Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups

Abstract

International audienceWe formulate Euler-Poincaré equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EP A ut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing ?-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group Aut vol(P) of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group. © American Institute of Mathematical Sciences

Momentum maps for mixed states in quantum and classical mechanics

“…In the classical case, a construction similar to (28) also leads to a dual pair of momentum maps [24,25,47] and this is reported in the following section.…”

“…In the case of classical mechanics, this fact led Marsden and Weinstein [47] to construct a dual pair of momentum maps underlying planar incompressible fluid flows. As reported also in the sections below, this construction has recently been developed further in [24,25], while the application to Liouville-type (Vlasov) equations was presented in [33]. The following section shows that an analogue construction also underlies quantum mixed states.…”

“…to observe that the generalized mixture (14) identifies a momentum map for the natural right action ψ(r) → Uψ(r) of unitary operators U ∈ U(H ). We remark that the symplectic form (15) is strictly related to a class of symplectic forms previously appeared in [24,25]; let S be a compact orientable manifold with volume form µ S and let (M, ω) be an exact symplectic manifold. One can endow the manifold F (S, M) of smooth functions S → M with the symplectic formω…”

“…Likewise, in the classical setting the same group acts by pullback on the space F (R n , R 6 ) of generalized coordinates q(r),p(r) from Section 2.2. Since this representation is also Hamiltonian, it leads to an equivariant momentum map that is expressed as [24,25,33] q(r),p(r) → −d p a (r)dq a (r) = dq a (r) ∧ dp a (r) ∈ dΛ 1…”

Dual Pairs for Non-Abelian Fluids

“…the EPAut vol case), a pair of momentum maps has been found in [10] that extends the dual pair for the ideal fluid found in [33]. Interestingly, the proper definition of one of the momentum maps needs the introduction of new infinite-dimensional Lie groups, such as the group of special Hamiltonian automorphisms and the group of Vlasov chromomorphisms, defined in [10]. In this case, one of the momentum maps yields a Clebsch representation that extends the classical Clebsch representation for ideal fluid to the non-abelian case, whereas the other momentum map recovers, as a particular case, the expression of the Klimontovich particle solution of the Yang-Mills Vlasov equations.…”