Enlarged Bound on the Measurability of Distances and Quantum κ-Poincarè Group

When quantum mechanical and general relativistic effects are taken into account in the analysis of distance measurements, one finds a measurability bound. I observe that some of the structures that have been encountered in the literature on the Quantum κPoincarè Group naturally lead to this bound. OUTP-96-54P gr-qc/9611016 August 1996 One of the greatest contemporary challenges for theoretical physics is posed by the incompatibility between Quantum Mechanics and (classical) General Relativity. It is likely that the solution of this puzzle, e.g. the construction of a quantum theory incorporating gravity, will require the development of a completely new understanding of physics and geometry. Hints on the structure of the sought new framework can come from the investigation of problems in which the incompatibility between Quantum Mechanics and General Relativity is more evident. Work in this direction has led to the expectation that in Quantum Gravity, unlike ordinary Quantum Mechanics, there might be bounds on the measurability of distances[1, 2, 3]. The most commonly[1, 2] expressed expectation, mostly because of its relevance[2] for the popular critical string theory, is that there should be a flat (i.e. L-independent) bound on the measurability of a distance L min [δL] = LP , (1) where LP is the Planck length. (The distinction between the Planck length and the string length is inessential to the line of argument here presented.) Based on the expected inadequacy of ordinary space-time concepts for scales smaller than the Planck length, this can be considered as a minimal bound on the measurability of distances in Quantum Gravity. Within the critical string theory framework it has actually been possible to find indications[2] that (1) originates from the modified uncertainty relation δx δP = h̄ + LP h̄ δP 2 . (2) As discussed in Ref.[3], an enlarged (more stringent and L-dependent) measurability bound is suggested by the observation that, once gravitational effects are taken into account, it is no longer possible to rely on the availability of classical agents for the measurement procedure (the limit of infinite masses leads to inconsistencies[3] associated with the formation of horizons). Based on this observation one arrives[3] at the measurability bound