Universal invariants of quantum-mechanical and optical systems

We give a brief review of the theory of quantum and optical universal invar iants, i.e., certain combinations of the second- and higher-order moments ( variances) of quantum-mechanical operators or the transverse phase-space co ordinates of optical paraxial beams that are preserved in time (or along th e axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that t he effective Hamiltonian is either a generic quadratic form of the generali zed coordinate-momenta operators or a Linear combination of generators of c ertain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we el ucidate the relation between the quantum invariants and the classical unive rsal integral invariants of Poincare and Cartan. The specific features of t he Gaussian beams are discussed as examples. (C) 2000 Optical Society of Am erica [S0740-3232(00)04112-0] OCIS codes: 000.1600, 350.5500, 030.6600, 060 .5530, 060.2310.