Quantum Field Theory on Quantum Spacetime

Condensed account of the Lectures delivered at the Meeting on {\it Noncommutative Geometry in Field and String Theory}, Corfu, September 18 - 20, 2005.Comment: 10 page

Minimal length in quantum space and integrations of the line element in Noncommutative Geometry

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime; on the other side, Connes' spectral distance in noncommutative geometry. Although on the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular on the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d'_L, which coincides exactly with the spectral distance d_D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator - together with their translations - d'_L and d_D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d'_L and d_D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.Comment: 29 pages, 2 figures. Minor corrections to match the published versio

Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime

We derive new space-time uncertainty relations (STUR) at the fundamental Planck length $L_P$ from quantum mechanics and general relativity (GR), both in flat and curved backgrounds. Contrary to claims present in the literature, our approach suggests that no minimal uncertainty appears for lengths, but instead for minimal space and four-volumes. Moreover, we derive a maximal absolute value for the energy density. Finally, some considerations on possible commutators among quantum operators implying our STUR are done.Comment: Final version published in "Class. Quantum Grav.

Braided Tensor Products and the Covariance of Quantum Noncommutative Free Fields

We introduce the free quantum noncommutative fields as described by braided tensor products. The multiplication of such fields is decomposed into three operations, describing the multiplication in the algebra M of functions on noncommutative space-time, the product in the algebra H of deformed field oscillators, and the braiding by factor Psi_{M,H} between algebras M and H. For noncommutativity generated by the twist factor we shall employ the star-product realizations of the algebra M in terms of functions on standard Minkowski space. The covariance of single noncommutative quantum fields under deformed Poincare symmetries is described by the algebraic covariance conditions which are equivalent to the deformation of generalized Heisenberg equations on Poincare group manifold. We shall calculate the covariant braided field commutator, which for free quantum noncommutative fields provides the field quantization condition and is given by standard Pauli-Jordan function. For ilustration of our new scheme we present explicit calculations for the well-known case in the literature of canonically deformed free quantum fields.Comment: 19 pages, v.4, final versio

Lorentz Covariant k-Minkowski Spacetime,

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