The real symplectic groups in quantum mechanics and optics
Abstract
We present a utilitarian review of the family of matrix groups Sp(2n, ℛ), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the socalled unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning tonmode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n, ℛ) action are delineated.
 Publication:

Pramana
 Pub Date:
 December 1995
 DOI:
 10.1007/BF02848172
 arXiv:
 arXiv:quantph/9509002
 Bibcode:
 1995Prama..45..471A
 Keywords:

 Symplectic groups;
 symplectic geometry;
 Huyghens kernel;
 uncertainty principle;
 multimode squeezing;
 Gaussian states;
 Quantum Physics
 EPrint:
 Review article 43 pages, revtex, no figures, replaced because somefonts were giving problem in autometic ps generation