Global Theory of Quantum Boundary Conditions and Topology Change

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 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(L 2 (Γ, C N )) of the Hilbert space of boundary data L 2 (Γ, C N ). 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.

Edge states: topological insulators, superconductors and QCD chiral bags

“…Hence Ψ approaches zero weakly in the Dirichlet limit µ → ∞. In fact the convergence to a null vector is strong in this limit [2].…”

“…Then, iγ 3 γ 4 is a generator of the so(4), or rather the spin(4) Lie algebra. The Lie algebra so(4) is the direct sum su(2) (1) ⊕su(2) (2) , where su(2) (j) are commuting su(2) Lie algebras.…”

“…al. [2], proved that as µ becomes large, and the Dirichlet condition is approached, the negative eigenvalues recede to −∞. At the same time, the corresponding eigenstates get progressively more localised at the edge and eventually become weakly zero as µ → ∞.…”

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

“…Some of the material presented here has already appeared published elsewhere (see for instance [5] where some of the preliminary ideas on the global topology of the space of self-adjoint extensions for the covariant Laplacian and its relation to topology change appeared for the first time), or will appear in various forms (see for instance [25] for a detailed discussion of 1D Schrödinger operators). The general theory of self-adjoint extensions from the point of view of quadratic forms is discussed in [26] and will not be considered here as well as the theory of self-adjoint extensions with symmetry that will be discussed elsewhere.…”

Boundary dynamics driven entanglement

“…We would also like to mention here the possibility of describing topology change as a boundary effect. This idea was already considered in [Ba95] and further elaborated in relation with specific boundary conditions in [As05], but it has gained new impetus because of Wilczek's et al [Wi12] recent contributions to it.…”