Three lectures on global boundary conditions and the theory of self--adjoint extensions of the covariant Laplace--Beltrami and Dirac operators on Riemannian manifolds with boundary

TL;DR

This paper explores the mathematical framework of self-adjoint extensions of covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary, emphasizing their physical significance and topological aspects.

Contribution

It provides a boundary condition-based description of self-adjoint extensions, linking geometry, topology, and physics, including topology change implications.

Findings

Describes global boundary conditions for self-adjoint extensions

Links boundary conditions to physical phenomena like the Casimir effect

Analyzes the topological structure of the space of boundary conditions

Abstract

In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation with the theory of global boundary conditions. Self-adjoint extensions of symmetric operators, specially of the Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as they determine either the energy of quantum systems and/or their unitary evolution. The well-known von Neumann's theory of self-adjoint extensions of symmetric operators is not always easily applicable to differential operators, while the description of extensions in terms of boundary conditions constitutes a more natural approach. Thus an effort is done in offering a description of self-adjoint extensions in terms of global boundary conditions showing how an important family of self-adjoint extensions for the Laplace-Beltrami and Dirac operators are easily describable in this way. Moreover boundary conditions play in most cases an significant physical role and give rise to important physical phenomena like the Casimir effect. The geometrical and topological structure of the space of global boundary conditions determining regular self-adjoint extensions for these fundamental differential operators is described. It is shown that there is a natural homology class dual of the Maslov class of the space. A new feature of the theory that is succinctly presented here is the relation between topology change on the system and the topology of the space of self-adjoint extensions of its Hamiltonian. Some examples will be commented and the one-dimensional case will be thoroughly discussed.