We explore some general consequences of a proper, full enforcement of the "twisted Poincaré" covariance of Chaichian et al [14], Wess [52], Koch et al [35], Oeckl [43] upon many-particle quantum mechanics and field quantization on a Moyal-Weyl noncommutative space(time). This entails the associated braided tensor product with an involutive braiding (or ⋆-tensor product in the parlance of Aschieri et al [3,4]) prescription for any coordinates pair of x, y generating two different copies of the space(time); the associated nontrivial commutation relations between them imply that x − y is central and its Poincaré transformation properties remain undeformed. As a consequence, in QFT (even with space-time noncommutativity) one can reproduce notions (like space-like separation, time-and normal-ordering, Wightman or Green's functions, etc), impose constraints (Wightman axioms), and construct free or interacting theories which essentially coincide with the undeformed ones, since the only observable quantities involve coordinate differences. In other words, one may thus well realize QM and QFT's where the effect of space(time) noncommutativity amounts to a practically unobservable common noncommutative translation of all reference frames.-Preprint 06-54 Dip. Matematica e Applicazioni, Università di Napoli; -DSF/2-2007
We apply one of the formalisms of noncommutative geometry to R N q , the quantum space covariant under the quantum group SO q (N ). Over R N q there are two SO q (N )-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of R 3 q . As in the case N = 3, one has to slightly enlarge the algebra R N q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N = 3 there is a conformal ambiguity in the natural metrics on the differential calculi over R N q . While in our previous article the frame was found 'by hand', here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of R N q with U q so(N ) into R N q , an interesting result in itself.
Guided by ordinary quantum mechanics we introduce new fuzzy spheres S d Λ of dimensions d = 1, 2: we consider an ordinary quantum particle in D = d+1 dimensions subject to a rotation invariant potential well V (r) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to 'freeze' the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutativeà la Snyder; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations -as Madore's fuzzy sphere -, but under the full orthogonal group O(D). Making the cutoff and the depth of the well dependent on (and diverging with) a natural number Λ, and keeping the leading terms in 1/Λ, we obtain a sequence S d Λ of fuzzy spheres converging to the sphere S d in the limit Λ → ∞ (whereby we recover ordinary quantum mechanics on S d ). These models may be useful in condensed matter problems where particles are confined on a sphere by an (at least approximately) rotation-invariant potential, beside being suggestive of analogous mechanisms in quantum field theory or quantum geometry. arXiv:1709.04807v3 [math-ph]
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