Minimal length and a new paper on deformed relativity

[marcus]

http://arxiv.org/abs/0708.3813
Non-Commutativity of Effective Space-Time Coordinates and the Minimal Length
Florian Girelli, Etera R. Livine
5 pages
(Submitted on 28 Aug 2007)

"Considering that a position measurement can effectively involve a momentum-dependent shift and rescaling of the "true" space-time coordinates, we construct a set of effective space-time coordinates which are naturally non-commutative. They lead to a minimum length and are shown to be related to Snyder's coordinates and the five-dimensional formulation of Deformed Special Relativity. This effective approach then provides a natural physical interpretation for both the extra fifth dimension and the deformed momenta appearing in this context."

The issues of minimal length and extensions of Lorentz invariance are currently of considerable interest (recall the Dittrich-Thiemann paper that just came out, and the MAGIC observational result that would appear to indicate some slight bending of conventional Lorentz symmetry.) That could make this paper a timely one, also they have some new ideas.

 

[marcus]

Extended Special Relativity (ESR)

The authors have a good idea to use a new name, ESR, because this is a new way to formulate symmetry in the case that both the speed of light and the Planck length are invariant.

===exerpt===
...This is exactly the 5d point of view on DSR, which we actually proposed to call Extended Special Relativity to emphasize the difference with the standard formulation of DSR [3].

We have considered a shift and a rescaling (both Lorentz covariant and momentum dependent) of the space-time coordinates, which shows how the notion of minimum length can appear at an effective level within special relativity. This formalism can naturally be recast as a five-dimensional framework and related to Snyder’s approach for a Lorentz invariant non-commutative space-time. Our point of view then provides the missing physical interpretation of the extra 5d coordinates (x₄, p₄): they precisely encode the information about the shift and the rescaling. As a consequence, the deformed addition of (effective) momenta, which is commutative, also encodes the natural rescaling of the deformation mass scale avoiding therefore the “soccer-ball problem” often met in theories with a minimal length such as DSR.
==endquote==

We have been seeing for several years that it is more natural to use 5 spacetime dimensions. A lot of what John Baez did with us last year here works in 5D. Cosmology with a positive Lambda (accelerating expansion) leads naturally to DeSitter space as the generic model, and both DeSitter space and the DeSitter symmetry group are most easily visualized (and presented for calculation) in 5D. This goes back several decades to Snyder, who is cited here by Girelli and Livine.

So it is inevitable that we come to think of our 4D spacetime as hypersurface in 5D (or at least locally approximated by that). The big question then is how to give a physical interpretation of the new coordinate. There have been several suggestions, if I remember correctly. Girelli and Livine have proposed a new physical interpretation. They could have the right one.

 

[Entropia]

I find this paper on deformed relativity and the concept of a minimal length to be intriguing and potentially significant in our understanding of space and time. The authors propose the idea of effective space-time coordinates that are naturally non-commutative, meaning that they do not follow the traditional rules of addition and multiplication. This leads to the emergence of a minimum length, which has been a topic of interest in recent studies.

The authors also draw connections between their effective space-time coordinates and other theories, such as Snyder's coordinates and the five-dimensional formulation of Deformed Special Relativity. This provides a deeper understanding and a potential physical interpretation for these concepts.

Furthermore, the authors mention the relevance of their ideas in light of recent observations that suggest a possible deviation from conventional Lorentz symmetry. This further highlights the importance of exploring alternative theories and considering the concept of a minimal length.

Overall, this paper presents a novel and thought-provoking approach to the issues of minimal length and extensions of Lorentz invariance. I believe it will be a valuable contribution to the field and look forward to seeing further developments in this area of research.