Global Theory of Quantum Boundary Conditions and Topology Change

TL;DR

This paper develops a global geometric and topological framework for boundary conditions in constrained quantum systems, revealing how topology change involves infinite energy transitions and implications for string theory.

Contribution

It introduces a novel geometric description of boundary conditions as a unitary group, linking topology change to singularities in the Cayley transform and energy divergence.

Findings

Topology change involves infinite quantum energy.

Boundary conditions form a unitary group with rich topology.

Implications for string theory with open and closed strings.

Abstract

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 $\Gamma=\partial M$. The space $\CM$ 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 $\CM$. The space $\CM$ itself can be identified with the unitary group $\CU(L^2(\Gamma,\C^N))$ of the Hilbert space of boundary data $L^2(\Gamma,\C^N)$. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, $\CC_-\cap \CC_+$ (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 condition reaches the Cayley submanifold $\CC_-$. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space $\CM$ 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 $\CC_-$ can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold $\CC_-$ is dual of the Maslov class of $\CM$.