The symplectic group and invariants of quantum systems

André Avez

Vladimir I. Man’ko

About 20 years ago almost simultaneously a number of authors had introduced the notion of a dynamic group or dynamic symmetry for quantum systems [ 1 ] (also known as spectrum generating algebra [ 2 ] and non-invariance group [ 3 ]. Introducing this notion required a clearer understanding what is to be called the symmetry of an equation describing a physical system [ 4 ] and to investigate concrete examples of quantum systems. In [ 4–7 ] it had been shown that the dynamic symmetry of hydrogen atom is 0(4,2), as a dynamic group describing Landau levels of an electron in magnetic field can serve either non-homogeneous symplectic group ISp(4,R) or U(2,1) [ 8 , 9 ].

Kay-Michael Voit

Guy Burdet, M. Perrin

We interpret the one-element central extension of the Lie algebra of the group of real inhomogeneous linear canonical transformations of phase space as a time dependent invariance algebra in the Schrödinger picture. By limiting to a two-dimensional space quantum system, we exhibit all the nonconjugate Hamiltonians contained in the symplectic algebra which, in the Heisenberg picture, plays the role of dynamical algebra and contains several known degeneracy and spectrum generating algebras.

Gabriel Catren

Abstract. We argue that the classical description of a symplectic manifold endowed with a Hamil-tonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry (notably, Souriau’s moment map and the Mardsen-Weinstein symplectic reduction formalism) and group representation theory (notably Kirillov’s orbit method). The proposed argumentative line strongly relies on the conjecture pro-posed by Guillemin and Sternberg according to which “quantization commutes with (symplectic) reduction”. By using the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric un-derpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle. Key words: quantum mechanics; gauge theories; moment map; symplectic reduction. 1.