Hessian and graviton propagator of the proper vertex

Abstract

The proper spin-foam vertex amplitude is obtained from the EPRL vertex by projecting out all but a single gravitational sector, in order to achieve correct semi-classical behavior. In this paper we calculate the gravitational two-point function predicted by the proper spin-foam vertex to lowest order in the vertex expansion. We find the same answer as in the EPRL case in the 'continuum spectrum' limit, so that the theory is consistent with the predictions of linearized gravity in the regime of small curvature. The method for calculating the two-point function is similar to that used in prior works: we cast it in terms of an action integral and to use stationary phase methods. Thus, the calculation of the Hessian matrix plays a key role. Once the Hessian is calculated, it is used not only to calculate the two-point function, but also to calculate the coefficient appearing in the semi-classical limit of the proper vertex amplitude itself. This coefficient is the effective discrete "measure factor" encoded in the spin-foam model. Through a non-trivial cancellation of different factors, we find that this coefficient is the same as the coefficient in front of the term in the asymptotics of the EPRL vertex corresponding to the selected gravitational sector.

Spinfoam cosmology with the proper vertex amplitude

“…The heat-kernel time t has been absorbed into the scaling of P . Furthermore, this ansatz is analogous to Rovelli's original proposal [24] used in the graviton propagator calculations [10,[25][26][27][28][29].…”

“…In the current paper we investigate quantum amplitudes using the Euclidean proper vertex. Thus, this work provides another test of the proper vertex which has been previously used (in its Lorentzian guise) to calculate the graviton propagator [10]. The calculation begins with fixing a graph.…”

Spin foam propagator: A new perspective to include the cosmological constant

“…While this limit works fine in models without cosmological constant (see e.g. [17,18,[24][25][26][28][29][30]) it is bound to break in models that include a cosmological constant. This can be easily seen from the following argument: In the spinfoam model with cosmological constant [7-10, 31, 32], one has to take an additional limit such that Λ → 0 in the same rate as j becomes large in order to recover the correct semiclassical behavior, i.e.…”

“…We explore this in detail for the recently proposed model [7] by Haggard, Han, Kaminski and Riello and discuss alternative definitions of a graviton propagator, in which the double scaling limit can be avoided.PACS numbers: 04.60.Pp * Electronic address: zhuang2014@fau.edu 1 See e.g. [17,18,[24][25][26][27] for the recent results on spinfoam graviton propagator and 3-point function. 2 f denotes a triangle in the 4-simplex.…”

“…PACS numbers: 04.60.Pp * Electronic address: zhuang2014@fau.edu 1 See e.g. [17,18,[24][25][26][27] for the recent results on spinfoam graviton propagator and 3-point function. 2 f denotes a triangle in the 4-simplex.…”