We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R 2 , we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple [19] is not a spectral metric space in the sense of [5]. This motivates the study of truncations of the spectral triple, based on M n (C) with arbitrary n ∈ N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n = 2.

Abstract. Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra M. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of M that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on M and a natural interpretation of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra M n .C/ and the algebra of matrix valued functions C 1 .M /˝M n .C/. The UV/IR mixing problem of the resultingYang-MillsHiggs models is also discussed. (2010). 81T75, 81T13, 81T15. Mathematics Subject Classification

Abstract. The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak * topology. The case of the noncommutative torus is also considered and a formula for the spectral distance between some states is also obtained.