Volume Operator in Discretized Quantum Gravity

In loop quantum gravity, basic geometric quantities have been quantized and found to have discrete spectra.

“Basic geometric quantities have been quantized [10][11][12][13][14] and found to have discrete spectra.”

The discreteness of the spectra of area and volume is a central result in loop quantum gravity, first obtained by Rovelli and Smolin in [20], and later confirmed and clarified by a number of authors [21-25].

“The discreteness of the spectra of area and volume -and the 'quantized' structure of space that these spectra suggests- is a central result in loop quantum gravity, first obtained by Rovelli and Smolin in [20], and later confirmed and clarified by a number of authors [21-25].”

The Ashtekar-Lewandowski quantum volume element was expressed using a differential structure introduced on A/G.

“It was written in terms of a differential structure introduced on A/G”

The same annihilation result holds for all the other volume operators [14,16,18] when each lattice vertex has no more than three intersecting edges.

“Remarkably, this result continues to be true for any of the remaining volume operators [14,16,18] which can be easily seen by using our formula for the Rovelli-Smolin volume.”

The Ashtekar-Lewandowski quantum volume element was the first compact analytic formula for a quantized volume.

“It was also the first 'compact', analytic formula for a quantized volume.”

The standard point splitting regularization scheme led Ashtekar and Lewandowski to define a quantum volume element.

“The standard point splitting regularization scheme led Ashtekar and Lewandowski to the definition of a quantum volume element presented in [12,14,15].”

The Ashtekar-Lewandowski quantum volume element produced a consistent family of differential operators defined on finite-dimensional spaces labeled by graphs.

“the operator gave rise to a consistent family of differential operators precisely defined on finite dimensional spaces labeled by graphs of”

The imaginary part of the Ashtekar action vanishes on the surface of the second-class constraints and the Gauss and Lorentz constraints.

“However, in [13] it was established that the imaginary part of the Ashtekar action (2) vanishes on the surface of the second-class constraints and the Gauss and Lorentz constraints.”

Barbero generalized the Ashtekar canonical transformation to a one-parameter scale transformation U(ι) on the phase space of general relativity.

“Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation U ( ι ) on the phase space of general relativity.”