No generalized statistics from dynamics in curved spacetime.

We consider quon statistics in a dynamically evolving curved spacetime in which prior to some initial time and subsequent to some later time is flat. By considering the Bogoliubov transformations associated with gravitationally induced particle creation, we find that the consistent evolution of the generalized commutation relations from the first flat region to the second flat region can only occur if q = ±1. PACS numbers: 05.30.-d, 04.60.+n ( To appear in Physics Review Letters ) Typeset using REVTEX 1 There has been an increasing interest in generalized commutation relations of the form aia † j − qa † jai = δij (1) where q is real, coming from two main directions. One motivation comes from the study of quantum groups ( for example see Refs. [1–5] ); in particular the quantum version [6,7] of SU(2). The other motivation is Greenberg’s study of particles with infinite statistics [8,9]. In either case, the relations (1) can be used to quantize the simple harmonic oscillator. ( The generalized commutation relations used in Refs. [6,7] can be put in the form (1) by an appropriate transformation. ) As q ranges from +1 to -1, it can be seen that the relations (1) interpolate between commutation relations and anticommutation relations ( appropriate for bosons and fermions respectively ). Given an operator ai and its adjoint a † i which satisfy (1), a vacuum state |0 > can be defined in the usual way by