A note on the secondary simplicity constraints in loop quantum gravity

Abstract

A debate has appeared in the literature on loop quantum gravity and spin foams, over whether the secondary simplicity constraints, reducing the connection to be Levi-Civita, should imply the shape matching conditions, reducing twisted geometries to Regge geometries. We address the question using a simple model with a flat dynamics, in which secondary simplicity constraints arise from a dynamical preservation of the primary ones. We find that shape matching conditions arise, thus providing support to an affirmative question. The origin of these extra conditions is to be found in the different graph localisation of the Hamiltonian and primary simplicity constraints. Our results are consistent with previous claims by Dittrich and Ryan, and extend their validity to Lorentzian signature and arbitrary cellular decompositions. We show in particular how the (gauge-invariant version of the) twist angle ξ featuring in twisted geometries equals on-shell the Regge dihedral angle multiplied by the Immirzi parameter, thus recovering the discrete extrinsic geometry from the Ashtekar-Barbero holonomy. Finally, we confirm that flatness implies both the Levi-Civita and the shape-matching conditions using twisted geometries and a 4-dimensional version of the vertex condition appearing in 't Hooft's polygon model.

Holographic description of boundary gravitons in (3+1) dimensions

“…[62][63][64][65], but most work is so far concentrated on BF-like TQFT's [66][67][68], which start from an enlarged space of generalized simplicial geometries [69][70][71][72][73]. A key problem is to devise either a mechanism to restrict back to proper geometric configurations [66,[74][75][76][77][78][79] or to find a dynamical principle for these generalized simplicial geometries [80]. It will be fruitful to explore alternatives, such as the one just discussed, even if these end up 'only' describing flat space.…”

Paving the Way to Space-Based Gravitational-Wave Detectors

“…The core instrument of LPF [5], consists of two quasicubic test masses, of size ð46.000 AE 0.005Þ mm and mass M ¼ ð1.928 AE 0.001Þ kg, formed from a high-purity goldplatinum alloy. During science operations, these masses are in free fall inside a single spacecraft with their centers separated by a nominal distance of ð376.00 AE 0.05Þ mm along a line that we take as the x axis (see Fig.…”

“…week ending 10 JUNE 2016 270 Hz, depending on the degree of freedom, which avoids the mixing of low-frequency electrical noise into force noise in the measurement band [5].…”

“…On LPF we thus correct the data by subtracting the centrifugal force time series, g Ω ðtÞ, calculated from the measured spacecraft angular velocity, as discussed in the next section. The third relevant difference is in the coupling of the two masses to the spacecraft motion through the static force gradients at the nominal positions of the two TMs [5]. In LISA, the TMs at the ends of one arm are coupled to their own distant spacecraft, while in LPF they share a common spacecraft.…”

“…A partial list of sources [5], and their ASD is ≃100 nT= ffiffiffiffiffiffi Hz p at 0.1 mHz. This, together with the fact that fluctuating field is uniform in space, indicates that the fluctuations arise from the interplanetary field.…”