Don't mess with the pass integral

[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.

 

[atyy]

@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 wasn't thinking about a problem with the causal restriction per se, but in the context of CDT being an approximation to AS.

 

[Fra]

My idea was not to interfere with other approaches

Mmm given the first line in my previous post, I don't know if you refer to what I wrote? If so, my comment was not directed to you, I was more having S.Daedalus post in mind, commenting on how discrete infomation process influence "counting histories".

I just made the statement that My comment was not interfering with the LQG discussion here, since my thinking questions the LQG abstractions which has been discussed before it was better to state that my comments doesn't refer to LQG or CDT, it's more referring to the general construction of physical measures, rather than mathematical ones.

/Fredrik

 

[tom.stoer]

Fredrik, it's OK; we were both too polite :-)

 

[Micha]

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

Ok, so that is a pretty strong claim. Where is the reason?

 

[tom.stoer]

b/c all calculations in perturbative gravity over the years failed due to several reasons

1) gravity is (by power counting) not perturbative renormalizable
2) perturbative gravity violates background independence
2b) perturbative quantization uses a fixed causal structure which cannot be subjetc to dynamical changes by construction; this is unphysical
3) all approaches to QG use some additonal input (strings, SUGRA, LQG, ...) or method (AS, CDT)

Clearly the third point is a response the 1+2). Perturbative non-renormalizability is a hint that for UV completion some essential input is missing. So either you complete the theory by introducing new concepts (like in SUGRA - which has not been proven to be well-defined, but were one has at least strong hints for finiteness of certain SUGRAs in D=4), or you use non-perturbative techniques.

The difference becomes clear in AS where a non-gaussian fix point is assumed. This is a generalization of asymptotic freedom (all couplings tend to zero in the UV).

 

[marcus]

Just a footnote on what was just said. The grav field is unusual in that it determines the causal relations among events. (4d geometry determines lightcone structure.)

Suppose you fix a background, then you are committed to a web of causality. Now you perturb, by superimposing a ripple on the background. Previously assumed causal relations are now strictly invalidated.

Perturbative treatment of gravity is in a fundamental way unphysical compared to perturbative treatment of other fields.

 

[tom.stoer]

I'll add this to my list in post #40

 

[Micha]

b/c all calculations in perturbative gravity over the years failed due to several reasons.

You mean all calculations in QFT. In string theory pertubation theory for gravity works just fine.

 

[Fra]

There are a lot of things that one could discuss about these things, but I'm not sure what we're discussing beyond commenting on Lubos "analysis" but...

Not too unlike the notion of B/I, perturbative or non-perturbative ways can be discussed at two levels I think. I'm not sure if this is obvious but it may help to point it out.

The very simplest view is that perturbation theory is simply a mathematical method, to find the solution to a given well defined problem that's hard to solve, from "perturbing" a known solution, and then somehow expand the full solution in terms of the order of perturbation. Now if the perturbation can't be shown to converge, then the whole technique fails. To save the scheme one might try to renormalize the known solution, to a different one and the see if we get convergence.

So far this can be discussed in terms of a mathematical method; there is no "physics" in this picture!

But the more interesting perspective, which does contain the physics, is that if we take an inference perspective to physics (like I do) then we also try to predict the future, given the present. In a sense one can abstract that as picturing "perturbations" of the present, so that we can produce expectations of the future from the perturbation of information. This is a physical picture (if you take the inference view; which btw isn't just silly comp sci analogies, it can be seen as an extension and purification of the essens of measurement theory as interaction or communication theory).

In this latter view, the perturbation, and the prior (which is to be perturbed) are physical. So the PHYSICS is actually in the perturbation details - and it's actually the non-perturbative idea that is non-physical. The non-perturbative formulation simply doesn't exists, and the interaction properties between two systems is encoded in their abilities or INabilities for that matter, for two perturbative expansion to negotiate and find an equiliribum.

This latter thing, would mean that what's sometimes called "problems" actually could explain interaction phenomenology (if developed properly in the future).

Actually what I would call unphysical non-perturbative techniques in the second view, is really pretty much a form of structural realism, which a lot of people subscribe to (except me).

/Fredrik

 

[tom.stoer]

You mean all calculations in QFT. In string theory pertubation theory for gravity works just fine.

I do not mean QFT, but exactly what I wrote - perturbative gravity - w/o any new formalisms, ingredients etc.

Regarding perturbative string theory: it is well-known in string theory that perturbative calculations are not sufficient; look at all the dualities, M-theory stuff, AdS/CFT, lack of a background independent formulation, ... all these ideas have been discussed in the literature (and here in this forum :-); they clearly indicate that even in string theory perturbation theory alone is not sufficient. I think that applies to maximal 4-dim SUGRA as well; even if finiteness can be proven order by order it is by no means clear that the whole series does converge.

Btw.: afaik it has not been proven that string theory is finite to all orders in perturbation theory; it has not been proven that the perturbation series as a whole does converge. That means that in string theory - as in any other QFT - perturbation theory is valid only in a very restricted regime.

 

[Fra]

I'm not disagreeing with what Tom wrote but I thought I'd just expand on this to illustrate further the subtle point I tried to make. I don't know if anyone appreciates the distiction though:

perturbation theory is valid only in a very restricted regime.

Seen as a mathematical method to solve a mathematical problem, this then of course means the method fails and simply isn't working. Just forget it, and try to find another solution method.

But, in the second perspective above, this isn't necessarily bad. Since it's not possible to have detailed expectations infinitely far into the future. The expectations are still bounded by the observers ability to formulate and encode possible future information states. So this "unability" to produce an expectation of an infinite future, is not a flawed technical method, it is (in the second view) how nature works(*). In particular this would suggest that the action of the observer (think PI) simply doesn't sum over all these unformulable possibilities, it is formed only over the physicall distinguishable and encodable. So this indeed makes the actions different.

I'm not suggesting here that this means that perturbative ST is OK, although no one has prooved convergence to all orders. This is because string theory does not IMO quite fully implement the second perspective properly.

But in other approaches, it could be that the "perturbation scheme" does not in fact correspond to physics, and not ONLY mathematical methods.

This is what I see a third way except sticking with perturbation theory in despite of it's possible non-convergence, or insisting on non-perturbative solutions. The third way could be to try to understand how even in the physical picture (future beeing a perturbation of the present; and the expected action sums over POSSIBLE expected futures only) what can be seen as perturbations. This latter unifies I think even more than renormalization as we konw today, interactions and observer-observer transformation and it should hold falsifiable predictions as this schemes puts constraints on possible interactions.

(*) IMO it's even the seed to understand causality and event order in nature as it imples action upon information at hand only.

/Fredrik

 

[Demystifier]

What about lattice QCD? It is a discrete path-integral theory, and yet it is in excellent agreement with observations. Would Motl say that lattice QCD is wrong?

 

[tom.stoer]

What about lattice QCD? It is a discrete path-integral theory, and yet it is in excellent agreement with observations. Would Motl say that lattice QCD is wrong?

Good question.

The point is that lattice QCD breaks certain symmetries (translational, rotational and Lorentz invariance) which should be (approximately) recovered in the continuum limit. So the issue could be that
a) in lattice QCD small violations of these invariances do not matter as they are small in the constinuum limit
b) lattice QCD is not a definiton of QCD but a calculational tool in a certain regime
c) breaking of the above mentioned theories is not as severe as the breaking of symmetries of QG (diff.-inv.?)
d) strictly speaking there is no continumm limit in the ordinary sense in QG as there is no artificial scale applied from the outside; the theory itself defines the scale and therefore sending something to zero is not possible

But in order to be sure one should ask Lubos

 

[Demystifier]

b) lattice QCD is not a definiton of QCD but a calculational tool in a certain regime

I think some lattice people would say lattice QCD *is* the definition of QCD, because QCD without a cutoff cannot be properly defined at all.

And LQG people have a similar reasoning about quantum gravity - that only discretization (in terms of networks or something alike) makes quantum gravity properly defined.

 

[tom.stoer]

Yes and no.

The lattice in lattice QCD is somehow artificial as the lattice people will eventually send the lattice spacing to zero which means they do not believe in a discretized spacetime. In that sense it's a calculational tool only. And it certainly does apply in all cases; e.g. scattering like DIS cannot be calculated using lattice QCD.

But I agree that regardign to the Wilsonian renormalization group approach requires some UV cutoff; lattice QCD is just one method to do that.

Lattice QCD is rather different from LQG. In lattice QCD the lattice is introduced as a calculational tool, whereas in lattice QCD the discretization is derived using loop space + diff. invariance. There is no parameter which can be controlled and send to zero at will (but I agree that in different approaches like triangulation / spin foams the discrete structure is introduced by hand - and that we are currently not sure about the final result of all these theories :-)