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 so-called unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning ton-mode 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G Racah,Ergeb. Exakt. Naturwiss. 37, 28 (1965)
B G Wybourne,Classical Groups for Physicists (Wiley, New York, 1974)
R Gilmore,Lie Groups, Lie Algebras and some of their applications (John Wiley and Sons, New York, USA, 1974)
F J Dyson,Symmetry groups in nuclear and particle physics (Benjamin, New York, 1966)
V Guillemin and S Sternberg,Symplectic techniques in physics (Cambridge University Press, 1984)
V I Arnold,Mathematical methods of classical mechanics (Springer, Berlin, 1978)
C L Siegel,Am. J. Math. LXV, 1 (1943), reprinted as C L Siegel,Symplectic Geometry (Academic Press, New York, 1964)
G Marmo, G Morandi and N Mukunda,La Rivista del Nuovo Cimento 13, No. 8, 1990
R Abraham and J E Marsden,Foundations of Mechanics (Benjamin-Cummings, Reading, Mass., 1978)
T F Jordan,Linear operators for quantum mechanics (John Wiley, New York, 1974)
G Lion and M Vergne,The Weil representation, Maslov Index, and Theta series (Birkhauser, Basel, 1980)
These are generalizations ton dimensions of definitions familiar from first order optics forn=1, 2. See for instance O N Stavroudis,The Optics of Rays. Wave Fronts and Caustics, (Academic Press, New York, 1972)
M Nazarathy and J Shamir,J. Opt. Soc. Am. B72, 356 (1982)
See for instance R Gilmore ref. [1] above
S Helgason,Differential geometry and symmetric spaces, (Academic Press, New York, 1962)
See for instance G. Racah in ref. [1] above
A Weil,Acta Mathematica 111, 143 (1964)
R Ranga Rao,The Maslov Index on the simply-connected covering group and the metaplectic representation, The metaplectic representation in the Fock model (Preprints, Dept. of Mathematics, Univ. of Illinois, Urbana, Illinois 61801) (1992)
R Simon and N Mukunda,The two-dimensional symplectic and metaplectic groups and their universal cover inSymmetries in Science VI: From the rotation group to quantum algebras, edited by B Gruber, (Plenum Press, New York, 1993) p. 659–689
For the case of Sp(2, ℛ) these are discussed in K B Wolf,Integral transforms in science and engineering (Plenum Press, New York, 1979)
R Simon, N Mukunda and B Dutta,Phys. Rev. A49, 1567 (1994)
Arvind, B Dutta, N Mukunda and R Simon,Phys. Rev. A52, 1609 (1995)
H Weyl,The theory of groups and quantum mechanics (Dover, New York, 1931)
E P Wigner,Phys. Rev. 40, 749 (1932); J E Moyal,Proc. Camb. Philos. Soc. 45, 99 (1949)
R Simon, E C G Sudarshan and N Mukunda,Phys. Rev. A36, 3868 (1987)
N Mukunda, R Simon, and E C G Sudarshan,Phys. Lett. A124, 223 (1987)
J R Klauder and E C G Sudarshan,Fundamentals of quantum optics (Benjamin, New York, 1968)
Arvind, B Dutta, C L Mehta and N Mukunda,Phys. Rev. A50, 39 (1994)
J Williamson,Am. J. Math. 58, 141 (1936)
D F Walls,Nature (London) 306, 141 (1983)
W Schleich and J A Wheeler,J. Opt. Soc. Am. B4, 1715 (1987)
B Dutta, N Mukunda, R Simon and A Subramaniam,J. Opt. Soc. Am. B10, 253 (1993)
R Simon, E C G Sudarshan and N Mukunda,Phys. Rev. A37, 3028 (1988)
Bialynicki-Birula, inCoherence Cooperation and Fluctuations, edited by F Haake, L M Narducci and D F Walls (Cambridge University Press, London, 1984)
R Simon, and N Mukunda,Phys. Rev. Lett. 70, 880 (1993)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arvind, Dutta, B., Mukunda, N. et al. The real symplectic groups in quantum mechanics and optics. Pramana - J Phys 45, 471–497 (1995). https://doi.org/10.1007/BF02848172
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02848172
Keywords
- Symplectic groups
- symplectic geometry
- Huyghens kernel
- uncertainty principle
- multimode squeezing
- Gaussian states