De Sitter space as an arena for doubly special relativity
Section snippets
The DSR theory
Doubly special relativity theory is a new attempt to approach the problem of quantum gravity. This theory was proposed about a year ago by Amelino-Camelia [1] and is based on two fundamental assumptions: the principle of relativity and the postulate of existence of two observer-independent scales, of speed identified with the speed of light c,2 and of mass κ (or length ℓ=1/κ) identified with the Planck mass. There are several theoretical indications that such a theory
DSR algebra and de Sitter space
Since the space–time algebra of Lorentz generators and positions given by (1), (10) and (11) is universal, it is worth to investigate it a bit closer. The first thing to note is that this algebra is the SO(4,1) Lie algebra with Lorentz generators belonging to its SO(3,1) Lie subalgebra (recall that in special relativity we have to do with the semidirect sum of SO(3,1) and R4, instead). Let us recall now that both Lorentz generators and positions can be interpreted as symmetry generators, acting
Conclusions
The main result of this Letter is that any DSR theory can be regarded as a particular coordinate system on de Sitter space of momenta. In addition, the Lorentz transformations have interpretations of stabilizers of the zero-momentum point in the de Sitter space, while positions are identified with the remaining four generators of SO(4,1). Moreover, one can single out the basis, which is most natural from the geometric point of view, and this basis turns out to be the bicrossproduct one.
The
Acknowledgements
The idea to investigate de Sitter structure of the momentum space of DSR theories arose in the course of many discussions with Giovanni Amelino-Camelia during my visit to Rome. I would like to thank him as well as the Department of Physics of the University of Rome “La Sapienza” for their worm hospitality during my visit.
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Research partially supported by the KBN grant 5PO3B05620.