Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for any arbitrary Lie algebra {\textbf g} of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space {\textbf g} $^\ast$ of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The SU(3) mean field theory is constructed explicitly in the coadjoint orbit framework.Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for any arbitrary Lie algebra {\textbf g} of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space {\textbf g}$^\ast$ of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The SU(3) mean field theory is constructed explicitly in the coadjoint orbit framework.

arXiv:nucl-th/9909072APS Division of Nuclear Physics Meeting AbstractsSep 29, 19990 citations

Symplectic models of n-particle systems

Karol Grudziński, B. G. Wybourne

Reports on Mathematical PhysicsOct 1, 19968 citations

Towards a Group-Theoretical Interpretation of Mechanics

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.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.

Nov 30, 20131 citation

Quantum Hamiltonian Mechanics on Symplectic Manifolds

Maciej Błaszak

Springer eBooksJan 1, 20190 citations

Mean field theory for usp(4) sime so(5)

George Rosensteel

The simple Lie algebra usp(4) ≃ so(5) is a ten-dimensional algebra that contains the angular momentum algebra su(2). In nuclear structure physics, the algebra so(5) describes beta-rigid collective modes in the Bohr–Mottelson and interacting boson models. The so(5) dual space consists of density matrices which are defined by the expectations of so(5) generators. A coadjoint orbit is a common level surface in the dual space of the two so(5) Casimirs. This paper develops mean field theory on any coadjoint orbit of so(5) densities. When the densities of a set of quantum states lie on one orbit, the system is said to have a weak dynamical symmetry. A Lax pair determines the dynamics of so(5) densities on each coadjoint orbit. Analytic solutions are reported for rotating so(5) densities in equilibrium for a particular energy function.The simple Lie algebra usp(4) ≃ so(5) is a ten-dimensional algebra that contains the angular momentum algebra su(2). In nuclear structure physics, the algebra so(5) describes beta-rigid collective modes in the Bohr–Mottelson and interacting boson models. The so(5) dual space consists of density matrices which are defined by the expectations of so(5) generators. A coadjoint orbit is a common level surface in the dual space of the two so(5) Casimirs. This paper develops mean field theory on any coadjoint orbit of so(5) densities. When the densities of a set of quantum states lie on one orbit, the system is said to have a weak dynamical symmetry. A Lax pair determines the dynamics of so(5) densities on each coadjoint orbit. Analytic solutions are reported for rotating so(5) densities in equilibrium for a particular energy function.

Journal of Physics A Mathematical and GeneralOct 5, 20053 citations

^\ast

of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The SU(3) mean field theory is constructed explicitly in the coadjoint orbit framework.