Spectral triplets, emergent geometry and entropy in Moyal plane

It was shown by Doplicher et.al. that the measurement of spacetime intervals of the order of Planck length scale is operationally impossible, as the process of measurement invariably gives rise to a black hole formation. This can be avoided by postulating non-vanishing commutation relations between the coordinates, which are now promoted to the level of operators. Formulation of quantum mechanics in these kinds of spaces through the introduction of Hilbert spaces of Hilbert-Schmidt operators is then shown to allow the construction of spectral triplets a la Connes naturally. The computation of spectral distance between pure and mixed states is then shown to exhibit a deep connection between entropy and geometry.