras instead of C-star algebra, reflecting the perturbation theory)
3. AQFT lore implies that this causally local net of observables is sufficient to fully define the quantum field theory.
   
4. ibut f desired, we may still take the limit now, not of the S-matrix, but of the local net of observables it induces, in the sense of limits over the functor assigning observable algebras. In the pAQFT literature they call this the "algebraic adiabatic limit" or similar. It may be used to construct operator representations of the quantum observables, but the main point of pAQFT is really that by and large it is not actually necessary to consider this.

 

[Urs Schreiber]

Could you please make a printable version of your slides https://ncatlab.org/schreiber/files/SchreiberTrento14.pdf, with the repetitions removed? (This is just an additional line in the latex before compilation.)

Ah, I didn't code this with the "beamer" package, but "by hand". Is there a tool that could extract from the pdf just those pages that have the screen completed, and put these together to a smaller file? Sorry for the trouble

The link (web) to Schenkel in the nlab article https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields is not working.

Thanks for the alert! I have fixed it now. The working link is here:

• Alexander Schenkel: "On the problem of gauge theories in locally covariant AQFT" (ncatlab.org/nlab/files/SchenkelTrento2014.pdf)

I recommend also Alexander's more recent exposition:

• Alexander Schenkel, "Towards homotopical AQFT" (web , pdf)[/QUOTE]

 

[Urs Schreiber]

How are pQFT and pAQFT related to lattice gauge theory? Would you accept lattice gauge theory, at least Hamiltonian lattice gauge theory, as a non-perturbative and physically relevant version of QED?

Sure, I was briefly referring to this in the paragraph starting with "Hence we will eventually need to understand non-perturbative quantum field theory."

I suppose the point is that Monte-Carlo evaluation of lattice gauge theory is more like computer--simulated experiment than like theory. It allows us to "see" various effects, such as confinement, but it still does not "explain" them in the sense that we could derive these effects structurally.

Another problem is that lattice gauge theory relies on Wick rotation, so it does not help with pQFT on general curved spacetimes.

Also, why do you say the path integral doesn't exist? At least in 2D and 3D, doesn't constructive field theory, which you mention at the end, show that the path integral exists?

Yes, that's what I meant by "toy examples" where I wrote "There is no known way to make sense of this integral, apart from toy examples"

Now of course it may be unfair to refer as a "toy example" to all the great effort that went into "constructive QFT". Mathematically it is a highly sophisticated achievement. But it remains a matter of fact that as far as the physical problem description is concerned, the real thing is interacting Lorentzian QFT in dimensions four or larger.

I should be careful with saying "the path integral does not exist in general", because there is no proof besides experience, that it does not. Maybe at one point people can make sense of it. But even so, it seems to me that the results of "constructive QFT" show one thing: even if one can finally make sense of the path integral, it does not seem all too useful. Very little followup results seem to have come out of the construction of interacting scalar field theory in 3d via a rigorous Euclidean path integral. If we follow the tao of mathematics, the path integral just does not seem to be the right perspective. Or so I think.

 

[Urs Schreiber]

Note to those that like me do not understand all the mathematical detail. Of course try to correct that, but still read it and try to get a feel for the issues at the frontiers of current physics.

That's the right attitude! Learning by osmosis.

And by asking questions! Feel invited to ask the most basic questions that come to mind.

 

[jordi]

Very interesting. I subscribe to this thread. Thank you very much for sharing, Urs.

 

[A. Neumaier]

The usual argument is that if there is an infinite number of terms to be chosen, then the theory is not predictive.

Yes. And as you hint at, this is a totally unfounded argument. In an asymptotic power series in two variables there are an infinite number of terms to be chosen (at each order a growing number more), but nobody concludes that therefore power series at low order are not predictive. They are highly predictive as long as one is in the range of validity of the asymptotic expansion at this order (i.e., typically as long as the first neglected order contributes very little).

Of course, for gravity at the Planck scale (and for QCD at low energies, etc.) one expects that one is outside this domain, so that the value of the expansion becomes questionable at each order. Thus a perturbative theory is in many respects not a substitute for a nonperturbative version of the theory.

 

[A. Neumaier]

I believe I did provide a pointer, to the section here , [...]
What you are referring to, and what remains unsolved in generality, is taking the adiabatic limit of the coupling constant.

The former only shows how to construct approximate observables at each order - This has nothing to do with the infrared limit. The latter is precisely the adiabatic limit. It is there (and only there) where the particle content of the theory (and hence issues such as confinement) would appear. For example, in QCD, the perturbative theory is in terms of quarks, but the infrared completed theory has no quarks (due to confinement) but only hadrons.

 

[Urs Schreiber]

Yes. And as you hint at, this is a totally unfounded argument. In an asymptotic power series in two variables there are an infinite number of terms to be chosen (at each order a growing number more), but nobody concludes that therefore power series at low order are not predictive. They are highly predictive as long as one is in the range of validity of the asymptotic expansion at this order (i.e., typically as long as the first neglected order contributes very little).

Right, the traditional lore highlights a would-be problem that does not actually arise because before it could, another problem kicks in (non-convergence of the perturbation series).

There is an interesting comment about this state of affairs in

• Suslov, "Divergent Perturbation Series" (arXiv:hep-ph/0510142):

Classical books on diagrammatic techniques describe the construction of diagram series as if they were well defined. However, almost all important perturbation series are hopelessly divergent since they have zero radii of convergence. The first argument to this effect was given by Dyson

[...]

Even though Dyson’s argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series

[...]

The modern status of divergent series suggests that techniques for manipulating them should be included in a minimum syllabus for graduate students in theoretical physics. However, the theory of divergent series is almost unknown to physicists, because the corresponding parts of standard university courses in calculus date back to the mid-nineteenth century, when divergent series were virtually banished from mathematics.

 

[Urs Schreiber]

The former only shows how to construct approximate observables at each order - This has nothing to do with the infrared limit.

Sure, but why do you say "only"? This is the point that the perturbative interacting observables, as long as they have bounded spacetime support, may consistently be computed in perturbation theory without passing to the adiabatic limit. This says that the perturbation theory is well defined, irrespective of infrared divergencies.

In this sense it seems correct to me to write that "pAQFT deals with the IR-divergencies by organizing the observables into a local net". Or maybe instead of "deals with" it would be better to write "circumvents the problem of". (?)

 

[vanhees71]

The former only shows how to construct approximate observables at each order - This has nothing to do with the infrared limit. The latter is precisely the adiabatic limit. It is there (and only there) where the particle content of the theory (and hence issues such as confinement) would appear. For example, in QCD, the perturbative theory is in terms of quarks, but the infrared completed theory has no quarks (due to confinement) but only hadrons.

That's why low-energy QCD, if not using lattice-QCD simulations (within their range of applicability), is usually treated in terms of various effective field theories. For the light (+strange) quark domain one uses chiral symmetry (ranging from strict chiral perturbation theory for the ultra-low-energy limit to more or less "phenomenological" Lagrangians constrained by chiral symmetry). Another example is heavy-quark effective theory (also combined with chiral models if it comes to light-heavy systems like D-mesons).

The naive phenomenological physicists approach is indeed that such effective non-renormalizable theories use some low-loop orders of the effective theory with the corresponding low-energy constants, and this provides also predictive power. Often one has to resum ("unitarization"). 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?

Another somewhat related question in my field (relativistic heavy-ion collisions) is the amazing agreement between relativistic viscous hdyrodynamics, derived from relativistic transport theory via the method of moments, Chapman-Enskog, and the like and full relativistic transport theory in a domain (of, e.g., Knudsen numbers around 1), where naively neither of these methods should work. On the other hand the finding of agreement suggest that two methods which are valid in opposite extreme cases (transport theory for dilute gases a la Boltzmann, where the particles scatter only rarely and otherwise are "asymptotically free" most of the time, i.e., large mean-free path vs. ideal hydrodynamics which is exact in the limit of vanishing mean-free path, i.e., the dynamics is slow compared to the typical (local) thermalization time).

 

[A. Neumaier]

Sure, but why do you say "only"? This is the point that the perturbative interacting observables, as long as they have bounded spacetime support, may consistently be computed in perturbation theory without passing to the adiabatic limit. This says that the perturbation theory is well defined, irrespective of infrared divergencies.

In this sense it seems correct to me to write that "pAQFT deals with the IR-divergencies by organizing the observables into a local net". Or maybe instead of "deals with" it would be better to write "circumvents the problem of". (?)

It avoids having to deal with it, just as standard renormalized perturbation theory does. The infrared divergences still show up (in both cases) when you try to calculate S-matrix elements. Indeed, the perturbatively constructed S-matrix elements cannot even have mathematical existence in case of QCD, because of confinement - there are no asymptotic quark states.

 

[Urs Schreiber]

It avoids having to deal with it, just as standard renormalized perturbation theory does.

Indeed this is standard renormalized perturbation theory, just done right.

Nothing in pAQFT is alternative to or speculation beyond traditional pQFT. It is traditional pQFT, but done cleanly. The observation that I have been highlighting, that the algebra of quantum observables localized in any compact spacetime region may be computed, up to canonical isomorphism, already with the adiabatically switched S-matrix supported on any neighbourhood of the causal closure of that spacetime region, is "just" the formal justification for why indeed it is possible to ignore the adiabatic limit in perturbation theory.

This is exactly like causal perturbation theory is "just" the formal justification for the standard informal construction of the perturbation series.

Anyway, we don't have a disagreement about the facts, maybe just about the wording.

 

[A. Neumaier]

Anyway, we don't have a disagreement about the facts, maybe just about the wording.

Yes. pAQFT removes cleanly all UV problems but none of the IR problems. The latter are resolved only by performing the adiabatic limit in causal perturbation theory - and there sit the constructive problems.

 

[Urs Schreiber]

Yes. pAQFT removes cleanly all UV problems but none of the IR problems.

The problem to be dealt with is that in the absence of the adiabatic limit, the perturbative S-matrix only exists in adiabatically switched form, which, taken at face value, does not make physical sense.

To make sense of causal perturbation theory in the absence of the adiabatic limit one needs to prove that the adiabatically switched S-matrix does, despite superficial appearance, serve to define the correct physical observables.

That proof is not completely trivial. It's result shows that the adiabatically switched S-matrix, while unable to define the global (IR) observables in the adiabatic limit, does, despite superficial appearance, induce the correct local net of localized physical perturbative observables. What is called pAQFT is just the name given to the result of this proof, the well-defined local net of perturbative observables obtained from unphysical switched S-matrices in absence of an adiabatic limit. This way pAQFT deals with the problem.

Without an argument like this you would have to make sense of the adiabatic limit in order to even define the perturbation theory. Which would essentially mean that you'd have to define the non-perturbative theory in order to define the perturbative theory. Which would be pointless.

I suppose the reason why we keep talking past each other is that you keep reading "deal with the IR problem" as "define the theory in the IR". But even before it gets to this ambitious and wide open goal, there is the problem of even defining the perturbation theory without taking the adiabatic limit. This second problem (which logically is the first one to consider) is what pAQFT solves.

 

[vanhees71]

But isn't the real solution of the IR problem in pQFT to use the correct asymptotic free states a la Kulish and Faddeev,

P. Kulish and L. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys., 4 (1970), p. 745.
http://dx.doi.org/10.1007/BF01066485

and many other authors like Kibble?

In the standard treatment one uses arguments a la Bloch&Nordsieck, Kinoshita&Lee&Nauenberg and soft-photon/gluon resummation to resolve the IR problems. It's of course far from being rigorous.

I've also no clue, how you can define proper S-matrix elements without adiabatic switching (in both the remote past and the remote present). Forgetting this leads to pretty confusing fights in the literature. See, e.g.,

F. Michler, H. van Hees, D. D. Dietrich, S. Leupold, and C. Greiner, Off-equilibrium photon production during the chiral phase transition, Annals Phys., 336 (2013), p. 331–393.
http://dx.doi.org/10.1016/j.aop.2013.05.021
http://arxiv.org/abs/1310.5019

All this is, of coarse, far from being mathematically rigorous, but maybe it's possible to make it rigorous in the sense of pAQFT?

 

[A. Neumaier]

induce the correct local net of localized physical perturbative observables.

but this has nothing to do with the infrared (i.e., low energy) behavior, so you shouldn't use the term IR in this connection.

The basic conflict in QCD (or quantum Yang-Mills) is that there are no physical quark fields although there are perturbative quark fields.

In QED, the conflict is less obvious but you may look at Weinberg's Volume 1, Chapter 13 for a discussion of IR effects in QED. These effects appear although the renormalized perturbative asymptotic series is already completely well-defined! The reason is that at a given energy the number of massless particles produced is unbounded, and to get physical results one must integrate over all these soft photon degrees of freedom. This is most correctly (but still in a mathematically nonrigorous way) handled by using coherent state techniques, as in the references given by Handrik van Hees.