Let $X$ be the interior of a compact manifold $\overline{X}$ of dimension $n+1$ with boundary $M=\partial X$, and $g_{+}$ be a conformally compact metric on $X$, namely $\overline{g}\equiv r^2{g}_{+}$ extends continuously (or with some degree of smoothness) as a metric to $X$, where $r$ denotes a defining function for $M$, i.e. $r>0$ on $X$ and $r=0$, $dr\ne 0$ on $M$. The restrction of $\overline{g}$ to $TM$ rescales upon changing $r$, so defines invariantly a conformal class of metrics on $M$, which is called the conformal infinity of $g_{+}$. In the present paper, the author considers conformally compact metrics...

In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the $d$-dimensional quantum harmonic oscillator...