2011

Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality

Martinetti, Pierre, Tomassini, Luca

Understand

We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view.

First, we compute Connes' spectral distance associated with the natural isometric action of R^2 on the algebra of the Moyal plane A.
We show that the distance between any state of A and any of its translated is precisely the amplitude of the translation.
As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane.

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