Universal unfolding of Hamiltonian systems: from symplectic structure to fiber bundles

Abstract

The problem of the global Lagrangian description of a dynamical system which has a global Hamiltonian description is considered; the motion of a charged particle in the field of a static magnetic monopole is a prototype of such dynamical problems. The symplectic structure on the (cotangent bundle of the) manifold is associated with a closed but not necessarily exact two-form. In conventional language this means that the tensor field inverse to the symplectic metric is divergence free but not necessarily expressible globally as the curl of a vector field; and hence the usual passage to Hamilton's principle and the Lagrangian can be carried out in patches but not necessarily globally. Generalizing techniques developed elsewhere, we show that the fundamental expression for the action integral is like the surface integral of the closed but not necessarily exact antisymmetric tensor field (two-form) rather than like the line integral of a vector field (one-form). This naturally leads to the path-space formalism in which the action functional depends not only on a point in configuration space but also a path from a chosen fixed point terminating in the relevant point in configuration space. We then show how this is a "universal unfolding" and how the topological obstructions in the configuration-space description are circumvented in the path-space description. The path-space description contains redundancies and they can be reduced; locally the new manifold is obtained by associating an angle of rotation [a U(1) fiber] with each configuration-space point, but globally there would be a nontrivial fiber-bundle structure. These questions naturally carry over to the quantum theory of such systems and it is shown how quantization conditions arise. The structure of the quantum-theory formalism for such systems is analyzed. The considerations are applied to the monopole problem where the enlarged space in which the Lagrangian description is possible is explicitly identified as the manifold of the group SU(2).