Global Theory of Quantum Boundary Conditions and Topology Change

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold M with regular boundary Γ = ∂M . The space M of self-adjoint extensions of the covariant Laplacian on M is shown to have interesting geometrical and topological properties which are related to the different topological closures of M . In this sense, the change of topology of M is connected with the non-trivial structure of M. The space M itself can be identified with the unitary group U(L2(Γ,C )) of the Hilbert space of boundary data L2(Γ,C ). This description, is shown to be equivalent to the classical von Neumann’s description in terms of deficiency index subspaces, but it is more efficient and explicit because it is given only in terms of the boundary data, which are the natural external inputs of the system. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, C−∩ C+ (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary conditions reaches the Cayley submanifold C−. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space M is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self–adjoint boundary conditions, the space C− can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold C− is dual of the Maslov class of M. The phenomena are illustrated with some simple low dimensional examples.