Noncommutative Geometry

Quantum gravity motivates the idea that space or spacetime should be considered as quantum objects. One possible realization of this idea is noncommutative geometry: essentially one generalizes the coordinates on a manifold to a nonabelian algebra. Mathematicians have considered this possibility since the 80's. I would mention here two of the physical motivations to take this possibility seriously. One is the argument by Doplicher et al. [1], which combines Heisenberg's microscope with gravitational effects, to deduce an uncertainty principle between the coordinates of a particle (as opposed to coordinates and momenta). A different motivation is the progress made by A. Connes with his `spectral triple' approach to noncommutative geometry, which has allowed him to deduce some of the structure of the standard model from first principles [2]. In my research, I studied a particular approach to noncommutative geometry based on a generalization of differential geometry and generalizations of Lie groups called `Quantum Groups', or Hopf Algebras. I believe these mathematical structures are worth studying for a physicist because they have a chance of being a description of some solutions of the yet unknown quantum theory of gravity. Moreover, they predict interesting new phenomena when used as a background on which particles and fields propagate (see, for example, the partial results like [3], in which the authors found a noncommutative spacetime structure as an effective description of particles propagating in 2+1-dimensional gravity). In the field of Noncommutative Geometry, I intend to develop the approach to field theory on a noncommutative background spacetime for which I laid the foundations in [4]. In that paper I developed the basic differential-geometric tools (exterior and interior derivatives, Lie derivative, hodge-*, integrals) needed to describe field theories in a way that respects the noncommutative structure of spacetime. The mid-term goal is to study quantum field theories on noncommutative spacetimes in a way that is free of inconsistencies. This has the potential to lead to testable predictions.

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