Quantized spacetime and Special Relativity

[Savant13]

How does special relativity affect a quantized spacetime? Specifically, how can time and space be quantized the same way for all observers?

 

[BenTheMan]

Well, if you quantize space, you break special relativity in general---discretizing space-time usually just ends up giving you a mess. I guess the loop quantum gravity people have some way around this, but I've never heard a good explanation as to why it's not a problem for them.

 

[Fra]

Specifically, how can time and space be quantized the same way for all observers?

I guess there is no mainstream answer to this.

Some personal thoughts.

First, I think it's not at all obvious to me at least what "beeing the same to all observers" means? Rovelli asks in his Relational QM - how can two observers even compare their measurements?? Ie how is the notion of comparasion defined? He argues that the answer is communication/interaction between observers. I think this is the best way of thinking of this.

I think the essence of general covariance is that observations from all observers, while sort of different, should at some level be consistent. IE. the laws pf physics deform along with the observer, so that the equivalence class of all descriptions and observers is the fundamental description. Indespite of it's deep beauty it's not hard to see that there is something strongly realist-minded about this. The point beeing that no real observer see the entire equivalence class. The whole reasoning IMHO rests on a realist vision of symmetry. I think this won't do when it comes to incorporating QM with this. I think probably both QM and GR needs change.

But the question is how to merge this into the big picture, which not only spacetime but also all other forces. One possibility IMO that any inconsistencies between observer view, manifest themselves as NEW forces. The mutual force is simply an mutual inconsistency, which is resolved as the interaction progresses, which results in a mutual relation which recovers the consistency of the differing views.

The question then is if one can continously resolve all inconsistencies that appear by adding new interactions? So that you end up with a total fundamental "symmetry". I don't think it's a prior obvious that such perfect symmetry exists. I personally don't think so, but that doesn't mean we are toast. It just means that perhaps the focus should be on the evolution of the symmetries? IE. to consider a true relative symemtry, rather than aiming for a universal fixed master symmetry, that may not even exist.

If this is so, then perhaps one might not expect perfect consistency, but OTOH the inconsistency may show us the way into the future.

/Fredrik

 

[BenTheMan]

I guess there is no mainstream answer to this.

There IS a mainstream answer, and it's that it's not consistent with special relativity.

A minimum length implies a breakdown of Lorentz Invariance at that scale. The reason is easy to see---in who's frame are you quantizing space? If I quantize space in my frame, and you come running by me really fast, then you see that my little chunk of quantized space is smaller by a factor of gamma. So "smallest chunk of spacetime" is a relative notion in special relativity.

 

[Tac-Tics]

I always wondered about quantized space, but if you have a square area with sides which are one quantized unit long, what is the length of the diagonal?

 

[marcus]

How does special relativity affect a quantized spacetime? ...

That would depend on what you mean by a quantized spacetime. For instance, talking about LQG, it is just a popular misconception that LQG divides spacetime up into little chunks .

LQG and allied approaches quantize the gravitational field. That is, they quantize geometrical measurements. Quantizing a measurement means to represent it as an observable, an operator on the state space. In some versions of LQG and related QG, the spectrum of the area operator is discrete, in other versions it is not discrete.

In neither case is there an implication here that space or spacetime has been divided up into little chunks. Just that if observers measure a physical area, like the are of your desk, the operation of measuring will, in some versions, have discrete spectrum. The expectation value of a measurement will in that case be an average of a countable set that includes zero, and the smallest positive eigenvalue, and the next smallest ...etc.
If one of the observers is flying very fast past your desktop, the expectation value of his measurement of the area could be very small----smaller than the smallest positive eigenvalue.

If you want, think of it this way. In some of the most active lines of QG research there is no minimal length (no minimum eigenvalue of the length measurement operator) or minimimal area or any such thing. A prominent example is Loll's approach, recently written up in the Scientific American. I have a link in my sig to that Loll SciAm article. Check it out. The size of the triangles goes to zero and there is no minimal length---they are explicit about this in their papers.

Loll's approach has a quantum spacetime, but they don't call it a quantized spacetime. That might mislead people and start them talking about dividing spacetime up into little chunks. What is quantized is the geometry. Gravity is geometry and QG stands for both quantum geometry and quantum gravity.

Loll's quantum spacetime is a continuum with a new kind of geometry, quantum geometry, in which geometric observables are quantum operators. The spacetime itself is not divided up into little bits, and it does not have a minimal length.

Now LQG is similar in some ways and different in others. It too is constructed on a continuum: so space itself is not divided up into little bits. But in some versions of LQG the area and volume operators have discrete spectrum. In other versions (spinfoam in particular) that I keep reading about, these geometric observable operators seen NOT to have discrete spectrum, at least they haven't been shown to have. I don't know how this is going to turn out. It will be interesting to see. But in either case there is no problem with Special Relativity

 

[Fra]

There IS a mainstream answer, and it's that it's not consistent with special relativity.

A minimum length implies a breakdown of Lorentz Invariance at that scale. The reason is easy to see---in who's frame are you quantizing space? If I quantize space in my frame, and you come running by me really fast, then you see that my little chunk of quantized space is smaller by a factor of gamma. So "smallest chunk of spacetime" is a relative notion in special relativity.

Yes regarding chunks I see your point, but I guess the way I see it the notion of "quantized spacetime" is not unambigous in what I think of as "mainstream". This was what I meant. And if you suggested that there IS a mainstream sense, that in turn is inconsistent, then that still feeds the question on howto resolve it.

Like others said, quantization doesn't equal make into chunks.

My comment tried to address the question in a general sense (admittedly non-mainstream), that suggests that an observer-observer inconsistencies in general might possibly be resolved, by identifying these as new interactions. Ie that the APPARENT inconsistency that two observers can not agree upon their observations, does exert a kind of influence/force in between them. Once you can classified this new "force" the consistency is recovered. That's how I personally see a possible way to resolve the "subjectivity problem" and howto maintain some level of objectivity in despite of a fundamental picture where each observer can not compare their measurements with some common standard, but only compare it by interaction with it's neighbours.

/Fredrik

 

[Savant13]

What I mean by 'the same to all observers' is that it should not depend on your velocity. Let's say we have two non-interacting particles moving very quickly with respect to each other over flat quantized spacetime. Since both have an equal claim to being at rest, they should observe the spacetime over which they move to be the same, despite relativistic effects.

 

[marcus]

What I mean by 'the same to all observers' is that it should not depend on your velocity...

I'm not sure I understand your question, if you have a question. LQG is a quantization of geometry which among other things is consistent with SR as far as we know. It does not involve thinking of space as consisting of little chunks That is not what "quantizing" geometry means.

There are other interesting QG approaches which also do not involve thinking of space as made of little chunks. Have a look at the Loll QG link in my sig. If you have some question, how about spelling it out in detail?---I'll be glad to do my best to answer.

 

[Haelfix]

Regge calculus is sort of the historical forebear to the whole atom of space idea. You can retain lorentz invariance provided that there is enough residual symmetry left over in the marginal operators, not unlike textbook lattice gravity.

The problem really is getting these five (not necessarily independant) conditions to mix:
1) Lorentz invariance or at least lorentz breaking effects but only up to very small factors (which is very constraining and hard to do)
2) Flat spacetime in at least some sort of limit (as opposed to crumpled up phases)
3) Manageable entropy densities (as opposed to Planckian entropy^4, which leads to ridiculous cosmologies)
4) Unitarity
5) Existence of a continuum limit.

All treatments known to date end up sacrificing 1 or 2 conditions, depending on the context.

 

[ensabah6]

Regge calculus is sort of the historical forebear to the whole atom of space idea. You can retain lorentz invariance provided that there is enough residual symmetry left over in the marginal operators, not unlike textbook lattice gravity.

The problem really is getting these five (not necessarily independant) conditions to mix:
1) Lorentz invariance or at least lorentz breaking effects but only up to very small factors (which is very constraining and hard to do)
2) Flat spacetime in at least some sort of limit (as opposed to crumpled up phases)
3) Manageable entropy densities (as opposed to Planckian entropy^4, which leads to ridiculous cosmologies)
4) Unitarity
5) Existence of a continuum limit.

All treatments known to date end up sacrificing 1 or 2 conditions, depending on the context.

Does string/M-theory satisfy all 5? thx

 

[Haelfix]

ST is not a discretization theory (atom of space idea) so it doesn't apply. Incidentally, LGQ is not really a discrete theory either. CDT is though!

 

[Savant13]

ST is not a discretization theory (atom of space idea) so it doesn't apply. Incidentally, LGQ is not really a discrete theory either. CDT is though!

LQG is definitely discrete

 

[marcus]

LQG is definitely discrete

Heh heh. In this case Savant, I would have to say that Haelfix knows a good deal more than you about it, so it seems odd for you to flatly contradict him without offering any link to peer-review literature. What technical article have you read about LQG, where it is defined? A simple one is Rovelli Upadhya LQG Primer. Look it up on arxiv. You will see that LQG is defined on a continuum representing spacetime. The operators representing geometric measurment have discrete spectrum, not the same thing as having a spacetime made of discrete bits. Please go back and read my post on this.

...LGQ is not really a discrete theory either. CDT is though!

I agree with you about LQG, certainly. But sometimes I'm not sure what you mean by the words you use, Haelfix. Could you explain what you mean by saying that CDT is a "discrete theory".

We know that CDT spacetime has no minimal length. Loll has mentioned that explicitly. It looks like a continuum (a manifold but without a metric to give it a metric geometry.)

The CDT continuum is not in any sense discrete.

In CDT triangulations are used to generate different possible geometries, but the size of the generalized triangles is taken to zero in the limit. This is why there is no minimal length in CDT. (By the way would you like an arxiv link for that?)

Perhaps you should explain what you mean by "a discrete theory" before people are misled and start thinking that the CDT continuum is divided up into little bits

 

[Savant13]

In Three Roads to Quantum Gravity, author Lee Smolin, one of the founder of LQG explicitly states that spacetime is quantized in LQG

 

[Haelfix]

The hope is that CDT is a continuum but this is a *hypothesis*, and its still done on a computer and very much discrete. So in other words you take the Einstein Hilbert action (or generalization), take some type of mini superspace approximation and stick it on a type of lattice (done in lorentzian signature, as opposed to euclidean, which is sort of the big success of the program), divide it up in a certain way, and see what comes out. The assumption is that at the end of the day, the details of the lattice drop out. And in practise they see that various quantities don't blow up and tend to the expected limits as they decrease the lattice spacing and increase the number of cells, the volume and so forth.

However there is no analytic *proof* that this continuum exists and therein lies the rub.

Lattice theories in general can give you very misleading indicators. Sometimes you can actually prove the continuum doesn't exist, even though the computer results naively looks like such a thing should. Other times a continuum theory exists, even when it shouldn't. So for instance lattice QED's UV continuum exists, but surely cannot be physical (b/c of the Landau pole or GUT unification etc). In other cases, there are several (perhaps infinite) amounts of different continuum theories possible.

CDT's main, as yet to be resolved problem is proving the existence/uniqueness of the continuum limit, whether the inclusion of matter spoils the results, and also the loss of unitarity b/c of the particular superspace approximation they use. So there's still a lot of work to be done to really convince people that they have a final admissible TOE.

 

[marcus]

In Three Roads to Quantum Gravity, author Lee Smolin, one of the founder of LQG explicitly states that spacetime is quantized in LQG

that was a non-mathematical popularization written for wide audience.

 

[ensabah6]

ST is not a discretization theory (atom of space idea) so it doesn't apply. Incidentally, LGQ is not really a discrete theory either. CDT is though!

in string theory, what are its predictions for a Planck-sized string accelerating toward c? Is string theory committed to infinitely continuous sub-planckian distances and time? In string theory, is spacetime classically smooth at arbitrary small distances?

 

[Haelfix]

I'm not the best one to ask about that sort of question, I am sure a stheorist could explain it better and with more authority.

From what I gather the question becomes illdefined. If you start probing the string (either via scattering or dumping energy into the free string) eventually past a certain point (not necessarily the Planck scale, but thereabouts) you no longer are probing strings, but rather black holes (b/c all that energy density eventually pushes the system past its Schwarzschild radius). At that point, transplanckian physics is no longer well described by string theory, but becomes quasi classical again (eg GR.. scattering of black holes and so forth).

In a certain sense, that's kinda what you want. It makes good sense that the degrees of freedom of a QG theory when pushed to the extreme breaking point eventually lose their significance b/c you can no longer ask questions about them since they lie behind horizons.

As for whether or not the spacetime is smooth. Well again, the question is a little fuzzy and only makes sense in certain limits. The metric is only part of the degrees of freedom of the whole (as yet to be understood) shebang, in fact its presumably not fundamental and therefore emergent. The main (string/brane) d.o.f as well as the precise nature of the moduli should in principle contribute to its dynamics, but like all emergent systems the technical details becomes really challenging. Still, since those d.o.f are decidedly quantum and fuzzy its a little hard to say with a straight face that spacetime is 'smooth'.. Its only 'smooth' when we make it so (by hand) as a sort of initial condition for calculational tractability.

 

[Savant13]

that was a non-mathematical popularization written for wide audience.

That does not mean that the statement was not correct on some level.

 

[Hurkyl]

That does not mean that the statement was not correct on some level.

But it does mean it's unlikely to be precise, and a Bad Idea to cite authoritatively.

 

[Savant13]

I'm not the best one to ask about that sort of question, I am sure a stheorist could explain it better and with more authority.

From what I gather the question becomes illdefined. If you start probing the string (either via scattering or dumping energy into the free string) eventually past a certain point (not necessarily the Planck scale, but thereabouts) you no longer are probing strings, but rather black holes (b/c all that energy density eventually pushes the system past its Schwarzschild radius). At that point, transplanckian physics is no longer well described by string theory, but becomes quasi classical again (eg GR.. scattering of black holes and so forth).

In a certain sense, that's kinda what you want. It makes good sense that the degrees of freedom of a QG theory when pushed to the extreme breaking point eventually lose their significance b/c you can no longer ask questions about them since they lie behind horizons.

As for whether or not the spacetime is smooth. Well again, the question is a little fuzzy and only makes sense in certain limits. The metric is only part of the degrees of freedom of the whole (as yet to be understood) shebang, in fact its presumably not fundamental and therefore emergent. The main (string/brane) d.o.f as well as the precise nature of the moduli should in principle contribute to its dynamics, but like all emergent systems the technical details becomes really challenging. Still, since those d.o.f are decidedly quantum and fuzzy its a little hard to say with a straight face that spacetime is 'smooth'.. Its only 'smooth' when we make it so (by hand) as a sort of initial condition for calculational tractability.

I'm not talking about strings. Can we please stop dragging other theories into this?

 

[marcus]

I'm not talking about strings. Can we please stop dragging other theories into this?

Hi Savant, now I am beginning to make better sense of what you are asking about. Your question seems to make sense in a particular context of Loop Quantum Gravity. You indicate that by citing a Smolin book and by saying not to drag in string. If we can just limit things to a specific theory like LQG, then we can get clear more easily.

It happens that Carlo Rovelli (a central LQG guy) published a paper in 2003 about just this thing. Why LQG and SR are compatible.

I have paraphrased what he said in some earlier posts in this thread, but I can also get the link.

Rovelli's article is technical---for publication in a professional journal. It is not something you need to try to read! But if I can find the link it will be a useful source.

A key realization is that when the experts say "quantum spacetime" they do not mean "quantized spacetime" in the sense of the continuum being split into little bits. That idea is only promoted in popular writing, like in Scientific American or mass-market simplifications. LQG is based mathematically on a continuum, a continuuous spacetime, and it is the geometry on the continuum that is quantum. Measurements of geometric stuff are uncertain and in some versions they assume a discrete range of values. Like the measured energy of a hydrogen atom can only be certain things, and the expectation value of a given measurement is an average of those distinct possibilities----so, measuring the area of something can have a discrete set of outcomes. Especially something very small. The spectrum of possibilities tends to blur together as you move up the scale.

If you are talking to a general readership audience, it is really difficult to make the distinction between measurement being discretized, and the actual underlying spacetime being grainy. Readers are apt to lose interest or get confused. You might just as well give up and say that space is comprised of atoms of space! So that is what popularizers typically do. Even Rovelli, when he writes non-mathematical articles will suggest that kind of grainy or particulate picture to people. What else can he do?

But the graininess is really up one level of abstraction, at the level of geometric observables---geometric measurements.

And a Special Relativity transformation just changes the weighted average of the discrete possible outcomes of a measurement---like the measurement of a very small area by a moving observer.

===============
Also I should say there is an interesting side issue that shows you are RIGHT to ask about the relationof LQG to SR. It's because there are some versions of LQG which favor a modification of SR called DSR (deformed special relativity) that is almost impossible to detect. It is extremely difficult to distinguish whether nature obeys standard SR or DSR, there is some hope that observations of Gamma Ray Bursts (GRB) will rule out one or the other. Although the effect is small, it would be exciting if it could be detected. Observing very high energy GRB that have traveled for astronomical times like a billion years, over very long distances, offers some hope of being able to distinguish. So far there was one false alarm but it was not subsequently confirmed.

Rovelli (as I say a central figure in the LQG community) has never suggested that DSR effects would be observed, but Smolin (also a leader) has done so, at least tentatively.

The effects are very small and can only be observed in very high energy photons so practically it seems of little consequence. But it is still interesting that at least some form of LQG is interpreted to favor a barely perceptible bending of the SR rules.

This is different, though, from the idea of space being divided up into little chunks. I will go see if I can find that article. Please let us know if this "quantized spacetime" business is still bothering you.

 

[marcus]

Yes Savant, it was easy to find. Here we go. It was published in Physical Review D in 2003.
The 2002 preprint is online (go here and click on PDF)
http://arxiv.org/abs/gr-qc/0205108
The brief summary is
"A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary."

 

[Savant13]

Okay, I see it. Thanks