  This report is an extension of previous one hep-th/9812189. Several quantum
mechanical wave equations for $p$-branes are proposed. The most relevant
$p$-brane quantum mechanical wave equations determine the quantum dynamics
involving the creation/destruction of $p$-dimensional loops of topology $S^p$,
moving in a $D$ dimensional spacetime background, in the quantum state $\Phi$.
To implement full covariance we are forced to enlarge the ordinary Relativity
principle to a $new$ Relativity principle, suggested earlier by the author
based on the construction of {\bf C}-space, and also by Pezzaglia's
Polydimensional Relativity, where all dimensions and signatures of spacetime
should be included on the same footing.
