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Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” *October 4, 2007*

*Posted by dorigo in Blogroll, internet, mathematics, physics, science.*

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**Marni D. Sheppeard** is a theoretical physicist in New Zealand. After 20 years of stints in experimental condensed matter physics, biomedical engineering, lattice QCD, electronics manufacture, programming, teaching, mountaineering and the real world, she has finally found some time to work on her main interest, quantum gravity. You’ll also find her serving excellent lattes in central Christchurch.

**IS CATEGORY THEORY USEFUL ?**

In the past, physics has made great progress with the mathematics of classical symmetries. For example, motion of a body in a plane can be decomposed into straight line translations and rotations.

The collection of all possible rotations is described by a circle in the plane, marked with a reference point which represents no rotation. Other points on the circle represent rotations by an angle corresponding to the angle between the given point and the reference point. Any rotation has a reverse operation, namely a rotation by the same angle in the opposite direction, which equals the rotation that moves from the original point around the remainder of the circle. That is, a pair of rotations is represented by a pair of arrows which make up a circle. These arrows combine to form a single arrow around the whole circle. Note that this is the same physical operation as the original marked reference point. Two shorter arrows could also be combined into a single rotation.

The idea of combining arrows in this way is what Category Theory is about. By definition, a category is just a collection of points along with arrows between them which can be combined to form new arrows. But points don’t have to represent actual points in a classical space. A point might represent a whole space, and the arrows ways of mapping one space into another. By studying these arrows we have a way to look at properties of spaces that doesn’t involve drawing complicated pictures in higher dimensions, because a space is represented by a single point and arrows are only one dimensional. But the concept of category is even more general than this. Points might represent sets, and the arrows functions between sets. Now much of 20th century mathematics is built upon the properties of sets, and here there is the possibility of using functions *and* sets together. But there seems to be a snag: surely the collection of points and arrows in any category must form sets! And it is true that the basic axioms for a category state the existence of such sets. However, by being a little bit more sneaky we can describe the category of *all* sets and then ask ourselves about other categories that might replace this one.

Why on earth would we want to do this in physics? First, observe that for rotations in the plane there is only one point, the marked reference. All classical symmetries, in any dimension, correspond to categories with only one point. In the quantum world, however, we secretly play with categories with more than one point. For example, an atom emits light only at certain frequencies determined by state transitions for its electrons. A transition may be pictured as an arrow between states. Two consecutive transitions combine to give a photon of frequency equal to the sum of its component transitions. This looks similar to the planar rotations, except that each state now gives a point in the category.

Another category associated to quantum physics takes points to be spaces of states. This category has a logical aspect, similar to the category of sets but also with glaring differences. Whereas elements of sets obey the rule that they either exist or do not exist, quantum matter only takes on this feature when it is observed. The logic of quantum mechanics is built from operators (projections) that take a space of states and pick out a specific choice of state. Such an operator, as an arrow in the quantum category, has the feature that doing it twice is the same as doing it once, because once a state is chosen the second arrow will just select the state again. This type of operation appears in many places in mathematics. In category theory, a map that selects the point at the start of an arrow is such an operator, because after the point is selected the second iteration of the map just reselects the point, which is viewed as an arrow from the point to itself that does nothing. Category theory is unique in the way it combines algebra, geometry *and* logic.

But the important feature of a quantum particle that is really not understood is its mass. To derive mass numbers correctly would require some knowledge about Quantum Gravity, the theory of quantized mass. Only the photon has zero mass and travels at the local speed of light relative to observers made of massive particles. In the late 1960s, Sir Roger Penrose considered spacetime events as collections of light rays incident on a point from the celestial sphere. Mathematically, this amounts to thinking of the sphere as a category of sets of light rays, with its own non-classical logic. This is still a one dimensional category. Categories in higher dimensions, with higher dimensional arrows, are currently been investigated as a language for gravity.

In three dimensions a basic arrow may be replaced by a cube. We imagine that the cube contains arrows on the faces and one arrow in the interior. An analogue of a projection operator is a map that sends the whole cube to one half of its boundary. Is it possible that particle masses could be derived from the simple algebra associated to such operators? Carl Brannen has been studying projection operator algebras associated to cubes. He has shown that the electron, muon and tau masses are described by a simple three-by-three array of numbers, as are the three neutrino masses. Current work is extending this analysis to baryons and mesons.

## Comments

Sorry comments are closed for this entry

20 years ago, I was enthusiastic about topoi as fundational pieces of mathematics, and then perhaps of physics.

Are Dynkin diagrams used such a case?

Figure 2. Clebsch’s Diagonal Surface

Check post on name. Some were with Carl on Asymptotia trying to understand how to use Latex to help us draw.

Your drawing of the cube is a bit misleading. If I look only at the arrows you drew, I only see a planar diagram. My guess is that you were trying to describe a 2-category in the sense of Baez, or perhaps a functor between categories.

[…] particles as having spins, not just masses. (Incidentally, I notice that Marni Sheppeard made a guest post on another blog arguing that category theory is useful to physics. Here is another example of how […]

Hi Arivero, Plato and others. Yes, I admit to having swept many, many things under the rug, but the cube here is definitely not to be associated with 2-categories. If you like, such a cube appears in the axioms for tricategories, as worked out by Gordon, Power and Street in 1995. Tricategories have a special property that lower dimensional structures lack, namely that not all tricategories are (weakly) equivalent to the strict ones.

This de facto definition of “useful” is less useful. It relates category theory as descriptive instead of constructive.

But Mark Chu-Carroll over at

Good Math, Bad Mathclarifies that it makes it possible “to talk about extremely sophisticated structures and transformations using a really simple, clean abstraction”. He also notes that a general algorithmic description in computer science, lambda calculus, can be realized by software based on the categorial semantics.Nitpicks: Some particles mass is described by the hypothesized Higgs mechanism, aren’t they? And all massless particles travel at lightspeed AFAIU, the hypothesized graviton does so.

Good points, TL. I was only referring to known, observed particles.

This really was a wonderful post in that I understood everything but the final paragraph. That’s putting the high jump bar mighty low.

Hi Plato,

believe it or not, Marni’s last name is exactly as reported above 🙂

As for using latex to draw diagrams, I gave up a long time ago. I rather use xfig, which is a linux utility which I find quite versatile.

Cheers,

T.

Tommaso:believe it or not, Marni’s last name is exactly as reported aboveI hope the translation here is correct, in that, what is laid out following her name, “describes all the points that I was asking of previously?

Okay. 🙂

…..and of course to further be inspired in the continued effort to look at Euler of course. Thanks, for such posts a these, as they are very educative.

Marni D. SheppeardFor example, motion of a body in a plane can be decomposed into straight line translations and rotations.That such a process would be linked to the abstract world and related to quantum gravity is part of the problem I was asking of you previously here in the colliders.

Your answer presents a trouble for me with regard to what Marni D. Sheppeard by association makes to quantum gravity.

Would it be wrong to make such an association to “rubber sheet geometry?” What do you see in the colliders that could compare all those tracks “as a Konigsbergs bridge problem?” Have I captured the essence?

Hi Plato,

either my understanding of English is rapidly deteriorating, or you write giving too many things for grante. The result is however that I have a hard time understanding what you mean! 🙂

Marni, maybe you can answer better than me ?

Cheers,

T.

Sorry Tammaso,

In future I will try to be more to the point. 🙂

Tommaso,

Allow me to make a further mess of Plato’s queries. It seems that Plato is referring to the post titled, “Meccablog,” of Sept. 29, 2007 where he questions you in Comment #4 and you answer him in Comment #5. It appears he is attempting to resolve and understand your answer by further associating it with Marni’s statement in this post, “For example, motion of a body in a plane can be decomposed into straight line translations and rotations.” I sense he believes there is a contradiction concerning quantum gravity with your answer in that post and her above statement in this guest post. If not, I’ll have to paraphrase Groucho, “Those are my (interpretations), and if you don’t like them… well, I have others.”

… wince

About the relationship of gravitons to Higgs bosons (responsible for particle masses), the Higgs bosons are massive and so travel at slow speeds. Gravitons go at light velocity because gravitational fields propagate at light velocity (this is one of the established parts of general relativity), so gravitons can’t have any rest mass.

A fermion only undergoes standard model interactions: electromagnetic, weak and strong. It doesn’t have any intrinsic mass of its own. All the mass of a fermion arises from the Higgs bosons in the vacuum field around the fermion particle core.

There is thus a two-step connection of the graviton to the fermion: gravitons interact with Higgs field bosons, which in turn interact with fermion particle cores:

Gravitons interact with Higgs bosons, which then mire Fermions.

Gravitons don’t need mass themselves, they merely need to interact with Higgs bosons which exhibit mass. There is no experimental evidence for the claim that gravitons must interact with themselves. They only need to interact with Higgs bosons in the vacuum. Higgs bosons also give mass to weak force gauge bosons (W and Z), and that all right-handed fermions have zero weak isospin charge and don’t undergo weak interactions. This is a clue maybe that Higgs bosons have a handedness and possibly they only can give mass to W and Z gauge bosons which one type of spin. Maybe the physical basis of the chirality in particle physics is that only bosons with a particular handedness can acquire mass from the Higgs bosons.

[However, maybe the U(1)xSU(2) electroweak unification is not right, and I’m not advocating the existing speculations about the Higgs field. It’s possible that SU(2) includes gravitation and electromagnetism if you see the Higgs boson as causing chirality: one handedness of the Z and W weak bosons acquire mass, producing the weak force. The other handedness of W and Z fails to acquire mass because it doesn’t interact with the Higgs bosons. Hence, half of the W and Z bosons could be massless, and these would travel at light velocity and have infinite range unlike the massive short ranged versions. The positive and negative massless W’s may mediate positive and negative electric fields, replacing U(1) with its special gauge boson ‘photons’ which have 4 not 2 ‘polarizations’, while the massless Z may be the graviton. There are many advantages in this scheme over the existing standard model: you do away with U(1)’s single charge whereby a positron is an electron travelling backward in time (instead, SU(2) applied to electromagnetism and gravity allows you two electric charges, as observed!), you include gravitons, you radically change the Higgs mechanism into something less speculative and you simplify the standard model from U(1)xSU(2)xSU(3) to SU(2)xSU(3) while having these extensions to gravity, etc.]

Hi Plato. I’m afraid we’re out of phase here with regard to daylight, unsurprisingly. To be honest, I’m not sure what you mean either. I don’t really like the rubber sheet picture, but I agree it may be helpful in introducing people to the idea of classical geometric dynamics, and, yes, the rubber sheet is a curved analogue of the flat plane here.

It is true that we are very interested in (fancier versions of) the Konigsberg bridge problem: Tom Leinster has been working recently on developing Euler characteristics for 1-categories. These, and higher dimensional analogues, will probably be very useful in physical computations.

15. nc – October 5, 2007 writes:

“There is thus a two-step connection of the graviton to the fermion: gravitons interact with Higgs field bosons, which in turn interact with fermion particle cores:”

This is not correct. Remember, gravity does not couple to mass, rather it couples to energy [technically it is the stress-energy tensor that appears in the Einstein equations]. Even photons [which are massless] couple to gravity; this is required by the equivalence principle, and is directly observed in the laboratory [eg. Pound-Rebka falling photon experiment], and in the cosmos [eg. the Eddington light-bending experiment, and the Shapiro time-delay experiment]. Photons also act as a source of gravity; for example, in the early universe, at red-shifts over 5000, the energy density is dominated by radiation. This source of gravity must be correctly taken into account in calculating cosmic deceleration, or calculations of Big-Bang nuclesoynthesis would be grossly wrong. So no, gravitons will couple to any fermion carrying energy, whether the fermion has mechanical mass from the Higgs mechanism or not.

15. nc – October 5, 2007 writes:

There is no experimental evidence for the claim that gravitons must interact with themselves.”

Black holes depend on the non-linear gravitational self-interaction; so all the astronomical evidendence for black holes is also evidence for this non-linear self-interaction.

Cowherd, re your second point: nobody would deny that GR predicted black holes, but gravitons are theorised

quantumparticles, about which GR says precisely nothing. Moreover, the applicability of GR over cosmological distances is certainly open to question. Consider, for example, recent evidence from the S5 run at LIGO, that failed to detect GWs when they really ought to have been seen.18. Kea – October 5, 2007 writes

“but gravitons are theorised quantum particles, about which GR says precisely nothing”

I don’t understand this comment. The Einstein-Hilbert action is the weighting factor that gives the phase in the weighting in the functional integral over histories [modulo Feynman-DeWitt-Faddeev-Popov determinants to ensure independence of gauge slicing]. Extrema of this action determine the classical histories [like black-hole spacetimes], and this same action governs the quantum fluctuations about the classical backgrounds [graviton propagation and interaction]. So the same interaction terms are responsible for the classical solutions and for the interactions of the quanta.

On an unrelated point. Astronomical black holes are NOT of cosmological size. Solar mass black holes have a Schwarzschild radius of a few kilometers. The black holes at the centre of galaxies have radii that are small fractions of the galactic radii. And most of our observational evidence concerning them comes from the electromagnetic radiation emitted in their proximity [X-rays etc.], so that our information concerning their existence and nature does not depend on understanding gravity on cosmological scales.

18. Kea – October 5, 2007 writes

“the applicability of GR over cosmological distances is certainly open to question. Consider, for example recent evidence from the S5 run at LIGO, that failed to detect GWs when they really ought to have been seen.”

Just to be clear: are you claiming that LIGO purports to be able to rule out the existence of gravitational waves based on their present limits?

Astronomical black holes are NOT of cosmological size.Gee, I’d never have guessed (dripping sarcasm). As for the functional integral over histories: this is NOT the only way to do cosmology. If you insist on LCDM (or some other conservative alternative) then we will just have to agree to disagree, because this is simply not what I’m talking about.

…are you claiming that LIGO purports to be able to rule out the existence of gravitational waves based on their present limits?No, of course not. This is, after all, the very first result of this nature, but worth taking on board. We eagerly await future results from em triggered events.

It’s widely believed that: “Gravitons go at light velocity because gravitational fields propagate at light velocity (this is one of the established parts of general relativity), so gravitons can’t have any rest mass.”

But there is no experimental evidence for the speed of gravitational fields being that of light. There was a big argument in the journals about this recently. I blogged an introduction to the issue, uh, from my twisted point of view of course.

Thanks Fred.

Tammaso:In theories contemplating a low quantum gravity scale, black holes could in principle be created in high energy collisions, but if a chance of detecting their creation exists, it is not by gravitational effects, which remain billions of billions of billions of billions (and then some) of times smaller than those caused by strong interactions.I was thinking that the energy valuation used in the collider process was itself of value in relation to gravity?

There is “no evidence of the geometry” in the blackhole no one knows of? Yet, it would not seem to be so unprovable that the complexity would of “held values”, in collision process, would have, been related to that energy.” How else could you have predicted the outcome?

Hence, “the description of particle manifestations” would have come from “opening a door” through the quark gluon plasma. “Cerenkov radiation” or muon production could not have been seen in IceCube or Gran Sasso respectively. This is the New Physics?

Well, this post is a success if we judge by the interest of the ensuing comments. I myself am learning something by reading them… So I thank everybody who is contributing to this column.

Fred, thank you for your explanation! I guess I understand the question better now. And I am happy Marni pitched in…

Plato, I believe that if -and when- you create a black hole in a particle collision, the energy that goes in its creation is then released by radiative processes (causing its evaporation) that have nothing to do with gravity, despite the importance of gravity in determining the space-time geometry of the point of space.

Cheers,

T.

Dear anomalous cowherd, regarding your comment #19, I venture to remind you that GR is a classical theory, which was noted some time ago to yield a non-renormalizable perturbative expansion with standard quantization approaches. Sure you can plug the Einstein-Hilbert action into a path integral, but the result is not a consistent quantum theory. That’s why there is (or was) much joy and merriment about string theory, just to mention the usual suspect.

Maybe you subscribe to the hope of Reuter et al. that a UV fixed point is lurking in quantized GR, but that’s just an unproved hope.

I presume that’s what Kea had in mind when she said that “gravitons are theorised quantum particles, about which GR says precisely nothing”. There are no gravitons in GR, just like there are no photons in classical electrodynamics. The latter we know how to quantize consistently, the former has proved a little harder.

“15. nc – October 5, 2007 writes:

There is no experimental evidence for the claim that gravitons must interact with themselves.”

“Black holes depend on the non-linear gravitational self-interaction; so all the astronomical evidendence for black holes is also evidence for this non-linear self-interaction. – anomalous cowherd”

Black holes don’t depend on gravitons having mass; they just rely on having a gravitational field strong enough to trap light. Light doesn’t have mass either; it has no gravitational charge whatsoever. It merely has energy in the form of electric and magnetic fields.

Mass is provided by the Higgs field, not directly by gravitons. Higgs bosons mediate massiveness between fermions (which have electromagnetic charge but not gravitational charge) and gravitons (which provide the basis for gravitation and inertia). The Higgs bosons are unique in that they interact

bothwith electromagnetic fields and with gravitons.Gravitons don’t require mass any more than photons do; gravitons are exchanged between Higgs bosons. The Higgs bosons cause light to bend in a gravitational field because they interact with electromagnetic fields as well as gravitons. Higgs bosons around an electron give that electron mass in the standard model. The interaction between a Higgs boson and an electron is electromagnetic because electrons have no intrinsic mass (gravitational charge); the gravitational interaction between two electrons thus consists of an electromagnetic interaction between the electron core and a Higgs boson, followed by graviton interactions between this Higgs boson and a Higgs boson near the other electron, followed by an electromagnetic interaction between the Higgs boson around the other electron and the core of that other electron.

‘… there is no experimental evidence for the speed of gravitational fields being that of light. There was a big argument in the journals about this recently.’ – Carl.

Hi Carl, thanks for that link. The key experimental evidence to me is is the gravitational contraction effect, the effect of spacetime being contracted around a mass (time dilation in gravitational fields is well established, as is accompanied by a spatial contraction, e.g. Earth’s radius is contracted by 1.5 mm although that can’t be measured).

When you analyse the amount of gravitational time dilation (and the spatial contraction of distance around a mass) in GR, it turns out to to be similar to the Lorentzian form in SR: (1-v^2 /c^2)^(1/2), but with v^2 = 2GM/R (the escape velocity law, you can explain this on the basis of the gravitational potential energy gained by a body falling from an infinite distance being equal to the kinetic energy it needs to escape a similar gravitational field). However with SR, only distance in the direction of motion gets contracted, while in GR 3 spatial dimensions (all radial dimensions) get contracted, so GR predicts that you get a contraction of (1/3)GM/c^2 = 1.5 mm for the Earth (approximately, using the binomial expansion) instead of GM/c^2 which would be the equivalent of SR’s v^2/c^2.

The point is,

gravity is producing exactly the same contraction as SR which is based on Maxwell’s equations with propagation velocity c. Because at least the time-dilation prediction of GR has been experimentally validated, there is experimental evidence that gravity is going at the same speed at light. The time dilation is similar to length contraction law. Both result from motion (SR) and gravitation (GR). If you are moving in space, you’re colliding with more gravitons on one side than on the other, so you’re contracted in the direction of your motion; a static mass in space is radially squeezed by the same graviton effect. Both depend on the speed of light via v^2/c^2; c occurs there because contraction depends on your velocity relative to that of the field (electromagnetic or gravitational both having the same value, c). Yes, it would be nice to directly measure the speed of gravitons or at least group effects of gravitons like gravity waves, but I think that there is good evidence since if gravitons went at a speed different to c, measurements of gravitational time dilation made with atomic clocks would have shown this to be the case.nigel, first of all anomalous cowherd never said that black holes “depend on gravitons having mass”. Second, the statement that “Higgs bosons mediate massiveness” does not make much if any sense (it suggests a picture of Higgs bosons flying around carrying mass from particle A to particle B; no, particles get their inertial mass by coupling to the Higgs field) and to say that “Higgs bosons cause light to bend in a gravitational field” is just plain wrong. Light bending occurs because light follows geodesic paths. This was understood half a century before anyone had even thought of the Higgs. It has absolutely nothing to do with Higgs bosons.

Guess Who: I quoted what anomalous cowherd said (he claimed that black holes observations demand that the gravitational field has a self-interaction).

Let me explain it so you might just understand it: for gravitational field quanta (gravitons) to self-interact, they need to have gravitational charge.

Gravitational charge is called mass. Have you grasped it now?

The Higgs field mediates massiveness in the way I described in detail: Higgs bosons interact with gravitons and with SM particles.

Because SM particles don’t have gravitational mass, they don’t have gravitational charge and can’t directly interact with gravitons. Higgs bosons mediate the interaction between gravitons and SM particles.

Got it?

“to say that “Higgs bosons cause light to bend in a gravitational field” is just plain wrong. Light bending occurs because light follows geodesic paths. This was understood half a century before anyone had even thought of the Higgs. It has absolutely nothing to do with Higgs bosons.”

Light bending occurs because photons interact with Higgs bosons which interact with gravitons. You’re confusing classical (non-quantum) GR with quantum gravity. Light follows geodesic paths because of the gravitational field, which is a quantum field. There are many failures in classical GR, for example it neglects quantum interactions and leads to precisely the confusion you are in.

Cheers!

#27

Is it not the realization of reaching a “cross over” point?

Just as a point of interest word process allows blockquote to be used as a direct link, so do not forget to mouse over the quotes.:) Changing url links on name is a viable source as well.

The ideas already exist in different “formulated idealizations” as theoretical positions. How have they been translated to experimental processes?

Being provide for the conditions for such radiation does not discount what will be released as the new physics?

umm sorry….word process…should read “word press” bloggery format.

nigel, gravity does have self-interactions, as you would know if you had ever laid eyes on the Einstein equations.

As anomalous cowherd already pointed out, gravitational “charge” is energy, not just mass. Clearly, you still haven’t grasped this.

The statement that the Higgs “mediates massiveness” is nonsensical. Yes, it’s reasonable to assume that Higgs bosons interact with gravitons and with SM particles; the same can be said of any SM particle.

Because particles have energy, they have “gravitational charge” and interact directly with gravitons. No, Higgs bosons do not “mediate the interaction between gravitons and SM particles”.

Got it?

You are confusing your fantasies about “quantum gravity” with the real thing. There are many failures in your understanding of physics, for example your refusal to learn the basics before pontificating about advanced topics leads to precisely the confusion you are in.

Cheers.

Good to see there was a nice wholesome discussion while I was asleep, albeit not directly related to Category Theory or M theory. Thanks for that gem of a quote, Plato:

We have made it a planet that notices things. We have made it an observant earth.“… gravity does have self-interactions, as you would know if you had ever laid eyes on the Einstein equations. As anomalous cowherd already pointed out, gravitational “charge” is energy, not just mass. Clearly, you still haven’t grasped this.” – Guess Who.

Gravity doesn’t have any observed self-interactions; regarding general relativity field equations, you haven’t even read my comments above – I’ve explained that gravitons and photons are energy.

You are confusing the gauge bosons with the charges.Charges are not the same thing as gauge bosons (the exchange radiation) in any Yang-Mills quantum field theory.“You are confusing your fantasies about “quantum gravity” with the real thing. There are many failures in your understanding of physics, for example your refusal to learn the basics before pontificating about advanced topics leads to precisely the confusion you are in.” – Guess Who.

Er, I do have proof of everything I’ve stated.

Kea:

“Whereas elements of sets obey the rule that they either exist or do not exist, quantum matter only takes on this feature when it is observed. The logic of quantum mechanics is built from operators (projections) that take a space of states and pick out a specific choice of state. Such an operator, as an arrow in the quantum category, has the feature that doing it twice is the same as doing it once, because once a state is chosen the second arrow will just select the state again. This type of operation appears in many places in mathematics. In category theory, a map that selects the point at the start of an arrow is such an operator, because after the point is selected the second iteration of the map just reselects the point, which is viewed as an arrow from the point to itself that does nothing.”

I hope you find the time to fully formulate quantum gravity interactions using category theory. I saw some Smolin lectures on the Perimeter site where LQG sets out to obtain the Einstein field equation without a metric (background independent) by summing interaction graphs in a spin network like a Feynman path integral (integrating actions over all possible routes).

It would be great if this general idea could be evaluated physically (the LQG work is not too physically successful in its present form) to tie it down completely to factual predictions. E.g., consider a three dimensional array of gravitational charges and then consider possible individual Yang-Mills interactions as exchange radiation between them. All gravitational charges should be exchanging gravitons with all others. The question is the best mathematical way to formulate and treat this problem.

Can category theory simplify the analysis of such situations, e.g. each interaction could be summarised by a Feynman diagram but all the diagrams would be fairly similar except for differences in the direction and strength of the coupling. If you want to calculate the field curvature for a given point in spacetime, you have to sum all the interaction graphs involved. If you can categorise the graphs so that those with opposite resultants can be cancelled out, it would simplify the summation greatly. Is there any rigorous way to formulate this?

24. Guess Who – October 6, 2007 writes:

“Dear anomalous cowherd, regarding your comment #19, I venture to remind you that GR is a classical theory, which was noted some time ago to yield a non-renormalizable perturbative expansion with standard quantization approaches. Sure you can plug the Einstein-Hilbert action into a path integral, but the result is not a consistent quantum theory. That’s why there is (or was) much joy and merriment about string theory, just to mention the usual suspect.”

The result IS a consistent Wilsonian effective quantum field theory. What this means is that we can not use this theory to answer physics questions at arbitrarily high energies [the Planck scale and above], and that at the Planck scale the Einstein theory needs to be replaced by a more inclusive theory, such as string theory. Conversely we can use this effective quantum field theory to answer physics questions that only involve gravity at energy scales much lower than the Planck scale [ie. distances much longer than the Planck length]. This includes the post-inflationary history of the universe, and astrophysical black holes.

For an excellent introduction to Wilsonian effective field theories see:

-Effective field theory and the Fermi surface.

Joseph Polchinski (Santa Barbara, KITP & Texas U.) . NSF-ITP-92-132, UTTG-20-92, Jun 1992. 40pp.

Lectures presented at TASI 92, Boulder, CO, Jun 3-28, 1992.

Published in Boulder TASI 92:0235-276 (QCD161:T45:1992)

e-Print: hep-th/9210046

for Wilson’s review of his original work on the subject see:

-The Renormalization group and the epsilon expansion.

K.G. Wilson (Princeton, Inst. Advanced Study & Cornell U., LNS) ,

John B. Kogut (Princeton, Inst. Advanced Study) . Jul 1973. 126pp.

Published in Phys.Rept.12:75-200,1974. TOPCITE = 1000+

or see his Nobel lecture reprinted in Reviews of Modern Physics

for a textbook treatment of Wilson’s theory see:

-Quantum Field Theory and Critical Phenomena

-by Jean Zinn-Justin

-Oxford University Press, USA; 4 edition (July 17, 2002)

For treatments of Einstein gravity as an effective quantum field theory

see:

-Introduction to the effective field theory description of gravity.

John F. Donoghue (Massachusetts U., Amherst) . UMHEP-424, Jun 1995. 26pp. Talk given at Advanced School on Effective Theories, Almunecar, Spain, 25 Jun – 1 Jul 1995.

e-Print: gr-qc/9512024

– Quantum gravity in everyday life: General relativity as an effective

field theory.

C.P. Burgess (McGill U.) . Nov 2003. 57pp.

Published in Living Rev.Rel.7:5,2004.

e-Print: gr-qc/0311082

– Les Houches lectures on effective field theories and gravitational

radiation.

Walter D. Goldberger (Yale U.) . Jan 2007. 45pp.

To appear in the proceedings of Les Houches Summer School – Session 86: Particle Physics and Cosmology: The Fabric of Spacetime, Les Houches, France, 31 Jul – 25 Aug 2006.

e-Print: hep-ph/0701129

The result IS a consistent Wilsonian effective quantum field theory….…which is a non-rigorous mathematical construct that might not describe Nature’s QG particle spectrum correctly.

Kea – October 6, 2007 writes

” The result IS a consistent Wilsonian effective quantum field theory….

…which is a non-rigorous mathematical construct that might not describe Nature’s QG particle spectrum correctly”

Actually, modern rigorous constructive quantum field theory is based on Wilsonian methods. For a textbook treatment see:

-Quantum Physics: A Functional Integral Point of View

by James Glimm and Arthur Jaffe

(Springer 1987).

Also if you look at the Polchinski article that I referenced above, he includes, for example, references to the rigorous construction of the continuum limit of the Gross-Neveu model by Gawedzki and Kupiainen, and to the rigorous construction of the continuum limit of D=4 Non-Abelian gauge theories by Balaban, each using Wilsonian methods.

However, the real use of Wilsonian effective actions is to discuss the infrared physics of theories in a way that does not presuppose knowledge of their eventual ultraviolet completion. This is important in theories such as quantum electrodynamics, or quantum Einstein gravity, where the ultraviolet completion is unknown but the observable physics we wish to explain [atomic spectra, gravitational radiation from pulsar binaries, etc…] occurs deeply in the infrared.

And gravity is particularly simple in this regard. Ignoring the cosmological constant [which needs to be fine tuned to be hierarchically close to zero to have a large classical spacetime], the leading Wilsonian operator is the Einstein-Hilbert density which is an irrelevant [in the Wilsonian sense] operator; ie. one whose effects scale to zero in the infrared as a power. Since the question of the low-energy spectrum is an infrared question, it is a question governed by physics we have under control. The infrared divergences in the theory were calcuated by Weinberg 40 years ago and are innocuous. See:

-Infrared photons and gravitons.

Steven Weinberg (UC, Berkeley) . Jun 1965.

Published in Phys.Rev.140:B516-B524,1965.

40 years ago? And yet to be observed? How many more decades should we wait before we begin to question the assumption that

the low-energy spectrum is an infrared question, a question governed by physics we have under control.Dear anomalous cowherd, if you have a consistent quantum field theory, you can use it to derive a low energy effective action. The reverse is generally not true: if all you have is a low energy effective action, you can not derive a consistent quantum theory from it. You can exclaim “damn the torpedoes!” and run your classical theory through the machinery anyway, but then you are just hoping (and maybe praying) that the result will bear some resemblance to the true underlying quantum theory. It’s a simple matter of information loss as you integrate out high energy modes on the way down to your effective action.

For instance, take classical hydrodynamics and “quantize” it. Does the result make any sense? Never mind “high energy hydrodynamics”, just ask yourself if your “quantum hydrodynamics” tells you anything about low energy hydrodynamics that wasn’t already in the classical theory.

nigel, are you drunk?

Kealbeit not directly related to Category Theory or M theoryIf the regime with which strings are considered, what use any phenomenological approach, if one can not find some relevance to the experimental processes currently being used?

It is not without my thinking(?) that such “genus figures” were seen in my mind, in relation to Euler’s approach?

Early, early, universe now in expression?

You are directing your attention backwards to the substance of reality but also explaining the universe at the same time.

Point particles seen in a different light?:)

I just think such “reductionist views” even within the realm of the quantum world, had to have a connection to the experimental one you use in the colliders?

Mathematically, it is an engagement right now with topoi, yet, early universe thinking had to also be connected to Guth’s inflationary Universe. Conformal field theory describing the inside of a blackhole? A fifth dimensional perspective? Gravitons in the bulk perspective now.

More on name.

Hi all,

I am happy that the discussion here is above my head – if not my understanding, at least my capability of arguing about it or teaching it. Because I can leave the discussion going without worrying about it. Kea, you are welcome to take over on leading the discussion here if you feel like it.

GW, please do not discuss the correlation between the opinions on physics issues and the ingestion of liquids by others here, we can have a post about that some other time.

Plato, the paper you link to appears a bit outdated, I think. The RHIC has called off early claims of being able of producing black holes, IIRC.

Cheers,

T.

And, Plato, maybe you can find some answers on the (absent) connection between string theory and experimental particle physics by reading a couple of excellent books, of which I especially recommend the first: “Not Even Wrong”, by Peter Woit, and “The Trouble with Physics”, by Lee Smolin. You can find a link to Woit’s blog in my blogroll column. My excuses in the quite likely event that you were already aware of them.

Cheers,

T.

T,

As a lay person I have been at it a couple of years and watched the progression of “not even wrong ” by instigating a discussion between Lumo and Peter.

Unfortunately it descended into something other then what I had desired. Thoughts and issues did materialize even in that atmosphere, which allowed me to see different sides to it. It was never my place to be judgemental about characters of people, yet it does have it’s effect.

It does not mean there are no things to learn, on the contrary, I have read so many books at this point, it is always trying to understand the latest that I try and keep really close to what is happening.

You have been a big help and those involved in science by committing themselves to spreading what you do know and helping others. To that effect Quantum Diaries was a good experiment and others have followed. Wonderful.

Outdated the dual blackhole issue written, I was following Steinberg, Giddings and Bee, Clifford of Asymptotia has been extremely helpful. I had been with the “superstringtheory.com” while it’s forum was developing, as well as her site.

Thanks for your time Tammaso.

Marni states in her first sentence of the last pargraph:

“In three dimensions a basic arrow may be replaced by a cube.”

There may be alternatve representations?

Let the corners of the cube be vertices or nodes.

1 – 4 vertices may be a face.

May there be strategic [or virtual] diagonal edges to opposite nodes?

2 – 8 vertices may be a volume.

May there be strategic [or virtual] diagonal edges to opposite nodes?

3 – Rather than use edges, consider arcs that may allow representation with sphere [=ball], cylinder or torus?

This may be more consistent with curved space-time?

38. Guess Who – October 7, 2007 writes:

“if you have a consistent quantum field theory, you can use it to derive a low energy effective action. The reverse is generally not true: if all you have is a low energy effective action, you can not derive a consistent quantum theory from it…..”

“…..but then you are just hoping (and maybe praying) that the result will bear some resemblance to the true underlying quantum theory.”

We seem to have a terminology problem in our communication. Whenever you write “consistent quantum field theory”, what you seem to actually mean is “UV complete theory”. You are correct that a Wilsonian effective quantum field theory does not, by itself, determine its ultraviolet completion [in general its UV completion(s) may not be unique, or even be field theories]. But to do physics, from the Wilsonian effective field theory point of view, we simply don’t care about the nature, or the existence, of possible UV completions.

What we ARE discussing is whether a Wilsonian effective field theory can give a consistent, quantum mechanical, description of the physics at energy scales (well) below the cutoff scale of the Wilsonian effective action. The answer to this is yes, and well described in the reviews and texts cited above [try the Polchinski for an accessible introduction]. If this were not the case, then the Nobel committee made a mistake when they gave Wilson the prize.

The theory of the low-energy modes is just the theory of the low-energy modes. For example, Einstein gravity could arise from string theory, dynamical lattice space-time, whatever. Similarly the pseudoscalar octet of the strong interactions could arise from quark confinement in QCD, a Nambu-Jona-Lasinio model, whatever. But once we know the low-energy field excitations and (approximate) symmetries, we can use effective field theory methods to do consistent QUANTUM calculations of the low-energy behaviour, at scales well below the cutoff where other degrees of freedom are integrated out. And the quantum dynamics are important. For example, in the Effective Chiral Field Theory of Pions and Kaons, agreement with experiment for many processes depends on resummation of the “chiral logs” that appear in the diagrammatic expansion of quantum corrections in the theory. Calculability of these quantum corrections does NOT depend on any assumed UV completion of the Chiral Effective Theory.

For good reviews of calculation of quantum corrections in low-energy effective field theories see:

-Effective field theories.

Aneesh V. Manohar.

e-Print: hep-ph/9606222

-Effective field theories.

David B. Kaplan.

e-Print: nucl-th/9506035

-Chiral perturbation theory.

A. Pich.

Published in Rept.Prog.Phys.58:563-610,1995.

e-Print: hep-ph/9502366

38. Guess Who – October 7, 2007 writes:

“ask yourself if your “quantum hydrodynamics” tells you anything about low energy hydrodynamics that wasn’t already in the classical theory.”

That depends on the fluid. Think of Feynman’s theory of superfluid Helium.

anomolous cowherd

We might just have to agree to disagree. With regard to gravitons in particular, all we are saying is that it is debatable whether we

“know the low-energy field excitations”.Dear anomalous cowherd, I am aware of how EFTs are constructed. Let’s continue with quantum hydrodynamics, which I brought up for the following reason: yes, there is such a thing, but no, it is not just classical hydrodynamics naively plugged into a path integral.

To construct quantum hydrodynamics, you start from your knowledge of microphysics, include the conservation laws which you know about from the classical theory, and end up with something which can be shown to reduce to the classical theory in the appropriate limit. It is not just the classical theory naively quantized.

Generally speaking, the degrees of freedom of a classical theory can not be directly identified with the degrees of freedom of the underlying quantum theory. 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. Under very special circumstances (weak, dimensionless coupling), the two descriptions are so close that you can identify the classical theory with the tree level of a perturbative expansion of the full quantum theory. Even then, this identification is not trivial; you must run it through the machinery of renormalization. If your low energy theory is not renormalizable, this will obviously not work; the identification of degrees of freedom fails.

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.

In both cases mentioned so far, hydrodynamics and QCD, we have the advantage of knowing the actual underlying theory. Wilson’s well-deserved Nobel prize was for techniques suitable to handle this situation.

With gravity, we are not so lucky. The thing is notoriously not renormalizable. You can proclaim that GR is the effective low energy theory of quantum gravity, and if GR is the correct low energy description of gravity (we don’t really know this for a fact beyond the solar system scale) then fine, it’s true; but this does not imply that quantizing its degrees of freedom makes any more sense than quantizing the degrees of freedom of classical hydrodynamics. What you really need is a quantum theory which you can show reduces to GR in the appropriate limit. It is only with such a UV completion in hand that you can impose the matching conditions which relate the low energy theory to the high energy one. In other words, it is only with a UV completion in hand that the statement “GR is a low energy EFT” contains some non-trivial information. Without one, all it does is beg the question: low energy ETF of what?

To bring up the usual suspect again, it is fine to say that string theory (a quantum theory) contains gravitons, and it is fine to say that string theory reduces to GR in some limit. This is not synonymous with the statement that GR itself, a classical theory, contains gravitons, i.e. quanta.

If you are very, very lucky, naively quantized GR has a UV fixed point. Then the (almost) straight identification of high and low energy degrees of freedom using Wilsonian techniques is still possible. This is the basis for the work by Reuter et al. which I mentioned way up in this thread (though the idea is originally due to Weinberg, if memory serves). But you don’t know that there really is such a UV fixed point. You may hope and pray for one (while the likes of Distler will tell you that there is no chance in hell of one), but if you could actually prove it you’d already be on your way to Stockholm, along with Reuter and collaborators.

“With gravity, we are not so lucky. The thing is notoriously not renormalizable. …” – Guess Who

You’re enelessly repeating the mainstream gibberish which has led nowhere. The claim that quantum gravity is not renormalizable has problems in that it depends on the assumption that gravitons possess gravitational charge, which isn’t an experimentally confirmed fact. So you’re inventing a problem with “quantum gravity” on the basis of prejudiced assumptions, but you’re not pointing out your prejudiced assumptions…

Look at it another way, renormalization in known QFTs like QED consists of using a running coupling, i.e. a changed value of the relative charge as a function of collision energy or distance between particles, when the collision energy lies between the IR and UV cutoff energies. The

physical explanationfor such a varying charge is that there is screening by radial polarization of vacuum charge pairs around the core of a particle. Pair production in an electric field results in virtual positrons moving on the average closer to a real electron core (because they are attracted) than virtual electrons. Hence, this vacuum polarization has a radial electric field which opposes the core charge, and thus shields the particle’s charge, and you get more cumulative shielding as you move to greater distances (until you get to the IR cutoff, beyond which the electric field is too weak to cause pair production, so beyond the IR cutoff the remaining observable electric charge remains constant).Quantum gravitation can’t have this

physical mechanismfor renormalization; there is onlyonetype of gravitational charge, mass.All masses fall the same way in a gravitational field. Quantum gravity cannot therefore be renormalized by the physical mechanism of virtual charges moving in opposite directions in a gravitational field, so the quantum gravity charge can’t be renormalized. (You can’t even try to get around this using “dark energy” plus armwaving arguments, because the repulsive effect that produces is only significant over cosmologically large distances.)The entire role of people like Guess Who to these discussions, beyond ad hominem aattacks, is that of physics dismissal. If you want to propose that renormalization has anothing to do with quantum gravity, first give us a proposed mechanism akin to that in QED for renormalization which isn’t complete trash, and then you can start to construct some mathematics.

“What you really need is a quantum theory which you can show reduces to GR in the appropriate limit. … To bring up the usual suspect again, it is fine to say that string theory (a quantum theory) contains gravitons, and it is fine to say that string theory reduces to GR in some limit. This is not synonymous with the statement that GR itself, a classical theory, contains gravitons, i.e. quanta.” – Guess Who

So you want a quantum gravity that reduces to GR in the classical limit, but that doesn’t contain gravitons. And you call me drunk!

nigel, the “gibberish” that naive quantization of GR yields a non-renormalizable result (barring the unproved existence of a UV fixed point) is a fact. That gravity has self-interactions, which implies gravitons carry “gravitational charge”, is a fact.

These are not “invented” problems. They are just problems which you do not know nearly enough to understand.

You keep repeating your silly analogy with QED, apparently oblivious to the fact that QED is an Abelian field theory, whereas the closest QFT analogy to GR is Yang-Mills theory, in which – contrary to your evident misguided belief – the gauge fields DO carry charge. Try looking up “confinement” and “anti-screening” sometime when you feel like learning something. Note the “anti” in “anti-screening”.

You have been told repeatedly that “gravitational charge”, to the extent that you can talk of such a thing, is energy, not just mass. Yet to persevere even with this basic misunderstanding.

As usual, I have no idea why you keep this up. It is evident to anyone who knows anything about physics that you simply have no clue as to what you’re talking about. None.

Guess Who: the charges carried by Yang-Mills vector bosons like the W’s and the gluons are

notanalogous to mass.For quantum gravity, you have a situation with one gauge boson and one quantum gravity charge, which looks closest to the abelian U(1) having one charge and one gauge boson. Yang Mills SU(2) has 3 gauge bosons and 2 charges; SU(3) has 3 charges and 8 gauge bosons. In any case, charges are distinct from the field quanta. Field quanta (gauge bosons) can carry charges but that does not make them the same as the charges; the gauge bosons are exchanged between charges.

Now, the

whole reason for antiscreening in SU(3) is that each gluon carries a color charge and an anti-colour charge, so that vacuum polarization effects strengthen the net strong force with increasing distances (up to a limit), offsetting electromagnetic force and confining the quarks and giving a asymptotic freedom over a small range (the hadron size).This can’t occur with gravitons, where you have one charge and one type of gauge boson.If you want to pontificate about Yang-Mills theories and anti-screening, consider where the energy comes from which makes the colour force get stronger with increasing distance. It’s clearly coming from the energy being lost from the electromagnetic field (screened energy). Conservation of mass-energy tells you that if a stream of electromagnetic gauge bosons flowing towards an electron is being screened at a short distance, the energy is being transformed. Pair production tells you it’s being converted into virtual particles. Pair production will create quark-antiquark pairs at sufficiently high energy, and these are accompanied by gluons. So the energy of screened out electromagnetic gauge bosons is partly converted into gluons, which mediate the strong force. This suggests a more physical route to unification than supersymmetry, but out of respect for Kea’s post I’ll resist responding to any more nonsense from people who don’t want to learn physics.

(I’m referring to GW, not anyone else.)

At this point I feel compelled to ask the blog owner what course of action he recommends when somebody repeatedly posts blatant nonsense and presents it as fact. Just ignore it and let the innocent be suckered or keep pointing out the most glaring absurdities?

Hi GW,

the policy here is that all comments stay and are welcome, as long as they are not aimed at insulting or targeting anybody, and they do not contain profanities in a measure above a certain threshold I can’t exactly quantify. I am glad if erroneous claims get countered, but I pick my battles and do not always choose to do it.

Science might not progress through endless arguing, but controversy is everywhere you look, and Physics is no exception. I tend to avoid forcing my point of view on scientific matters, and rather try to provide innocent readers with the keys to discern, if I have a chance. Presenting Physics as wholly uncontroversial is not what I aim at…

Of course, politics is a totally different matter, as you know 😉

Cheers,

T.

Controversy about the unknown is one thing, “factual” statements about what’s known another. If somebody posts a claim that the top quark mass is 10 MeV, or that CERN is located in Batavia, do you let that stand uncountered? The statements posted by nigel are equivalently grotesque misrepresentations of known facts. They are not controversial stands about whether a theory is a correct description of the physical world or about how it should be extended. They are simply, trivially, embarassingly incorrect statements about the workings of existing theories. I can see no legitimate reason to let such bizarre misrepresentation go on. But, it’s your blog.

GW, I agree with you in general, but there are differences between arguing about an experimental measurement and a theory. The name “theory” is revealing: it describes things, but it is not truth “by construction”, i.e. just because it’s there.

And then, there are differences in the level of my personal competence in arguing about general relativity or top quark physics. I can do better in the latter case, which does not mean I would not be able to discuss the correct way to interpret anti-screening and yang-mills theories either: but I like to pick my battles, as I said.

Finally, three other miscellaneous reasons:

– arguing about some things triggers me more than about others

– I enjoy arguing with some guests more than with some others

– if somebody is into a discussion with somebody else, I let things evolve naturally here, if they do not degenerate.

I guess what I am trying to say is that this blog is not a bible book on Physics: it may contain inconsistencies, useful as well as useless comments, endless arguments, wrong statements. It is all part of the entertainment value. If I report on something, I try to be accurate – and may fail. Others take their own responsibility. ARivero said he likes this blog because it is a-hierarchical. This is part of the recipe.

Cheers,

T.

“The statements posted by nigel are equivalently grotesque misrepresentations of known facts.” – GW

Since Tommaso agrees with GW to some extent, can I make clear GW consistently claims that because GR states that gravitational fields have energy and are a source of gravitation, that means gravitons must have mass (gravitational charge) as well as energy.

In the SM,

allparticles with mass acquire that mass via the Higgs field. There is absolutely no experimental, factual basis for claiming that gravitons behave differently and have intrinsic mass. You can get the whole of the classical approximation for GR without them. The simplest theory which fits the facts is that gravitons pick up mass from the Higgs field, just like all the known particles in the SM do.GW’s claim may be correct if he states “according to certain existing mainstream prejudices about gravitons acquired from a misunderstanding of GR” beside each of his claims. He/she does not, and persistently confuses the facts for speculation, and then responds with insults when his confusions are pointed out. Thanks.

TD, regarding theory vs. experiment, that’s why I emphasized that nigel’s statements “are not controversial stands about whether a theory is a correct description of the physical world or about how it should be extended. They are simply, trivially, embarassingly incorrect statements about the workings of existing theories.”

To claim that GR is not the correct classical theory of gravity can be a legitimate stance (if the proposed alternative is not insane); to claim that GR is an Abelian field theory (!) is not. The latter is a statement about a known property of the theory. Such statements are unambiguously right or wrong, and anybody who knows the theory knows whether they are right or wrong, independently of whether the theory is a correct description of physics. They are statements about properties of the theory, not of physics.

This has nothing to do with hierarchies (which I thoroughly dislike myself), it’s just about correctness.

Arrrgh! Nigel, for the umpteenth time: “gravitational charge” is not just mass. You are confused by a ridiculous analogy with QED which you picked up in some popular book. The closest gravitational analogy to an ordinary field theory is a Yang-Mills theory based on the Poincare’ group, which is anything but Abelian. That means gravitons carry gravitational chargeS (no, if you insist on the QFT analogy there is not only one kind of “gravitational charge”) and self-interact. It does not in any way imply that gravitons are massive. That’s just your immense confusion talking.

Why, oh why, do you keep doing this?!?

“The closest gravitational analogy to an ordinary field theory is a Yang-Mills theory based on the Poincare’ group, which is anything but Abelian.” – Guess Who

I wrote in comment 15, that the Yang-Mills group SU(2) seems to be the gauge group for not just weak interactions via isospin, but also for electromagnetism and gravity. You don’t like that theory (presumably because it has experimental evidence behind it) but now you are claiming that yes, a Yang Mills group is the closest thing to gravity. Presumably the Yang Mills group you have in mind is not SU(2), but something more exotic like an unproved, no-evidence GUT.

However, you then draw nonsense conclusions:

“That means gravitons carry gravitational chargeS (no, if you insist on the QFT analogy there is not only one kind of “gravitational charge”) and self-interact. It does not in any way imply that gravitons are massive.”

Shows the problem you are creating. You are not building rigorously on solid facts, you don’t predict the coupling constant for gravity to check your theory, you just use prejudice and assumption backed by arrogance and pleading. If spin-2 gravitons have solid experimental proof, you don’t have a basis to argue theoretically from that theory’s landscape of possibilities, if you claim to be scientific. You are saying that some unknown quantum gravity theory looks good although it has no evidence, and using that claim to dismiss facts which do have evidence that you don’t want to know.

typo: “If spin-2 gravitons have

nosolid experimental proof,…”In short, nigel just admitted to not having even ever heard of the Poincare’ group, which he assumes to be “an unproved, no-evidence GUT”, yet he keeps posing as an expert on field theory, general relativity and quantum gravity.

This is just too much.

See, Nigel, I tried to explain I did not want to get involved in your argument with GW, and still you two are dragging me in!

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.

Sorry Tommaso! GW now reveals he was thinking about a 10-dimensional Lie group (I’m not expert on the Poincare group, nor am I expert on many things in life), rather than a GUT. It makes little difference as far as I’m concerned, and BTW, I’m not “posing as an expert”, merely stating facts that I have backed up with calculations (unlike the “real” mainstream “experts” in quantum gravity…). I’ll stop reading this thread now and won’t comment any more despite what GW says.

48. Guess Who – October 8, 2007 writes:

“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.

48. Guess Who – October 8, 2007 writes:

“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.

Dear anomalous,

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.

Consider, for example, recent evidence from the S5 run at LIGO, that failed to detect GWs when they really ought to have been seen.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

NOTbeen seen. That they haven’t is thus not surprising (albeit disappointing).anomalous cowherd mentions “… Coleman’s proof that the Sine-Gordon theory is quantum equivalent to the Thirring model …”,

and

Tommaso said (I think applicable to that, but I could be misreading) that he “… need[s] to read some documentation …”.

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

Dear anomalous cowherd, ChPT is a low energy effective theory *of 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 :).

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.

Re #44: Sorry for the slow reply. Good insights, Doug. As you know from AF, we like to play with many dualities and trialties represented by symmetries of simple cubical graphs.

anomolous said:

As another example, consider Coleman’s proof that the Sine-Gordon theory is quantum equivalent to the Thirring model …. This is only possible if the Sine-Gordon soliton [a collective mode] is itself quantized.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!

To: 70. Guess Who – October 9, 2007

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.

> I take this text-book material as understood and stipulated.

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?

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

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.

75. dorigo – October 10, 2007 writes

“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

That’s quite ok Anomalous! Actually, weekends should be left for time away from the keyboard 😉

Cheers,

T.

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