# Is spacetime quantized or relative?

I was wondering how spacetime could be relative and quantized. It doesn't make sense to me. I am especially interested in how this works in causal dynamical triangulation. I think it is a very interesting theory, and it sometimes doesn't get enough credit. But, I can't understand how at a tiny level, all spacetime is quantized, where at a macroscopic level, spacetime is continuous and relative. If anyone could help explain what I'm missing here, that'd be great!

atyy
I was wondering how spacetime could be relative and quantized. It doesn't make sense to me. I am especially interested in how this works in causal dynamical triangulation. I think it is a very interesting theory, and it sometimes doesn't get enough credit. But, I can't understand how at a tiny level, all spacetime is quantized, where at a macroscopic level, spacetime is continuous and relative. If anyone could help explain what I'm missing here, that'd be great!

It depends on how their parameters are tuned. In some phases the spatial geometry changes randomly with time. Take a look at Fig 1 of http://arxiv.org/abs/1205.1229 .

Sorry, I'm sure this was a perfect explanation, but I don't really understand it. What I got from this was that there are three phases: A, B, and C. They can transition from one to another.

I'm new at this. Anyways, these three phases are placed in a phase diagram. However I'm not quite sure what the differences between these phases are.

My current understanding is that depending on the phase spacetime is in, its geometry could be relative or quantized. It all depends.

However, I'm not even certain if I am looking at this the right way.

Yeah, so basically I have no idea. :p

marcus
Gold Member
Dearly Missed
As a beginner interested in CDT you should have already read Renate Loll's SciAm article.

I'll bet you already have. But if not, let us know. I'll try to find link to online copy. Or at least the date so you can find it in library.

"Quantized" does not mean what you may be thinking. A continuous medium can have a quantum geometry. If angles, lengths, areas etc are uncertain. And if there are some restrictions on what happens when you measure them. Like the energy levels of an atom, a measurement of an area or a volume may only have some discrete possible outcomes.

When geometry is "quantized" it does not mean that space is "made" of little "grains" or "chunks". Space is not a substance.

Saying "quantized" is saying something about geometry (a web of relations among measurements of lengths, areas, angles, volumes etc).
The uncertainty and the discreteness applies to outcomes of measurements, not to some imaginary material or "fabric".

In CDT they do not say that spacetime is made of little blocks, they MODEL the geometry that way, as if it were made of little blocks.

Renate Loll's SciAm article was really good. Let me know if you havent read it yet.

wait, I have a reference, I think it was February 2007 and a free online copy is here:

www.signallake.com/innovation/SelfOrganizingQuantumJul08.pdf [Broken]

try this and see if it works. Have to go.

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That looks like a really good article, I'll take a good look at it.

Thank you so much for that article, it helped me a lot. I had seen Renate Loll's lecture, (which can be found here: ) but this described more of the history of quantum gravity.

So, just like particles, spacetime can go into superposition. This means that different geometries can overlap with one another. To describe these geometries, we can approximate with four-simplexes. If we give each of these four-simplexes a direction of time, we get causality, and a universe that is similar to our own. However, quantum spacetime isn't really made of these simplexes, we just used them to give us an approximation of the quantum geometry. In real life, the spacetime isn't split into chunks, but it does go into superposition, and it does have an arrow of time. This way, we still have our relative, continuous, macroscopic spacetime.

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atyy
Thank you so much for that article, it helped me a lot. I had seen Renate Loll's lecture, (which can be found here: ) but this described more of the history of quantum gravity.

So, just like particles, spacetime can go into superposition. This means that different geometries can overlap with one another. To describe these geometries, we can approximate with four-simplexes. If we give each of these four-simplexes a direction of time, we get causality, and a universe that is similar to our own. However, quantum spacetime isn't really made of these simplexes, we just used them to give us an approximation of the quantum geometry. In real life, the spacetime isn't split into chunks, but it does go into superposition, and it does have an arrow of time. This way, we still have our relative, continuous, macroscopic spacetime.

In the tentative conception of CDT, space is split into little chunks at first, and they form superpositions. Depending on some parameters, the little chunks assemble into a nice spacetime that is smooth on large scales. In other parameter ranges, the little chunks don't assemble into anything like our universe. The different behaviours of their model in different parameter ranges are called diffrerent phases - just like liquid, solid and gas are different phases of water depending on parameters like temperature and pressure. Because of the phase behaviour of their model, they hope that the chunks can be made smaller and smaller until the theory is completely smooth on small and large scales. However, this remains conjectural.

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So they hope to get closer and closer approximations of quantum spacetime geometry, eventually getting rid of the 4-simplexes all together? They just haven't gotten there yet.

Thanks for the help on phases, I was just completely lost there.

atyy
So they hope to get closer and closer approximations of quantum spacetime geometry, eventually getting rid of the 4-simplexes all together? They just haven't gotten there yet.

Thanks for the help on phases, I was just completely lost there.

Yes, that is related to a conjecture called "Asymptotic Safety".

If spacetime can go into superposition, does that mean that there is a small probability that the distance between me and my computer could change slightly?

atyy
If spacetime can go into superposition, does that mean that there is a small probability that the distance between me and my computer could change slightly?

It is unclear to me in CDT what a measurement of a distance is.

In LQG, the answer is tentatively yes, since there are proposed measurement operators. However, having the distance change each time you measure it seems to require multiple identical preparations of you and your computer, so I'm not sure.

some details:
here are some good discussions on continuous versus discrete spacetime. Relativists often don't much like the idea of continuous spacetime [because that's not the perspective Einstein developed] but when you stick in 'h' for quantum mechanics formulations of the worold just about everything gets quantized...

From Wikipedia:

Planck discovered that physical action could not take on any indiscriminate value. Instead, the action must be some multiple of a very small quantity (later to be named the "quantum of action" and now called Planck's constant). This inherent granularity is counterintuitive in the everyday world, where it is possible to "make things a little bit hotter" or "move things a little bit faster". This is because the quanta of action are very, very small in comparison to everyday human experience. Thus, on the macro scale quantum mechanics and classical physics converge. Nevertheless, it is impossible, as Planck found out, to explain some phenomena without accepting that action is quantized.

http://pirsa.org/09090005/
Spacetime can be simultaneously discrete and continuous, in the same way that information can.

http://arxiv.org/abs/1010.4354

“The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any band limited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the band limit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possess an ultraviolet cutoff.”

and
http://arxiv.org/abs/0708.0062
On Information Theory, Spectral Geometry and Quantum Gravity
Achim Kempf, Robert Martin
4 pages
(Submitted on 1 Aug 2007)
We show that there exists a deep link between the two disciplines of information theory and spectral geometry.

"argument for the discreteness of spacetime",

Ben Crowell posted this question...
The following is a paraphrase of an argument for the discreteness of spacetime, made by Smolin in his popular-level book Three Roads to Quantum Gravity. The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.

Lee Smolin says this in THREE ROADS TO QUANTUM GRAVITY

"On the Planck scale space seems to be composed of fundamental discrete units. String bits are one view of this, the Bekenstein bound from black hole thermodynamics is another. (LQG sees these units as spin networks.) It’s possible these are three different approaches to the quantum world..maybe there is a way of unifying them within a single theory.

The Holographic principle was inspired by the Bekenstein bound. Einstein’s equations of relativity can be derived by using the Bekenstein bound and laws of thermodynamics..."

If you search these forums you'll find many interesting discussions on your topic.

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"If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete."

I'm not quite certain how this conclusion is drawn. Is there any way we know how to describe spacetime using fields?

I'll take a look at that information theory stuff.

atyy
The gauge/gravity duality appears to be provide a mathematically sensible theory of quantum gravity in which spacetime is continuous. It probably does not describe our universe because of the matter content and the cosmological constant. But it appears to be a consistent theory of quantum gravity.

http://www.sns.ias.edu/~malda/sciam-maldacena-3a.pdf

On the Planck scale space seems to be composed of fundamental discrete units. String bits are one view of this, the Bekenstein bound from black hole thermodynamics is another. (LQG sees these units as spin networks.) It’s possible these are three different approaches to the quantum world..maybe there is a way of unifying them within a single theory.

The Bekenstein bound just has to do with the amount of entropy in an area, not the quantum structure of spacetime, right?

http://en.wikipedia.org/wiki/Bit-string_physics

Thanks for all the help, guys. I have a lot of questions. :shy:

Yeah, I thought that it probably wasn't, but that's what came up when I looked for it. Thanks, I never would have found that information.

The gauge/gravity duality appears to be provide a mathematically sensible theory of quantum gravity in which spacetime is continuous. It probably does not describe our universe because of the matter content and the cosmological constant. But it appears to be a consistent theory of quantum gravity.

http://www.sns.ias.edu/~malda/sciam-maldacena-3a.pdf
It is my understanding from this article that this is a good way to describe certain aspects of reality, but it can't be a consistent version of quantum gravity, because our universe isn't anti-de Sitter space. To quote the article:
In particular, does anything similar hold
for a universe like ours in place of the
anti–de Sitter space?
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I think it was February 2007 and a free online copy is here:
www.signallake.com/innovation/SelfOrganizingQuantumJul08.pdf [Broken]
It says that in LQG, spacetime is thought of as "chunks." However in this thread:
Marcus makes a point that in LQG, spacetime isn't thought of like that. He said that there is a
smallest measurable area
but
it does not consist of separate points."
How can is not consist of separate points (like marcus said), and at the same time, consist of chunks (like the article said)?
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If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.
Sorry, still not sure where they are getting infinitely many degrees of freedom and how they are drawing that conclusion. Maybe it's because I don't know enough about fields?

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atyy
It is my understanding from this article that this is a good way to describe certain aspects of reality, but it can't be a consistent version of quantum gravity, because our universe isn't anti-de Sitter space.

When I used the word consistent, I meant "mathematically consistent". The gauge/gravity duality does indeed seem inconsistent with observations. It is being studied to see whether its ingredients can be used to construct more realistic theories of quantum gravity.

It says that in LQG, spacetime is thought of as "chunks." However

Marcus makes a point that in LQG, spacetime isn't thought of like that.

The discrepant views are discussed in this review by Dupuis, Ryan and Speziale.

"Truncating the theory to a given graph captures only a finite number of degrees of freedom, and these in turn may be used to describe a discretization of general relativity. Indeed, from the viewpoint of LQG, there is a priori no need to interpret this set as discrete geometries."

"On the other hand, it has been shown that the same holonomies and fluxes describe certain discrete geometries, more general than the one used in Regge calculus, called twisted geometries"

"Historically, Loop Quantum Gravity and discrete Regge calculus remained somewhat detached from each other, despite several superficial similiarities. Ultimately, spin foam models facilitated the development of a precise link between the two since they both provide a dynamics for the LQG on a fixed triangulation and approximate, in the large spin limit, exponentials of the Regge action."

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LQG is a continuous theory of quantum gravity, defined as a projective limit/direct sum
over graphs. Truncating the theory to a given graph captures only a finite number of degrees of freedom, and these in turn may be used to describe a discretization of general relativity.
So, even though LQG involves spin networks, it is still continuous? It can just be used to get a discretization of GR?
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Sorry, still not sure where they are getting infinitely many degrees of freedom and how they are drawing that conclusion. Maybe it's because I don't know enough about fields?
Ah, I found this:
Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other independent physical variables on which the field depends. Usually this is done by writing a Lagrangian or a Hamiltonian of the field, and treating it as the classical mechanics (or quantum mechanics) of a system with an infinite number of degrees of freedom. The resulting field theories are referred to as classical or quantum field theories.
So, basically, a field is treated as a system with an infinite number of degrees of freedom? If you can describe gravity through fields like how they described it using "strings" of gluons in gauge/gravity duality, you could argue that spacetime is discrete? That is basically just saying that if we use gauge/gravity duality to help us find our quantum gravity theory, then spacetime is discrete, because the "strings" of gluons are discrete?

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So, even though LQG involves spin networks, it is still continuous? It can just be used to get a discretization of GR?
Or does LQG involve discrete geometry, but not separate "chunks?"
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So, basically, a field is treated as a system with an infinite number of degrees of freedom? If you can describe gravity through fields like how they described it using "strings" of gluons in gauge/gravity duality, you could argue that spacetime is discrete? That is basically just saying that if we use gauge/gravity duality to help us find our quantum gravity theory, then spacetime is discrete, because the "strings" of gluons are discrete?
From the article:
It turns out that one type of gluon chain behaves in the four-dimensional spacetime as the graviton, the fundamental quantum particle of gravity. In this description, gravity in four dimensions is an emergent phenomenon arising from particle interactions in a gravityless, three-dimensional world.
Well, at least they said that some "strings" of gluons behave as a gravitons. I'm just trying to figure out what this has to do with the original idea, which was:
The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.

julian
Gold Member
I think Rovelli in his book talks about something related to the original question...He asks if areas and volumes come in discrete values and in particular take on a minimum eigenvalue them how can this (discrete quantum nature) be compatible with relativity because in relativity you can make a distance smaller by going to another frame in motion?

To adress this he draws an analogy with quantized angular momentum in ordinary QM...Say we are dealing with the Hydrogen atom. The operator, $J_z$, which corresponds to the z component of angular momentum has a minimum eigenvalue. But then if you can perform continuous rotations doesn't that imply we can make the supposedly smallest einvalue smaller?! Contradicting its quantized discrete nature? The way out is this is...under a rotation where z maps to z', the new angular momentum operator, $J_{z'}$, correponding to the new z' doesn't commute with the original operator $J_z$. This implies that they dont have simultaneous eigenstates!! Under a rotation the original eigenstate is transformed into a superposition of eigenstates of $J_{z'}$. What does transform continuously under a rotation is the expectation value of the eigenvalue - the two operators still have the same spectrum and the same same smallest eigenvalue.

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