nLab
self-adjoint extension

Idea

Given a symmetric operator on a domain of some Hilbert space, there may be several extensions of it to a self-adjoint operator. These typically correspond to choices of boundary conditions (WeiJiang).

Applications

In quantum mechanics

In quantum mechanics the observables are supposed to be self-adjoint operators, in particular the Hamiltonian. The physical input often directly provides only a symmetric operator, which encodes local information about the dynamics of the system . The choice of its self-adjoint extension corresponds to choices of boundary conditions on the states of the system, hence global information about the kinematics .

See the References on applications in quantum mechanics below.

References

General

• Guangshenh Wei, Yaolin Jiang, A characterization of positive self-adjoint extensions and its application to ordinary differential operators Proceedings of the American Mathematical Society, Volume 133, Number 10 (jstor) (pdf)

For applications in quantum theory

An exposition and motivation by means of the simple case of a quantum particle in an infinitely deep well potential is given in

• Guy Bonneau, Jacques Faraut, Galliano Valent, Self-adjoint extensions of operators and the teaching of quantum mechanics (arXiv)

A discussion in the context of AQFT is in

• H. J. Borchers, Jakob Yngvason, Local nets and self-adjoint extensions of quantum field operators , Letters in mathematical physics (web)

Applications to quantum field theory of anyons is discussed in

• G. Amelino-Camelia, D. Bak, Schrödinger Self-adjoint Extension and Quantum Field Theory (arXiv:hep-th/9406213)