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jakob1111

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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.)