de Vaucouleurs-Ikeuchi Diagram and Commutation Relations among Phase Space Coordinates

Abstract

We consider the relations between de Vaucouleurs-Ikeuchi diagram and generalized commutation relations among the coordinates and momenta. All physical objects in the Universe ranging from elementary particles to super cluster of galaxies are confined within the Triangle of the de Vaucouleurs-Ikeuchi diagram on the matter density versus scale length plane. These three boundaries are characterized by the quantum uncertainty principle, gravitational event horizon, and cosmological constant. These are specified by the non-zero commutation relations $[x_\mu,p_\nu], [x_\mu, x_\nu]$ (strictly $[x_i,t]$) and $[p_\mu,p_\nu]$, respectively. The canonical commutation relation $[x_i,p_j]$ are slightly modified, which preserves the self consistency as a whole.