This week I have continued to study the paper ‘A new realization of quantum geometry‘. In particular I am interested in the proposed forms of the translation operator and the Area Operator in both the Abelian group U(1)³ and Non-Abelian group SU(2) cases.
In this article the authors obtain a new realization of quantum geometry by quantizing the recently introduced flux formulation of loop quantum gravity. The authors discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. They find that the area operator is bounded and that there are several diferent ways in which the Barbero-Immirzi parameter can be taken into account.
Spectrum of the translation operator
.For normalized eigenvectors of the form
associated eigenvalues are:
The k-representation of the eigenvectors vα,κ, with ψα(k) = e-ikα is
In this section the authors introduce the area operator and discuss some of its properties. For this they focus on the case with d = 3 spatial dimensions.
The area of a surface can be expressed in terms of the fluxes. This surface can be triangulated to form a surface ΔS with elementary triangles by t.
In order to quantize the area operators have to approximate the fluxes with difference operators. To do this choose a parameter μ> 0, along with a basis τi ∈ su(2), and define μi±∈SU(2) as:
Spectrum of the area operator
The quantized area operator is given by
Note that the squared area operator is a linear combination of bounded operators, and therefore is itself bounded. The bound does however grow with 1/μ.
Abelian group U(1)³
In the case U(1)³ we have generalized eigenvectors:
where wα,ρ is given by
This leads to a discrete spectrum:
Non-Abelian group SU(2)
In the more complicated case of the gauge group SU(2),the action of the square of the area operator is given by the matrix:
The general behaviour of the eigenvalues is similar to the case of
the gauge group U(1)³ discussed above, but for small μ and j, the
matrix Aj() is nearly diagonal with almost equal eigenvalues, approximating well the Casimir eigenvalues j(j+1). For growing spin, the approximate degeneracy of the eigenvalues is lifted, and the eigenvalues spread more and more. Moreover, the eigenvalues are bounded and an oscillatory behaviour sets in.