Global Theory of Quantum Boundary Conditions and Topology Change (original) (raw)
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 nontrivial 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.
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The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space mathcalM\mathcal{M}mathcalM of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions mathcalMellip\mathcal{M}_\ellipmathcalMellip is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian mathbfGr\mathbf{Gr}mathbfGr. The topology of mathcalMellip\mathcal{M}_\ellipmathcalMellip is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of extension of self-adjoint extensions.
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We show how to use boundary conditions to drive the evolution on a quantum mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schrödinger equation. In particular, we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study the topology change and to obtain necessary conditions to accomplish it in a dynamical way.
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