Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity

Abstract

Although the physical Hamiltonian operator can be constructed in the deparametrized model of loop quantum gravity coupled to a scalar field, its property is still unknown. This open issue is attacked in this paper by considering an operatorĤ v representing the square of the physical Hamiltonian operator acting nontrivially on a two-valent vertex of spin networks. The Hilbert space H v preserved by the graphing changing operatorĤ v is consist of spin networks with a single two-valent non-degenerate vertex. The matrix element ofĤ v are explicitly worked out in a suitable basis. It turns out that the operatorĤ v is essentially self-adjoint, which implies a well-defined physical Hamiltonian operator in H v for the deparametrized model.

Loop quantum Schwarzschild interior and black hole remnant

“…Thus ψ(µ) is unnormalizable. Moreover, since ψ(µ) behaves asymptoticly as a plane wave of ln(|µ|), we can show by using the same techniques as in [50] and the Weyl's criterion [49], that the spectrum of p b sin(δ b b) is the entire real line R. Furthermore, the two leading-order functions in (3.19), denoted by…”

Bouncing evolution in a model of loop quantum gravity

“…introduced in [32,44]. However, instead of adding any new edge to the graph γ by the operator kinĤ E v,ee as defined in [33], we use the loops constituting γ to regulate the curvature and ignore the limit process in the classical expression. According to the framework, operator…”

“…Several proposals of graph-changing quantum Hamiltonian operator were considered in the literatures [1,4,23,28,31,32]. Self-adjointness of such operators is addressed in [33], where a specific graph-changing Hamiltonian operator is proven to be self-adjoint in a domain of certain special states. The second category, that is the graph persevering action, is the one considered in the current paper.…”

Fermion coupling to loop quantum gravity: canonical formulation

“…The spectra of these operators take discrete values, which implies the fundamental discreteness of the spacetime. The purely gravitational Hamiltonian constraint is regularized and promoted to an operator in the kinematic Hilbert space by [25,26,[37][38][39][40][41][42][43][44][45]. The Hamiltonian constraint comprises of the curvature of the connection A.…”