# Algebraic mean field theory

Given a many-fermion system and a set of relevant physical observables that close under commutation to form a Lie algebra g* , a mean field theory is determined naturally on the algebra's dual space g* . The dual space consists of generalized density matrices, defined as the expectation of each element (or observable) in the Lie algebra. An algebraic mean field theory is a Hamiltonian dynamical system on a coadjoint orbit of the Lie group. Coadjoint orbits are even-dimensional surfaces in g* with a non-degenerate Poisson bracket defined on them. The energy function on a coadjoint orbit is determined from physical considerations. In essence, a generalized mean field theory for a Lie algebra g optimizes the predictions for the expectations of the algebra's generators. No prediction is made for observables that are not elements of the algebra. The advantages of the algebraic mean field theory, in comparison to the corresponding models based on irreducible representations, are generally lower dimensions and simplicity of analysis This generalized mean field construction is defined and applied to two paradigm cases: the dynamical symmetry models ROT(3) and SU(3) of collective rotational motion in nuclei. The eight-dimensional algebra rot(3) is generated by the angular momentum and the microscopic mass quadrupole operator. It is shown that its mean field theory describes rigid rotation. The eight-dimensional algebra su (3) is spanned by the angular momentum and the Elliott quadrupole operator. Its mean field theory is related to the cranked anisotropic oscillator. Both rotational mean field theories share many properties with the corresponding irreducible representations