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Towards a first-principles approach to spacetime noncommutativity

Our main thesis in this note is that if spacetime noncommutativity is at all relevant in the quantum gravitational regime, there might be a canonical approach to pinning down its form. We start by emphasizing the distinction between an intrinsically noncommuting ‘manifold’, i.e., one with noncommuting coordinate functions, on the one hand, and particles with noncommuting position operators, on the other. Focusing on the latter case, which, we feel, more adequately reflects the experimental nature of our knowledge of spacetime properties, we find that several complementary considerations point to a spin-dependent noncommutativity, which is confirmed in the single-particle sector of Dirac's theory, as well as in Fokker's relativistic ‘center-of-mass’ prescription. Finally, we propose an extension of Jordan and Mukunda's work to gain a glimpse on the effect of curvature on position operator noncommutativity.