# The de Sitter group and minmal length?

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• jakob1111
In summary: That question is too general and vague. You need to ask more specific questions about specific paper(s) that do this. Note that your 1st cited paper (https://arxiv.org/abs/hep-th/0207279 ) does not consider a deSitter position space, but rather a deSitter energy-momentum space, iiuc.

#### jakob1111

Gold Member
The de Sitter group is often used as an extension of the Poincaré group, because its a simple group and preserves, in addition to a velocity c, a length L.

A natural candidate for this length scale is the Planck length. Thus it seems to make sense to think about the invariant Planck length as some kind of minimal length scale, comparable to how the invariant speed $c$ is a maximal velocity. ( Just two out of many, many possible references: https://arxiv.org/abs/hep-th/0207279 and https://arxiv.org/abs/0805.2584 )

However, the de Sitter group contracts to the Poincare group in the limit L -> ∞. This means, the Poincare group is a good approximation of the de Sitter group as long as we are dealing with lengths much shorter than the invariant length scale: $l << L$. This is analogous to how the Galilei group is a good approximation of the Poincare group, as long as we are only dealing with slow velocities $v << c$, i.e. in the c -> ∞ limit.

The invariant velocity $c$ is a large velocity and thus I would suspect that the invariant length $L$ is a large length scale, not a small one. Therefore I'm wondering how this fits together with the interpretation that the invariant length is the Planck length, which is a very small length scale.

If we take the idea a quantized spacetime serious, we need an invariant length as a fundamental building block of space. However it seems that the de Sitter group is the wrong group to use in this context, because it contracts in the "wrong" limit to the Poincaré group. In my understanding a theory with a quantized spacetime, we would need to use a group that contracts in the L -> 0 limit to the Poincare group. This would mean that the Poincare group is a good approximation as long as we are only dealing with length scales that are large compared to the invariant length scale. In contrast the de Sitter group contracts in the L -> 0 limit to a quite strange spacetime, called cone spacetime.

1.) Is there some flaw in my line of thought? I've seen a lot of papers that go in this direction (doubly special relativity etc.), but I'm still unsure how the de Sitter group can make sense in the context of a minimal length (= maximal energy) scale.

2.) Is there a simple group that contracts to the Poincare group in the L -> 0 limit, where L is an invariant length?

3.) How do theories with a quantized spacetime (LQG etc.) deal with the "minimal invariant length scale" problem? (Not in general, I'm just curious which group they consider and why.)

More likely, the ##L## is something like the Hubble Length. See also De Sitter Relativity. It's incompatible with the ideas involved in doubly-special relativity, for reasons explained on that Wiki page.

strangerep said:
More likely, the ##L## is something like the Hubble Length. See also De Sitter Relativity. It's incompatible with the ideas involved in doubly-special relativity, for reasons explained on that Wiki page.

Thanks a lot for your reply! Yes, I think taking ##L## as the Hubble length would make sense. However I would like to understand how and why people still consider the de Sitter group in models with a minimal length scale/ maximal energy scale.

Moreover, I wasn't able to find something on the Wiki page that says that De Sitter Relativity is incompatible with doubly-special relativity. In the section on DSR it says "de Sitter relativity can be interpreted as an example of the so-called doubly special relativity"...

jakob1111 said:
Yes, I think taking ##L## as the Hubble length would make sense.
That's what your 2nd cited paper seems to start off doing (https://arxiv.org/abs/0805.2584 ). But then it seems to change its mind. It starts off with ##\ell## as a fundamental cosmological length constant, but then says (p4, 2nd para) that: "Since in de Sitter relativity there is a free length parameter ##\ell##, it is natural to assume that its minimum value is the Planck length ##\ell_P##." But you can't have it both ways: either it's an invariant under the chosen symmetry group, or it's not.

However I would like to understand how and why people still consider the de Sitter group in models with a minimal length scale/ maximal energy scale
That question is too general and vague. You need to ask more specific questions about specific paper(s) that do this. Note that your 1st cited paper (https://arxiv.org/abs/hep-th/0207279 ) does not consider a deSitter position space, but rather a deSitter energy-momentum space, iiuc.

Btw, I seem to recall that doubly-special relativity was discredited some time ago (since it involved an energy-dependent speed of light, iirc). But don't quote me on that -- it's been a while since I studied it, and subsequently discarded it.

Moreover, I wasn't able to find something on the Wiki page that says that De Sitter Relativity is incompatible with doubly-special relativity. In the section on DSR it says "de Sitter relativity can be interpreted as an example of the so-called doubly special relativity"...
Apparently, you didn't read the next sentence, which says:
Wikipedia said:
There is a fundamental difference, though: whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry.

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strangerep said:
That's what your 2nd cited paper seems to start off doing (https://arxiv.org/abs/0805.2584 ). But then it seems to change its mind. It starts off with ##\ell## as a fundamental cosmological length constant, but then says (p4, 2nd para) that: "Since in de Sitter relativity there is a free length parameter ##\ell##, it is natural to assume that its minimum value is the Planck length ##\ell_P##." But you can't have it both ways: either it's an invariant under the chosen symmetry group, or it's not.

Good to hear that I'm not the only one who finds this confusing :D

strangerep said:
That question is too general and vague. You need to ask more specific questions about specific paper(s) that do this. Note that your 1st cited paper (https://arxiv.org/abs/hep-th/0207279 ) does not consider a deSitter position space, but rather a deSitter energy-momentum space, iiuc.

Yes, you're probably right. I was just curious what people think about this topic in general. A minimal length seems to be a quite popular thing. For example, the first paper I cited says: "First of all both loop quantum gravity and string theory indicate appearance of the minimal length scale." That's why I thought there would be some "canonical" thoughts on the "correct" group that preserves such a minimal length scale (e.g. in string theory or LQG).

Regarding the Wikipedia article: I'm still not sure if I understand what the author tries to communicate. On the one hand it "can be interpreted as an example of the so-called doubly special relativity". And on the other hand "in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry." To me doubly special relativity is the umbrella term for models that preserve in addition to a velocity, a length scale. With this definition, de Sitter relativity is an example of it and therefore not all doubly special relativity models violate Lorentz symmetry...

jakob1111 said:
To me doubly special relativity is the umbrella term for models that preserve in addition to a velocity, a length scale. With this definition, de Sitter relativity is an example of it and therefore not all doubly special relativity models violate Lorentz symmetry...
...except that doubly special relativity tends to be interested in a maximum energy scale (hence minimum length scale), whereas deSitter relativity has a maximum length scale (most likely cosmological).

strangerep said:
...except that doubly special relativity tends to be interested in a maximum energy scale (hence minimum length scale), whereas deSitter relativity has a maximum length scale (most likely cosmological).

Ah okay I see :) To me the term "doubly" simply referred to the property that we consider two invariant quantities instead of one in normal special relativity. But it really seems that doubly special relativity people are exclusively interested in a minimal scale and deSitter relativity contains "naively" only a maximal scale. However there are people who try to combine both ideas, although it remains unclear to me how this works in pratice. For example, in this paper the author claims "any DSRtheory can be understood as a particular coordinate system on four dimensional de Sitter space of momenta imbedded in five dimensional Minkowskispace."

jakob1111 said:
Ah okay I see :) To me the term "doubly" simply referred to the property that we consider two invariant quantities instead of one in normal special relativity. But it really seems that doubly special relativity people are exclusively interested in a minimal scale and deSitter relativity contains "naively" only a maximal scale. However there are people who try to combine both ideas, although it remains unclear to me how this works in practice.
I think it does not work satisfactorily. That's why it hasn't really gone anywhere and seems to be mostly ignored these days.

For example, in this paper the author claims "any DSR theory can be understood as a particular coordinate system on four dimensional de Sitter space of momenta imbedded in five dimensional Minkowski space."
Note: "de Sitter space of momenta ...". That means they can work with a large scale (high energy limit). I.e., their 4-momentum space is de Sitter, hence has constant positive curvature, unlike ordinary momentum space which is flat. As a consequence, they have a noncommutative spacetime (i.e., position operators do not commute) which is a pest to work with.

[Edit: Sabine Hossenfelder discussed one of the problems with DSR in her more recent paper about the "soccer ball" problem, (i.e., the problems associated with nonlinear composition of momenta).]

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## 1. What is the de Sitter group?

The de Sitter group, also known as the de Sitter algebra, is a mathematical group that describes the symmetries of de Sitter space, which is a solution of Einstein's equations in general relativity. It is closely related to the Lorentz group, which describes the symmetries of Minkowski space.

## 2. What is de Sitter space?

De Sitter space is a solution of Einstein's equations in general relativity, named after the mathematician Willem de Sitter. It is a maximally symmetric space with a positive cosmological constant, meaning that it has a constant and positive curvature in all directions.

## 3. How does the de Sitter group relate to minimal length?

The de Sitter group is closely related to the concept of minimal length in physics. This is because de Sitter space has a minimum length scale, known as the Planck length, where quantum gravity effects become important. The de Sitter group is used in some theories of quantum gravity to describe the symmetries of this minimal length scale.

## 4. What is the significance of the de Sitter group in physics?

The de Sitter group is important in theoretical physics because it plays a role in various areas of research, such as cosmology, string theory, and quantum gravity. It is also used in the study of black holes and the properties of spacetime at the smallest scales.

## 5. How is the de Sitter group used in cosmology?

In cosmology, the de Sitter group is used to study the symmetries of de Sitter space, which is believed to be a good approximation of the universe's early expansion. It is also used in inflationary models of the universe, which suggest that the universe underwent a period of rapid expansion in its early stages.