The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Abstract

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 M 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 M ellip is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of M ellip 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

Hyperspherical δ-δ′ potentials

“…Physically one would just think on V G (x − x 0 ) as a potential term in the same way as the Dirac-δ potential [8]. In this view, once the operator H 0 is fixed, the selfadjoint extensions can be seen as potentials supported on a point, and the other way around because of the one-to-one correspondence demonstrated in [15] (and recently reviewed in [14]).…”

“…The key point to understand it is that only for d = 2 and = 0 the centrifugal potential in (14) is attractive (centripetal), since d + 2 − 3 = −1 < 0. Proof.…”

“…From what has been mentioned above, it is intiutively clear that quantum boundaries and contact interactions are almost the same. The rigorous mathematical framework to study them is the use of selfadjoint extensions to represent extended objects and point supported potentials (see [14][15][16] for a more physical point of view), such as plates in the Casimir effect setup [17,18], or contact interactions more general than the Dirac-δ in quantum mechanics and effective quantum field theories [19][20][21][22][23][24][25]. The theory of selfadjoint extensions for symmetric operators has been well known to mathematicians for many years.…”

Self-adjoint extensions and unitary operators on the boundary

“…Recently, Asorey, Marmo and Ibort [5,6] proposed on physical ground a different parametrization of the self-adjoint extensions of differential operators in terms of unitary operators U on the boundary. This description relies more directly on physical intuition and in the last years it has been applied to several systems ranging from one dimensional quantum systems with changing boundary conditions [3] or with moving boundaries [15,17], to the Aharonov-Bohm effect [14], to field theories [4], and in particular to the investigation of vacuum fluctuations and the Casimir effect [8,7].…”

Modelling quantum black holes

“…[20][21][22] The well known potential of this kind in one dimension is the δ(x) which introduces discontinuity in the derivative of the wavefunctions. More general potentials of the same type is the δ (x) potential which has the new feature of introducing discontinuities in the wavefunction it-self.…”