Quantum Space-Time Symmetries: A Principle of Minimum Group Representation

Abstract: We show that, as in the case of the principle of minimum action in classical and quantum mechanics, there exists an even more general principle in the very fundamental structure of {\it quantum space-time}: This is the principle of {\it minimal group representation} that allows to consistently and simultaneously obtain a natural description of the spacetime dynamics and the physical states admissible in it. The theoretical construction is based on the physical states, average values of the Metaplectic group $Mp(n)$ generators: the double covering of $SL(2C)$ in a vector representation, with respect to the {\it coherent states} carrying the spin weight. Our main results here are: (i) A connection between the Metaplectic symmetry generators and the physical state dynamics. (ii) The ground states are coherent states, of Perelomov-Klauder type of the Metaplectic group dividing the Hilbert space into {\it even} and {\it odd} states. (iii) The physical states have spin contents $s = 0,\; 1/2, \;1,\; 3/2$ and $2$. (iv) The generators introduce a natural supersymmetry and a superspace whose line element is the geometrical Lagrangian of our model. (v) A coherent physical state of spin 2 is obtained naturally related to the metric tensor. (vi) This is {\it naturally discretized} by the discrete series in the $n$ number representation, reaching the classical (continuous) space-time for $n$ $\rightarrow\infty$. (vii) A relation emerges between the coherent state metric eigenvalue $\alpha$ and the black hole entropy through the Planck length. The lowest level of the quantum space-time spectrum, $n = 0$ and its characteristic length, yields a minimum entropy for the black hole history.