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Journal of Mathematical Physics

On equations of motion on compact Hermitian symmetric spaces

Stefan Berceanu, Andrei H Gheorghe

Published Mar 1, 1992• 24 citations
Publisher

Abstract

The classical equations of motion on compact coherent state manifolds that have Hermitian symmetric space structure are considered. In the case of Hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states, the classical motion and the quantum evolution are both described by the same first-order second degree differential equation. The explicit forms of the classical equations of motion as Matrix Riccati equations on the Grassmann manifold GA(Cn) and the coset manifold SO (2n)/U(n), attached to the Hartree–Fock and, respectively, Hartree–Fock–Bogoliubov problem are obtained.

DOI
10.1063/1.529700

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The classical equations of motion on compact coherent state manifolds that have Hermitian symmetric space structure are considered. In the case of Hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states, the classical motion and the quantum evolution are both described by the same first-order second degree differential equation. The explicit forms of the classical equations of motion as Matrix Riccati equations on the Grassmann manifold GA(Cn) and the coset manifold SO (2n)/U(n), attached to the Hartree–Fock and, respectively, Hartree–Fock–Bogoliubov problem are obtained.
The classical motion and the quantum evolution determined by linear Hamiltonians on coherent state manifolds of compact and noncompact Hermitian symmetric space structure are considered. A matrix Riccati equation, with opposite signs of the quadratic terms in the compact and noncompact cases, determines the classical motion and the quantum evolution. The geometric meaning of equations of motion as flow on a complex Grassmann manifold and his noncompact dual is emphasized. Possibilities of generalizing the results to flag manifolds are pointed out.
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The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).
We study the total (dynamical plus geometrical (Berry)) phase of cyclic quantum motion for coherent states over homogeneous Kähler manifolds X = G/H, which can be considered as the phase spaces of classical systems and which are, in particular cases, coadjoint orbits of some Lie groups G. When the Hamiltonian is linear in the generators of a Lie group, both phases can be calculated exactly in terms of classical objects. In particular, the geometric phase is given by the symplectic area enclosed by the (purely classical) motion in the space of coherent states. I.
This thesis is concerned with a concept of geometrising time evolution of quantum systems. This concept is inspired by the fact that the Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider, in this thesis, coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This, in particular, generalises the geometric dynamics of a harmonic oscillator in the Fock-Segal-Bargmann (FSB) space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and exhibit explicit solutions for such systems
We study the total [dynamical plus geometrical (Berry)] phase of cyclic quantum motion for coherent states over homogeneous Kähler manifolds X=G/H, which can be considered as the phase spaces of classical systems and which are, in particular cases, coadjoint orbits of some Lie groups G. When the Hamiltonian is linear in the generators of a Lie group, both phases can be calculated exactly in terms of classical objects. In particular, the geometric phase is given by the symplectic area enclosed by the (purely classical) motion in the space of coherent states.