We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper.In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle T S 2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere.

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Starting from the principles of local relativistic Quantum Theory without long range forces, we study the structure of the set of superselection sectors (charge quantum numbers) and its implications for the particle aspects of the theory. Without assuming the commutation properties (or even the existence) of unobservable fields connecting different sectors (charge-carrying fields), one has a particle-antiparticle symmetry, an intrinsic notion of statistics for identical particles, and a spin-statistics theorem. Particles in "pseudoreal sectors" cannot be their own antiparticles (a variant of Carruthers' theorem). We also show how scattering states and transition probabilities are obtained in this frame. * Partly supported by CNR. 1 The observables which can be measured in a space-time region & generate the subalgebra %(Θ). The principle of locality is expressed in terms of these; it requires that two observables commute if they can be measured in spacelike separated regions. Therefore the anticommuting fields occurring in conventional quantum field theory are not affiliated with the algebra of observables. 2Mathematically a "state" means an expectation functional over the abstract algebra 51.

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincare covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.Research supported by Ministero della Pubblica Istruzione and CNR-GNAFA 9 Our gauge group G acts faithfully on g by definition. This might, for example, mean using SU(3)/Z 3 rather than SU(3), but has the invaluable merit of making G unique up to isomorphism. 10 G need not, of course, be a connected Lie group and it might even be a o-dimensional compact Lie group, i.e. finite group