Don't mess with the pass integral

[Micha]

Here is a clear and I think deep article from Lubos Motl about the path integral:
http://motls.blogspot.com/2010/10/dont-mess-with-path-integral.html

Looking forward to Marcus reaction.

 

[atyy]

I do like very much many of Motl's physics posts, but Causal Dynamical Triangulations is not intended to be a discrete theory of gravity. It is hoped to be the low energy limit of some continuum theory. It is of course unclear if this is the case, but some candidates are Asymptotic Safety, scale invariant gravity, or Horava-Lifgarbagez gravity (it's also not clear that any of those are UV complete quantum theories of gravity), but none of those are discrete. In fact, the second paper below cautions comparing their results with spin foams because they assume a continuum limit, whereas the most common spin foam formulations are discrete. Nonetheless, I think Lubos is right that the censoring in CDT is not properly derived, and this is pointed out in the CDT papers too. But basically, they have an interesting computational result, whose meaning is intriguing but unclear.

http://arxiv.org/abs/0906.3947
http://arxiv.org/abs/0911.0401

 

[tom.stoer]

Of course Lubos is always right in his blogs b/c he never writes down the issues where he is wrong.

It is not the case that one can always start with a classical action on a smooth manifold and then "derive" the PI or simply use this action to write down the PI. There are two problems:
1) I don't think that one can derive the PI containing the action from the Hamiltonian PI if the Hamiltonian is not a Gaussian in the momenta b/c then the p-integral can't be done.
2) I don't even think it's always necessary to derive the PI

Look at classical mechanics: you do not "derive" the action. You just write down an action and derive some predictions; if they fit to the experiments you found the correct action. Fine.
You can do the same thing in QM. You simply write down a PI. If it works you found the correct one. You need not care about the derivation (of course in most cases you will start with a classical action but that's not required by the formalism).

Ontologically it might even be nonsense to start with a classical theory and quantize it. Perhaps one should turn things round, write down a quantum mechanical PI and "classicalize" it.

Of course CDT (and many other theories) are somehow ad hoc. Most theories are not proved (nor disproved) mathematically but physically via experiments. As soon as this happens nobody cares about the derivation, the axioms and the proofs (the QCD PI is ill defined but it works in the perturbative regime; there it has produced reasonable physical results plus some Nobel prizes; funny, isn't it?)

One should ask Lubos to write down the full PI for string theory; afaik this is not known so far.

 

[marcus]

BTW some people seem to think there is only one kind of discreteness in QG and that it always means the same thing, but as Atyy just showed us there are different kinds. In CDT the triangulation is regularization discreteness. It is like doing the conventional particle path integral, with paths divided into piecewise linear segments.

The test of a good regularization for the path integral is (I think) pragmatic. The piecewise linear paths are a tiny, perhaps quite unrepresentative, subset of all paths---and the triangulated geometries are a small jagged subset of all geometries. The proof of the pudding is in the eating, as the saying goes---the important question is do you get interesting results.

With LQG (I think, tell me if you disagree) there is inferential discreteness. One imagines making finite measurements and inferring from them some predictions and then making more finite measurements of what was predicted. The information is discrete.

One is not interested in what space "IS" or what it is "made of". One is interested in how geometry responds to measurements---and how measurements are correlated. So in this sense, Lqg is an inferential theory (in the making). It is information centered.

There is no reason, for example, that its schemata cannot be locally Lorentz invariant.

I was watching the second Causets lecture (Fay Dowker and Rafael Sorkin) and I was struck by the fact that in Causets there is even a third kind of discreteness. Or seems to be. It seems not to be entirely analogous to the other two---the discreteness here seems to be more literal than in Lqg and Cdt. And yet the theory also has Lorentz invariance.

Well, that may be just a side comment--not immediately related to the path integral topic.

 

[inflector]

I'm in no position to judge Luboš' post on the merits but his tone certainly doesn't make me inclined to bother trying to figure out what he is saying.

I've run across his type in various arenas over the years and it has never been accompanied by actual deep expertise. I suppose he could be the exception but I find the bluster usually covers over some inadequacies or insecurities at the very minimum. People like him don't help their cause they harm it.

 

[atyy]

A different take on CDT:

http://golem.ph.utexas.edu/~distler/blog/archives/000713.html

 

[atyy]

I was watching the second Causets lecture (Fay Dowker and Rafael Sorkin) and I was struck by the fact that in Causets there is even a third kind of discreteness. Or seems to be. It seems not to be entirely analogous to the other two---the discreteness here seems to be more literal than in Lqg and Cdt. And yet the theory also has Lorentz invariance.

I just found out Zohren's supervisor was Dowker, even though his work was with Ambjorn and colleagues. http://arxiv.org/abs/0905.0213

 

[Micha]

Of course CDT (and many other theories) are somehow ad hoc. Most theories are not proved (nor disproved) mathematically but physically via experiments. As soon as this happens nobody cares about the derivation, the axioms and the proofs (the QCD PI is ill defined but it works in the perturbative regime; there it has produced reasonable physical results plus some Nobel prizes; funny, isn't it?)

How do the CDT people ensure that their path integral is unitary and does not contain
superluminal effects?
If these conditions are not met there is no hope that the theory is confirmed by experiment.

 

[tom.stoer]

I think they do not calculate anything that "propagates"; therefore there is no "speed" on these triangulations. Look at spacetime e.g. during inflation: it can expand with superluminal speed w/o causing any problems; speed of light applies only if something is propagating _on_ a predefined spacetime, but this is not what they are interested in (not yet)

 

[Micha]

What about gravitational waves?

What about the even more fundamental requirement of unitarity?

 

[tom.stoer]

They do not look at gravitational waves. A gravitational wave is an artificial object b/c it splits the geomeztry into a background (defining a metric and a light cone) and a wave that propagates on it. The strength of CDT is that it does not introduce this split.

Regarding unitarity I am not sure. Usually you need unitarity if you want to calculate something like <out|U|in>; then U must be unitary. But I do not know how CDT defines these states |..> and how these matrix elements are related with the path integral Z.

 

[marcus]

I just found out Zohren's supervisor was Dowker, even though his work was with Ambjorn and colleagues. http://arxiv.org/abs/0905.0213

To expand on your comment, there is a lot of sharing between Causets and CDT. After Joe Henson had done Causets for some years and co-authored with Sorkin and with Dowker, he took a postdoc with Renate Loll at Utrecht. Then moved on to Perimeter where he has done both. Loll has had Dowker come to Utrecht to give seminars. David Rideout has programmed cluster/supercomputer tools for several QG models including both of these (so I understand from Steve Carlip). Perhaps Causets and CDT have the kinship of both being "path integral" or sum over histories (SoH) approaches.

Zohren certainly has co-authored a lot on CDT with Loll, Ambjorn et al. I hadn't realized that he was a Dowker PhD. You've probably noticed how Loll PhDs and former postdocs have crossed over into Asymptotic Safety too.

 

[Micha]

Great, so these guys do quantum gravity but they don't calculate the properties of gravitons. Regarding background independence have these guys ever heard of pertubation theory?

Unitary is a basic requirement of any theory which claims to be quantum mechanical. It just means that probabilities add up to 1. So it has to be there no matter how you write things down.

 

[tom.stoer]

Great, so these guys do quantum gravity but they don't calculate the properties of gravitons. Regarding background independence have these guys ever heard of pertubation theory?

It is a common misconception that in order to study quantum gravity one needs gravitons; neither gravitons nor perturbation theory is required. It's just the other way round: especially CDT and LQG try to avoid using gravitons as perturbative degrees of freedom defined on a background; that is the core message of background independence!

Unitary is a basic requirement of any theory which claims to be quantum mechanical. It just means that probabilities add up to 1. So it has to be there no matter how you write things down.

I think "these guys" know perfectly well what they are doing; I only said that "I am not sure ..." and that "I don't know ..."; please note the difference :-)

 

[Micha]

It is a common misconception that in order to study quantum gravity one needs gravitons; neither gravitons nor perturbation theory is required. It's just the other way round: especially CDT and LQG try to avoid using gravitons as perturbative degrees of freedom defined on a background; that is the core message of background independence!

Right, background independence is the new religion.

300 years of successful pertubation theory are thrown out of the window.

I think "these guys" know perfectly well what they are doing; I only said that "I am not sure ..." and that "I don't know ..."; please note the difference :-)

That's ok but I think it is not only a problem with you. :-)

I am following the blog of Lubos Motl for quite a while and I am convinced he is a top
shot physicist. You can check by scanning the archive what and with whom he has published. If you are really interested in fundamental physics I can only give you the advice to read his blog carefully and you will start to see things differently.

 

[tom.stoer]

Right, background independence is the new religion.

300 years of successful pertubation theory are thrown out of the window.

I studied QCD for years and I learned that perturbation theory is only valid in a certain regime; it is neither helpful to study the low-energy limit (confinement, bound states, hadron masses, form factors, ...) nor to study conceptual issues (gauge fixing ambiguities, ...). Unfortunately reading standard textbooks one could get a different impression :-(

Regarding Lubos: I agree that he is a very talented physicist, but I think he is rather arrogant and even he has certain blind spots. He knows a lot but he is certainly not Doctor Know. Regarding conceptual discussions I usually trust the experts in a certain field; as Lubos is not an expert in CDT ...

 

[Fra]

What I'm about to say isn't in defense of CDT specifically, it's rather a strong objection against the simplicity of reasoning, and in particular IMO confusing confidence in current QM formalism with the more important foundations of QM - that we don't yet understand - that may survive even into QG.

I was about to comment earlier when I skimmed Lubos text but decided not to as I've done it before to no good. IMHO he misses the most important and difficult questions in his analysis, and confuses the success of QM formalism for particle physics, with the success of QM formalism for a generic general measurement theory (general inference), that applies also to observer/system scenarious that we are forced into when discussing unification and cosmological models.

Unitary is a basic requirement of any theory which claims to be quantum mechanical. It just means that probabilities add up to 1. So it has to be there no matter how you write things down.

Well probability adds up to one by definition, but the question is more complicated than that IMO. That's not even an argument.

The question is what the physical meaning of "probability" is in nature! Or rather what purpose the measure we call "probability" is supposed to have, and wether the mathematics of classical logic, QM logic is even properly understood?

Now, there are some people that just don't care. They don't understnad or see why this is a relevant question.

But, some people may think that probability is really just a way to count evidence, or rate an expectation. In particular ina way that is subjective = observer dependent. Furthermore this process of counting evidence, seens as processing and storing information, and noting that these are physical processes, certainly suggests that the measures we THINK that probability answers to as per it's axiomatiation, MAY not be quite the most general.

And in the light in such reconsiderations, violation of unitarity really is not so strange, if with it you mean that the set of possibilities does not fact change at a significant rate relative to the input processing during the COURSE of the processing of data and computation of the measures.

Sure Lubos is knowledgeable but from my perspective his overly exaggerated confidence in current QM formalism without considering what it's supposed to mean, comes out silly.

It works but many people - including people often considered to be physicists - completely misunderstand the necessary yet somewhat counter-intuitive conditions and subtle mechanisms that make it work
...
how these fundamental features are being messed with in the "discrete" approaches to quantum gravity, among various other memes whose goal is to rape and distort the basic principles of physics.

Like Marcus mentioned with causets, there are much deeper ideas on discreteness, whos ambition is to even derive QM logic (rather than just "play with the PI"). And these are the things I find interesting, but in that context Lubos comments just makes it entirely clear to me who it is that is misunderstanding something - and it's not the guys Lubos thinks is trying to "rape and distort the basic principles of physics".

(I do agree that CDT is simple in the sense that a deeper attack could and should also explain the PI.) But Lubos comments generealizes his critique beyond sense.

/Fredrik

 

[Micha]

Regarding Lubos: I agree that he is a very talented physicist, but I think he is rather arrogant and even he has certain blind spots. He knows a lot but he is certainly not Doctor Know. Regarding conceptual discussions I usually trust the experts in a certain field; as Lubos is not an expert in CDT ...

The problem with "trust the experts in a certain field" is that it won't work if the whole field is flawed. That it is what Lubos claims and I would like to encourage you to seriously consider the possibility.

I agree Lubos is not an easy character. But I claim you would get pretty much the same answers from another top shot, but very polite physicist, which is Nima Arkani-Hamed.

 

[tom.stoer]

But Lubos comments generealizes his critique beyond sense.

as usual - unfortunately. I think he is blind in a certain sense: he is not willing to see that others are much more open minded regarding weaknesses of their approaches than HE is with HIS approach. I bet asking Loll regarding open issues in CDT you will get a list of questions; try the same with Lubos and strings ... (this could be an interesting experiment; who is willing to start?)

 

[Micha]

I studied QCD for years and I learned that perturbation theory is only valid in a certain regime; it is neither helpful to study the low-energy limit (confinement, bound states, hadron masses, form factors, ...) nor to study conceptual issues (gauge fixing ambiguities, ...). Unfortunately reading standard textbooks one could get a different impression :-(

I know a little bit of QFT too. Of course if you study it in the regime where the coupling is of order one like the low energy regime of QCD then pertubation theory breaks down pretty much by definition.

It works well in QED where the low energy coupling is week. As we all know gravity is weak in the low energy limit as well so pertubation theory should work here too.

In fact every textbook about GR derives gravitational waves as pertubations of the metric tensor around the flat space Minkowksi metric.

 

[atyy]

How do the CDT people ensure that their path integral is unitary and does not contain
superluminal effects?
If these conditions are not met there is no hope that the theory is confirmed by experiment.

If CDT is a low energy approximation to Asymptotic Safety, then the issue of unitarity is unresolved. http://people.sissa.it/~percacci/

 

[Micha]

as usual - unfortunately. I think he is blind in a certain sense: he is not willing to see that others are much more open minded regarding weaknesses of their approaches than HE is with HIS approach. I bet asking Loll regarding open issues in CDT you will get a list of questions; try the same with Lubos and strings ... (this could be an interesting experiment; who is willing to start?)

Physics is not a contest in who is more polite or more politically correct but about what is true. So this argument is a non-starter.

 

[tom.stoer]

The problem with "trust the experts in a certain field" is that it won't work if the whole field is flawed. ... But I claim you would get pretty much the same answers from another top shot, but very polite physicist, which is Nima Arkani-Hamed.

CDT is able to reproduce a macroscopic deSitter space; afaik no other QG approach was able to come as close to a reasonable classical limit w/o fine tuning. I think CDT will not be the holy grail in QG, but I am sure it is not flawed.

String theorist had their 25 years to try to find a reasonable theory and they claim that they will need some decades more; so we should be patient and let a few people work for a few years in the CDT field; time will tell.

 

[atyy]

To expand on your comment, there is a lot of sharing between Causets and CDT. After Joe Henson had done Causets for some years and co-authored with Sorkin and with Dowker, he took a postdoc with Renate Loll at Utrecht. Then moved on to Perimeter where he has done both. Loll has had Dowker come to Utrecht to give seminars. David Rideout has programmed cluster/supercomputer tools for several QG models including both of these (so I understand from Steve Carlip). Perhaps Causets and CDT have the kinship of both being "path integral" or sum over histories (SoH) approaches.

Zohren certainly has co-authored a lot on CDT with Loll, Ambjorn et al. I hadn't realized that he was a Dowker PhD. You've probably noticed how Loll PhDs and former postdocs have crossed over into Asymptotic Safety too.

Between CDT and AS the link was clear to me - both very good old fashioned let's take renormalization seriously quantum field theory approaches.

But causal sets is visionary and far out (Bilson-Thompson's the only other comparably adventurous attempt) - so I was very surprised.

 

[atyy]

The problem with "trust the experts in a certain field" is that it won't work if the whole field is flawed. That it is what Lubos claims and I would like to encourage you to seriously consider the possibility.

I agree Lubos is not an easy character. But I claim you would get pretty much the same answers from another top shot, but very polite physicist, which is Nima Arkani-Hamed.

We should note that the standard text on string theory begins by saying that Asymptotic Safety is not ruled out.

Lubos has told me he thinks CDT cannot be an approximation to Asymptotic Safety, but I didn't quite understand the reason. He also admits that AS is not ruled out, although many arguments, especially about black hole entropy, suggest that it cannot work.

I do agree the link between CDT and AS is handwavy, so Lubos has a point. (I think Bendetti's recent results suggest that Lubos is right.) But historically, people working on AS have taken CDT results as suggestive that AS can work.

 

[tom.stoer]

It works well in QED where the low energy coupling is week. As we all know gravity is weak in the low energy limit as well so pertubation theory should work here too.

In fact every textbook about GR derives gravitational waves as pertubations of the metric tensor around the flat space Minkowksi metric.

Standard textbooks do not care about quantizing gravity. Perturbation theory works only if you either restrict to tree graphs or if you accept an unphysical cutoff that spoils all symmetries of the quantum theory. "Low-energy" in the classical sense is no longer applicable in QG or QFT if you go beyond tree-level.

I would say that the lessons we have learned in the last 50 years is that perturbative quantum gravity is nonsense.

 

[tom.stoer]

Lubos has told me he thinks CDT cannot be an approximation to Asymptotic Safety, but I didn't quite understand the reason.

Can you tell us more about these ideas?

 

[Micha]

Standard textbooks do not care about quantizing gravity. Perturbation theory works only if you either restrict to tree graphs or if you accept an unphysical cutoff that spoils all symmetries of the quantum theory. "Low-energy" in the classical sense is no longer applicable in QG or QFT if you go beyond tree-level.

I would say that the lessons we have learned in the last 50 years is that perturbative quantum gravity is nonsense.

Ok, great, so pertubation theory only works on tree level. Feynman should give back his Nobel Price and Feynman graphs never existed.

You should have made a good argument specifically for gravity. But unfortunately you didn't and wrote about QFT in general.

 

[atyy]

Can you tell us more about these ideas?

Here was his comment:

https://www.physicsforums.com/showpost.php?p=2312345&postcount=42

Edit: OK, now I see, his point is that the "causal" restriction is not derived in a principled way from any putative AS action.

There is a competing, less ad hoc approach to computational AS than CDT http://arxiv.org/abs/1002.0813

My own suspicion that CDT and AS aren't going together comes from http://arxiv.org/abs/0911.0401 where the spectral dimension only seems to match that from AS in a certain spacetime dimension. On the other hand, the types of fixed points that can exist do depend on dimension, so who knows?

Wouldn't it be ironic if CDT turned out to be Horava-Lifgarbagez?

 

[tom.stoer]

Ok, great, so pertubation theory only works on tree level. Feynman should give back his Nobel Price and Feynman graphs never existed.

You should have made a good argument specifically for gravity. But unfortunately you didn't and wrote about QFT in general.

Of course I was talking about quantum gravity. In standard QFT perturbation theory can make sense, but even there we have to be careful: given that each individual term in the perturbation series is finite (after renormalization) there is no reason why the perturbation series as a whole shall converge. That means that on a fundamental level the whole perturbation series is flawed; again this is something which is omitted in standard textbooks.

Let me repeat my last sentence: Perturbative quantum gravity is nonsense.

 

[tom.stoer]

@atyy: I don't think that there is a problem with the "causal restriction". This is true if you do ordinary QFT on a given background. But that's not the case; you try to construct the basic building blocks and there is no reason why you should use a building block X and not use a building block Y. Look at ordinary QFT: you just select a few fields (scalar, spinor, vector), write down the PI and check if it works. Of course you made a selection, but in the end nature will tell you if the selection was correct.

I see this issue of "local causal restriction", but I don't think it's an a priory issue for CDT in the narrow sense. What I coul imagine is that it becomes an issue as soon as you put matter on the triangulation and study its propagation.

The concept of causality is tricky in these discrete models and I agree that CDT may be too limited in order to study it properly. Let's focus on LQG (even if we know that according to Lubos it is flawed, too :-)

In LQG (or in general in any approach based on graphs) one has to distinguish between local and global causality. Local causality means that there is a local "nearest neighbor law" which says that during one "clock tick" only connected vertices are dynamically related. In LQG this is satisfied; the Hamiltonian respects such a local causality. But at the same time there could be non-local effects (where locality is now defined with respect to the existing spin network). Let's assume that you have a spin network which is dual to a triangulation; of course it's possible to construct a "non-local" link connecting two vertices which are separated by a huge number of cells (according to the triangulation). This "non-local" link does not violate local causality as long as it is guratantueed that it was created via local interactions. If the Hamiltonian produces such a non-local link connecting two vertices which are separated by 1000 cells during 1000 clock ticks everything is fine. So the existence of such non-local links doesn't automatically imply that there theory is flawed.

Of course there is a further requirement, namely that these non-local links must be dynamically supressed simply b/c we do not observe them in nature! There are no shortcuts from here two the Andromeda galaxy!

That means the theory (even if it allows for non-local links) must produce an emergent global causality which fits to the observed causality on large length scales. The whole concept of light cones doen't exist locally. It emerges at largeer scales and of course it must fit to what we observe in nature. That means that what we call "causality" is "global causality" which emerges from the dynamics of the theory and which is not an ingredient of the theory.

The problem with CDT is that it may be a too narrow framework as it focuses on triangulations which can only be defined using an underlying concept of a manifold (of which are at least not independent from these concept). In CDT one cannot study non-local links; they simply do not exist. Every triangulation is topologically a three-space whereas in LQG based on spi networks it cannot even be guarantueed that the graph is dual to a 3-dim. triangulation. So in LQG (or in any other theory based on graphs) the consistency check is not so much that the resulting long-range geometry is Minkowski or something like that, the check is if the long-range limit is a 3-geometry at all!

My conclusion is that CDT is only a restricted calculational tool which sits somehow in between of the concept of smooth 3-manifolds and an underlying discrete structure. That does not mean that CDT is wrong, it only means that CDT does not provide a deeper understanding of conceptual issues. Look at QCD and the non-relativistic quark model (based on constituent quarks). Of course this model is "wrong" in some sense, but in a certain regime it generates reasonable results. It serves as a calculational tool which incorporates some principles from full QCD (color, flavor) but which has somehow "integrated out" current quarks and gluons. In that sense CDT uses a ceratin concept of causality w/o explaining the emergence of this principle.

 

[S.Daedalus]

Does a discrete background geometry automatically imply the 'censoring' of certain histories? Intuitively, it would seem to me that one can just as well define a path sum across finitely (or countably infinitely) many paths that respects unitarity; the regular path integral then would be merely an approximation -- like taking Feynman's checkerboard to be fundamental, and considering the limit of vanishing spacing as a sort of macroscopic view.

A bigger potential problem would then be the cancellation of unphysical histories, though even here it seems that it might be the case that on average, you can find a history to cancel every sufficiently unphysical one.

In the end, however, I'm much more sceptical of a continuous spacetime -- that we shouldn't be able to model with the greatest supercomputers accurately what happens in the smallest regimes is something that is just too weird for my taste (and I generally think I can stomach a lot of weird).

 

[Fra]

I won't interfere with the LGQ or CDT discussion as I see there are still other options, but to just att briefly my view of how discreteness (in some sense) does censor histories.

Say consider how a given computer can compute an expectation, then obviously the summation is not over the set of all possible mathemtical histories, but only over the set of the encodable and distinghishable histories; ie the actual computation must be physicall possible!

The big confusion I see, is that we imagine all sorts "mathematical possibilities" "mathematical universes" but the only sensible way to construct an expectation is to count only the distinguishable possibilities. And what is physically distinguishable, is constrained by the nature of the computational system and it's memory; in particular computation systems and finiteness of memory.

A physical interaction can then loosely speaking be abstracted as two communicating computers, and obviously all notions of expectations or probabilities msut be evaluated with respect to a comptuer.

Some discrete ideas, mean that the physical action of a system, is constrained in similar ways to the discrete and finite nature of the information it encodes.

In a sense, each computer is it's own background. And any computers communication with another one can only be measured, counted and rated with respect to yet another one.

In these sense, it's just not acceptable to talk about probability and continuums as if it's obvious what it means. It's not, IMHO.

/Fredrik

 

[tom.stoer]

My idea was not to interfere with other approaches but simply to explain that there is not one unique concept of causality or locality. It should be clear that causality (in the macroscopic sense) is emergent (!) in LQG. So talking about locality, causality and light cones in the LQG framework is meaningless. Therefore it is not appropriate to pick some QFT / string theory concepts and apply them to other theories (LQG, CDT, ...).

It's not that other theories violate these "well-established" concepts, but that these concepts are valid only in a restricted class of theories. Theories not belonging to this class are not flawed but just different.

That was my main point!

 

[MTd2]

r. But I claim you would get pretty much the same answers from another top shot, but very polite physicist, which is Nima Arkani-Hamed.

If you read Jacques Distler's blog, you will know this is not true. Even other string theorists used to disagree strongly with him when it came to areas of string theory which lacked research.