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Non-commutative space-time of Doubly Special Relativity theories

Jerzy Kowalski-Glikman·Sebastian Nowak
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Abstract

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there is infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the $κ$-Poincaré quantum (Hopf) algebra. Then we use the co-product of this algebra and the known construction of $κ$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space-time of the DSR theory is unique and equivalent to the theory with non-commutative space-time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space-time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space-time, its intrinsic length parameter $\ell$ becomes observer-independent.

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Jerzy Kowalski-Glikman, Sebastian Nowak. 2002-04-29. Non-commutative space-time of Doubly Special Relativity theories. https://doi.org/10.1142/s0218271803003050

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