Cheers,
T.

“Wow you two, keep working on it and you will make this the longest thread in the history of this blog. ”

Sorry, but you’re going to wait for the weekend for the next installment. These posts are time-consuming to write, and this week is crazy at my university. Sorry about that.
Cowherd

Now seriously: I am happy to read your discussion, because I am learning stuff, 80% of which I had once known and long forgotten. The remaining 20% I do not fully grasp, but you are making me wish I was reading one of the five or six QFT books I have had on a shelf getting dust for the last 15 years.

Thank you for your effort, and if either of you thinks you can to make this a tad simpler and explain the discussion that took place here in a separate guest post, please just ask.

Cheers,
T.

OK. Let’s just underscore that non-renormalizable theories inevitably depend on their UV cutoff whereas renormalizable ones have no corresponding dependence, and that this problem of non-renormalizable theories can be more or less severe. There are decoupling EFTs, in which contributions from the high energy, cutoff-dependent part are suppressed by powers of energy/cutoff, so you can ignore them at sufficiently low energy; and there are non-decoupling EFTs, in which this is not the case. You are implicitly thinking of (or hoping for) the former, not-so-bad kind.

>>Dear anomalous cowherd, ChPT is a low energy effective theory
>>the Goldstone bosons* of QCD below the chiral symmetry breaking
>>scale. As you correctly note, it does not include nucleons, other
>>than as *classical* objects.
[...]
>-No! I described the inclusion of nucleons as being in an effectively static approximation.

Since the perturbative series in ChPT is a momentum expansion, treating the nucleons as static means treating them at tree level.

>They are NOT classical. As a Skyrme soliton they have to be quantized in the quantum theory
>of the non-linear Sigma Model [Chiral Lagrangian].

The skyrmion is a phenomenological model of hadrons; the Skyrme Lagrangian contains higher order derivative terms not present in the non-linear sigma model. Without them, the skyrmion could not be stable (Derrick’s theorem).

>The Chiral Lagrangian description of the low-energy hadronic interactions is a QUANTUM field
>theory description, including baryons as quantized Skyrmions.

I am sorry, but at the very least we seem to be using different terminology here. I do not think of ChPT and Skyrme model as one and the same. I can see how you might come to this point of view though: when you perform a derivative expansion of the sigma model you get stabilizing terms a la Skyrme. However, AFAIK this is not how baryons are included in ChPT (that’s done by a separate expansion in the baryon masses).

> I presently have on my desk the Gasser-Leutwyler paper:
>-Chiral Perturbation Theory To One Loop
>-J. Gasser and H. Leutwyler
>-Ann. Phys. 158, p142-210 (1984)
>Nowhere do they claim that the non-linear sigma model is renormalizable.

That is a very old paper. If you look up his more recent (Arxiv era) writings, you will see him emphasizing this point repeatedly.

>>I think I understand the source of the confusion now: you are
>>implictly assuming that effective field theory and quantum effective
>>field theory are synonymous terms. They are not. Example: the
>>venerable sigma model is an effective low energy theory of pions and
>>nucleons. It is not the same as ChPT, other than in the classical limit
>>of the latter. So if all you know is the classical sigma model, you know
>>nothing about quantum effects. Same for classical hydrodynamics vs.
>>quantum hydrodynamics.
>-The venerable [linear] sigma model is a quantum field theory that has been used to describe
>the low-energy interactions of pions and nucleons. It is a renormalizable theory that represents
>one possible ultraviolet completion of the non-linear sigma model [ChPT].

You are right. So am I: if I take the sigma mass to infinity, I am left with the non-linear sigma model, which is non-renormalizable. But in ChPT I do not stop there: I perform a derivative expansion and find that the higher order terms contain the counterterms I need to kill the divergences by coupling constant renormalization (which is what Leutwyler is fond of pointing out).

> This proof is described in detail in:
> -Chiral Dynamics
> -by B.W. Lee
> -Gordon and Breach (1972)

A more accessible presentation is probably Itzykson-Zuber.

> to go from a full UV theory to its low energy effective field theory you functionally
> integrate out *some* of the fields. The fields that you don’t integrate out remain to
> be integrated in the path integral, and as such are fields in a QUANTUM field theory.

Yes, I agree. My point is that this requires (1) a quantum theory (field or other, e.g. string, fine) which (2) lets you integrate out the high energy modes. Point number 2 requires the theory to be renormalizable (or finite, if not a field theory). If all you have from the outset is a non-renormalizable low energy theory, you don’t have what you need to carry out this program.

>My comment specifically addressed whether the infrared effective field theory is classical
>or quantum; it is QUANTUM.

I think what you mean to say is that the modes which you did not integrate out are quantum. However, whether those modes dominate observable low energy physics depends on whether your EFT is the good, decoupling kind or the bad, non-decoupling kind. If you have the bad kind, the cutoff-sensitive terms are not negligible and you have no way to separate them from the surviving quantum terms. In a situation where you can only observe low energy physics, how do you tell which kind it is?

>A UV complete continuum quantum field theory of gravity is a worthwhile objective; but it
>is NOT a prerequisite to the quantum infrared description of Einstein gravity as an effective
>quantum field theory.

What I know about gravity is what’s observed at (very) low energy. The mainstream theory of gravity deduced from those observations, GR, is a classical theory. I believe it to be merely an effective theory (because all other known interactions are quantum) but if I try to quantize it I get a non-renormalizable result. Is GR a good, decoupling EFT, in which case it still makes sense to try quantizeíng it, or is it a bad, non-decoupling EFT, in which case trying to treat
it as a quantum theory is nonsensical?

How can I tell, without the UV completion in hand?

I will reply to each of your points below. I will indent your text “pine style” to indicate what I am replying to, and put quotations around it.
Before I get to the reply, let us establish some standard notation, so there is no possibility of ambiguity. Whenever I write that a theory is “renormalizable” I will mean in the standard Dyson-Salam-Weinberg sense of power counting; ie. that there are a finite number of distinct 1PI vertices which develop divergences in perturbation theory, which require counterterms. If I say that a theory is non-renormalizable, I mean that there are an infinite number of 1PI vertices which will develop divergences in the course of the perturbation expansion. This occurs whenever we have operators in the Lagrangian whose engineering dimension is greater than the spacetime dimension in which we define the field; iteration of insertion of these operators will produce divergences in 1PI vertex functionals with higher and higher numbers of legs.

Before we start we should note the following: EVEN NON-RENORMALIZABLE THEORIES REQUIRE RENORMALIZATION.
This is one of the main points of Wilson’s work. For Wilson, quantum field theories contain, in the action, local operators of arbitrary dimension constructed from the fields of the theory, consistent with the symmetries of the theory. Each of these interaction operators renormalizes, and induces a renormalization group “flow” on the space of coupling constants of the theory. All but a finite number of the couplings are for “irrelevant” operators and flow towards zero (in the infrared), but they all renormalize and they all flow. This is standard text-book material, and is explained in detail in:
-Quantum Field Theory [Chapters 12 and 13]
-by Peskin and Schroeder
-Westview (1995).
and in:
-Quantum Field Theory and Critical Phenomena
-by Jean Zinn-Justin
-Oxford University Press, USA; 4 edition (2002)
For a more traditional “power-count and subtract the divergences” treatment of the same point see:
-The Quantum Theory of Fields; Volume 1 [Chapter 12.3]
-by Steven Weinberg
-CUP (1995).
I take this text-book material as understood and stipulated.

> “Dear anomalous cowherd, ChPT is a low energy effective theory
>the Goldstone bosons* of QCD below the chiral symmetry breaking
>scale. As you correctly note, it does not include nucleons, other than >as *classical* objects. It is therefore not correct to describe it as a
>quantum effective theory of the relevant degrees of freedom of low
>energy QCD (as somebody made of mostly nucleons, I am a bit
>touchy on this point .”
-No! I described the inclusion of nucleons as being “in an effectively static approximation”. They are NOT classical. As a Skyrme soliton they have to be quantized in the quantum theory of the non-linear Sigma Model ["Chiral Lagrangian"]. I am frankly perplexed that you keep coming back to this point. Quantization of solitons in non-linear quantum field theories was worked out in detail in the mid-1970’s, and is standard textbook material. See for example:
-Aspects of Symmetry
-by Sidney Coleman
-CUP (1988)
- [Chapter: "Classical Lumps And Their Quantum Descendents"]
or:
-Solitons and Instantons
-by R. Rajaraman
-North Holland (1989)
Since we are here discussing the nucleon as a Skyrme soliton, I will give you the reference where the statistics, spin, and isospin of the nucleon are determined by the QUANTIZATION of the collective coordinates of the Skyrme soliton. It is:
-Current Algebra, Baryons and Quark Confinement
-by Edward Witten
-Nuclear Physics B223 (1983) p.433-444.
The Chiral Lagrangian description of the low-energy hadronic interactions is a QUANTUM field theory description, including baryons as quantized Skyrmions.

> “BTW, I am a little surprised that you do not mention the standard
>references by Leutwyler on ChPT. If you care to look at his texts (just
> look him up on the Arxiv), you will find that he makes a point of the >renormalizability of ChPT. It is what makes it a consistent quantum
>theory”
Chiral perturbation theory [ChPT] is the perturbation theory of the non-linear sigma model. I presently have on my desk the Gasser-Leutwyler paper:
-Chiral Perturbation Theory To One Loop
-J. Gasser and H. Leutwyler
-Ann. Phys. 158, p142-210 (1984)
Nowhere do they claim that the non-linear sigma model is renormalizable. In fact if you look at their Appendix B. “Renormalizable Sigma Model”, they specifically introduce the LINEAR sigma model to have a “specific renormalizable model” [their words], to compare to the NONRENORMALIZABLE non-linear sigma model, whose chiral perturbation theory they have spent the previous 60 pages analyzing and renormalizing. This goes back to the remark at the start: even non-renormalizable quantum field theories require renormalization. The non-linear sigma model whose chiral perturbation theory [ChPT] encodes the low-energy dynamics of the strong interactions is manifestly non-renromalizable; its Lagrangian is non-polynomial, and when Taylor expanded involves operators of arbitrarily large dimension. In short the non-linear sigma model is non-renormalizable for exactly the same reason the the Einstein-Hilbert gravitational theory is non-renormalizable. Gasser and Leutwyler explicitly demonstrate how to renormalize the non-renormalizable non-linear sigma model; in an exactly analogous manner the reviews by Donoghue, and Burgess, that I quoted above, discuss how to renormalize the non-renormalizable Einstein-Hilbert theory.
The correct statement is that: they make a point of the RENORMALIZATION of ChPT. It is what makes it a consistent
EFFECTIVE quantum theory EVEN THOUGH THE THEORY IS NONRENORMALIZABLE.

> “I think I understand the source of the confusion now: you are
>implictly assuming that “effective field theory” and “quantum effective
> field theory” are synonymous terms. They are not. Example: the
>venerable sigma model is an effective low energy theory of pions and
>nucleons. It is not the same as ChPT, other than in the classical limit
>of the latter. So if all you know is the classical sigma model, you know
> nothing about quantum effects. Same for classical hydrodynamics vs.
> quantum hydrodynamics.”
-The venerable [linear] sigma model is a quantum field theory that has been used to describe the low-energy interactions of pions and nucleons. It is a renormalizable theory that represents one possible ultraviolet completion of the non-linear sigma model [ChPT]. The proof of the renormalizability of the linear sigma model, in the spontaneously broken phase, is due to Gervais and Lee, and independently due to Symanzik. This proof is described in detail in:
-Chiral Dynamics
-by B.W. Lee
-Gordon and Breach (1972)
If you take the [quantum] linear sigma model, and functionally “integrate out” the sigma field, you are left with the [quantum] non-linear sigma model, which is a non-renormalizable quantum effective field theory that reproduces the infrared dynamics of the [quantum] linear sigma model, its ultraviolet completion. This is a quantum infrared equivalence between two theories, a UV complete one and an effective theory describing the same quantum infrared behaviour [incuding the quantum "chiral logs"]. Again, this is general behaviour; to go from a full UV theory to its low energy effective field theory you functionally integrate out *some* of the fields. The fields that you don’t “integrate out” remain to be integrated in the path integral, and as such are fields in a QUANTUM field theory.

> “To complete your closing statement in #66: QUANTUM effective
>field theories are derived by a matching of QUANTUM field theories.
>Yes: you start with a quantum field theory, integrate out all modes
>above a cutoff Lambda to obtain your low energy quantum effective
>field theory and impose matching conditions at the cutoff. In order to
>be able to do all this, YOU MUST HAVE A CONSISTENT QUANTUM
>FIELD THEORY to start with.”
My comment specifically addressed whether the infrared effective field theory is classical or quantum; it is QUANTUM. However, the quantum theory, whose IR dynamics the infrared field theory is matching, need not be a field theory. This is already apparent from Wilson’s Nobel prize work. Wilson describes the IR behaviour of statistical LATTICE models by [Euclidean] continuum field theories. In his case the UV completions of his IR quantum field theories often are actually discrete lattice models. His discovery was that the infrared behaviour was universal over many models, and determined only by the effective theory in the infrared. In other words the critical behaviour of the three dimensional Ising, and lattice gas, models is governed by the IR fixed point of an effective 3-D $\phi^4$ theory, even though neither of these two UV completions is a continuum field theory. The infrared behaviour of these theories [eg. critical exponents] is independent of the microscopic structure of the model [ie. of the UV completion]. To understand the IR behaviour we only need the IR effective theory. The UV theory may not even look like a continuum field theory; we don’t care because it doesn’t matter!

Please don’t misunderstand my point of view. I find work on the “asymptotic safety” program of Weinberg interesting. [if I remember correctly Weinberg's original discussion of this was in an article he contributed to the Einstein Centenary Volume edited by Hawking and Israel].
-Ultraviolet Divergences In Quantum Theories Of Gravitation.
-Steven Weinberg (Harvard U.) . 1980.
-In *Hawking, S.W., Israel, W.: General Relativity*, 790-831.
Some related (?) work you might be interested in is:
-Lee-Wick Indefinite Metric Quantization: A Functional Integral -Approach.
-David G. Boulware, David J. Gross
-Published in Nucl.Phys.B233:1,1984.
and
-Unitarity In Higher Derivative Quantum Gravity.
-E.T. Tomboulis
-Published in Phys.Rev.Lett.52:1173,1984.
Also recently Mark Wise and collaborators have done much to revive and develop Lee-Wick type theories.
But one must be clear on what one is trying to accomplish with one’s theory. A UV complete continuum quantum field theory of gravity is a worthwhile objective; but it is NOT a prerequisite to the quantum infrared description of Einstein gravity as an effective quantum field theory.

It was in fact a study of the quantum sine-gordon system that originally led physicists to quantum symmetries (and hence categorical techniques via Hopf algebras, which as you know are now used to describe renormalisation in QFT via Connes-Kreimer type algebras) back in the 1970s and 1980s. I would highly recommend Reshitikhin’s original papers. Now, what does this have to do with gravitons? Nothing!

BTW, I am a little surprised that you do not mention the standard references by Leutwyler on ChPT. If you care to look at his texts (just look him up on the Arxiv), you will find that he makes a point of the renormalizability of ChPT. It is what makes it a consistent quantum theory.

I think I understand the source of the confusion now: you are implictly assuming that “effective field theory” and “quantum effective field theory” are synonymous terms. They are not. Example: the venerable sigma model is an effective low energy theory of pions and nucleons. It is not the same as ChPT, other than in the classical limit of the latter. So if all you know is the classical sigma model, you know nothing about quantum effects. Same for classical hydrodynamics vs. quantum hydrodynamics.

To complete your closing statement in #66: QUANTUM effective field theories are derived by a matching of QUANTUM field theories. Yes: you start with a quantum field theory, integrate out all modes above a cutoff Lambda to obtain your low energy quantum effective field theory and impose matching conditions at the cutoff. In order to be able to do all this, YOU MUST HAVE A CONSISTENT QUANTUM FIELD THEORY to start with.

At the risk of being guilty of promotion of my physics stuff, maybe my pdf paper at
http://www.valdostamuseum.org/hamsmith/sGmTqqbarPion.pdf
(sorry, no arXiv reference due to blacklisting)
uses the Coleman equivalence, and the model of fundamental fermions (in this case quarks) as Kerr-Newman black holes,
to get a realistic pion mass from up and down quark constituent masses.
Whether or not you like all of my model stuff in the paper, you might find the Coleman equivalence stuff interesting.

Tony Smith

Considering that the astrophysical event rates at the sensitivity of the S5 run corresponds to a rate of about 1 event every 10 – 100 years, they really ought to have NOT been seen. That they haven’t is thus not surprising (albeit disappointing).

thank you for your insightful comment. I am happy I did the right thing in not “moderating” or steering this discussion: indeed what you write above is very interesting to me – er, the first part. I feel I need to read some documentation before I understand the second part!

Cheers,
T.

“They are collective degrees of freedom of the quantum theory. Quantizing them makes no sense because their classical equations of motion already include all quantum effects.”

This is the nub of the disagreement! Collective modes DO need to be quantized in an effective field theory. They need to be quantized in the Chiral Effective Lagrangian to give the “chiral logs”. If you don’t quantize them you don’t correctly reproduce the low-energy physics that is in QCD. Think about this point; if composite pions were not quantized, we could have decided whether a pion was elementary or composite by just examining the presence, or absence, of quantum correlations in low-energy pion interactions [for example RHIC looks for Hanbury-Brown-Twiss correlations in pairs of emitted pions] . No quantized pions means no loop corrections; what happens now to the optical theorem in pion-pion scattering?

Similary, it is the quantization of the collective modes of solitonic states that is responsible for many of their most interesting properties [eg. the spin and statistics of the Skyrmion in the Skyrme model, following Witten]. As another example consider Coleman’s proof that the Sine-Gordon theory is quantum equivalent to the Thirring model. This equivalence relates a collective mode [the Sine-Gordon soliton] to the fundamental fermion in the Thirring model which by definition is quantized. This is only possible if the Sine-Gordon soliton [a collective mode] is itself quantized.

As explained in the references above [see especially the Polchinski, Manohar, Kaplan, and Pich reviews] effective field theories are derived by a matching of QUANTUM field theories.

“Consider the well known case of QCD. A straight identification of low energy degrees of freedom with high energy ones is not possible because the coupling blows up at low energy. The relevant degrees of freedom at low energy are not quarks and gluons but pions and nucleons. Given QCD, you can derive the low energy theory of pions and nucleons. Given only the effective theory of pions and nucleons, you can not derive QCD; you can only make an educated guess that QCD might be what’s underlying it.”

To be clear, I did NOT argue for the identification of the low-energy degrees of freedom with the high-energy ones. This is the problem of finding the ultra-violet completion of a given theory, and as I noted above it may not be uniquely specified by the low-energy theory, and may not even be a field theory.

Taking your example quoted, the low-energy degrees of freedom of the strong interactions [below the confinement/chiral-symmetry breaking scale] are pions and kaons and etas [Baryons are not normally treated as part of the low-energy effective field theory; if necessary they can be included as solitons of the Chiral Lagrangian (Skyrmions) and matrix elements calculated in what is effectively a static approximation]. The dynamics of the low-energy pion, kaon, and eta modes is encoded in an effective quantum field theory, referred to in the literature as the “Chiral Lagrangian”. This is a non-renormalizable effective quantum field theory defined in terms of the pion, kaon, and eta fields, imposing only the chiral and flavour symmetries of the low-energy hadronic interactions. This is a QUANTUM theory; quantum corrections which are necessary for agreement with experiment [and in particular the dominant "chiral logs"] are calculated directly in this non-renormalizable effective quantum field theory. No knowledge of the ultraviolet completion of the Chiral Lagrangian is required for these computations. For detailed treatments see:
-Chiral perturbation theory.
A. Pich.
Published in Rept.Prog.Phys.58:563-610,1995.
e-Print: hep-ph/9502366
-Effective field theory
Antonio Pich. FTUV-98-46, IFIC-98-47, Jun 1998. 106pp.
e-Print: hep-ph/9806303

It is correct that you cannot deduce the ultraviolet completion [above the confinement/chiral-symmetry breaking scale] of the Chiral Lagrangian theory [which we know to be QCD from HIGH energy experiments] from the low-energy effective chiral theory alone. This is exactly like gravity, where there is no way we can deduce the ultraviolet completion [above the Planck scale] of gravitational dynamics, from knowledge of the low-energy effective Einstein theory alone.

On the other hand, the low-energy [below the confinement/chiral-symmetry breaking scale] quantum dynamics of the strong interaction may be encoded in the Chiral Lagrangian, considered as a [non-renormalizable] QUANTUM effective field theory.
Similarly the low-energy [below the Planck scale] quantum dynamics of the gravitational interaction may be encoded in the Einstein-Hilbert Lagrangian, considered as a [non-renormalizable] QUANTUM effective field theory. In both cases low-energy calculations in the effective field theory do NOT require knowledge of an ultraviolet completion for the theory.

Let’s put it this way: I agreed with GW that CERN is not located in Batavia. Please leave me out of the rest – I think I explained why I prefer to avoid arguing on things I do not know very well. My knowledge of GR+QG is not solid enough and my studies of QFT are more than a decade old. I think I have my own ideas on what is right and wrong of the things you two discussed, but I think entering the fray would be useless here.

Cheers,
T.

This is just too much.