Some consequences of spacetime fuzziness Kourosh Nozari * , Behnaz Fazlpour Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-1467 Babolsar, Iran Accepted 16 March 2006 Abstract Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some conse- quences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction The problem of reconciling quantum mechanics with general relativity is one of the task of modern theoretical phys- ics which until now has not yet found a consistent and satisfactory solution. The difficulty arises since general relativity deals with the events which define the world-lines of particles, while quantum mechanics does not allow the definition of trajectory; in fact the determination of the position of a quantum particle involves a measurement which introduces an uncertainty into its momentum. These conceptual difficulties have their origin in the violation, at quantum level, of the weak principle of equivalence (universality of free fall) on which general relativity is based [1,2]. Such a problem becomes more involved in the formulation of quantum theory of gravity, owing to the non-renormalizability of general relativity when one quantizes it as a local quantum field theory [3]. Nevertheless, one of the most interesting conse- quences of this unification is that in quantum gravity there exists a non-vanishing minimal observable length on the order of the Planck length, l p ¼ ffiffiffiffi G h c 3 q  10 33 cm. One cannot set up a position measurement with uncertainty less than this minimal value. The existence of such a fundamental length is a dynamical phenomenon due to the fact that, at Planck scale, there are fluctuations of the background metric, i.e. a limit of the order of Planck length appears when quantum fluctuations of the gravitational field are taken into account. Existence of minimal length scale has been moti- vated by several promising candidates of quantum gravity [4–7] and its consequences have been studied extensively [7– 17]. This natural cut-off guarantees the renormalizability of underlying quantum field theory [18–20]. Also existence of this minimal cut off results in the modification of usual Heisenberg algebra to incorporate gravitational uncertainty from very beginning [21–24]. 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.066 * Corresponding author. E-mail address: knozari@umz.ac.ir (K. Nozari). Chaos, Solitons and Fractals 34 (2007) 224–234 www.elsevier.com/locate/chaos