Corrigendum: The Plebanski sectors of the EPRL vertex

We correct what amounts to a sign error in the proof of part (i.) of theorem 3 in Class.Quant.Grav.28 225003 (arXiv:1107.0709). The Plebanski sectors isolated by the linear simplicity constraints do not change --- they are still the three sectors (deg), (II+), and (II-). What changes is the characterization of the continuum Plebanski two-form corresponding to the first two terms in the asymptotics of the EPRL vertex amplitude for Regge-like boundary data. These two terms do not correspond to Plebanski sectors (II+) and (II-), but rather to the two possible signs of the product of the sign of the sector --- +1 for (II+) and -1 for (II-) --- and the sign of the orientation $\epsilon_{IJKL}B^{IJ} \wedge B^{KL}$ determined by $B^{IJ}$. This is consistent with what one would expect, as this is exactly the sign which classically relates the BF action, in Plebanski sectors (II+) and (II-), to the Einstein-Hilbert action, whose discretization is the Regge action appearing in the asymptotics.Comment: Corrigendum to arXiv:1107.0709, 4 page

Purely geometric path integral for spin foams

Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for a path integral to be at least formally equivalent to the corresponding canonical quantization, at each point in the space of histories it is important that the integrand have not only the correct phase -- a topic of recent focus in spin-foams -- but also the correct modulus, usually referred to as the measure factor. The correct measure factor descends from the Liouville measure on the reduced phase space, and its calculation is a task of canonical analysis. The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst formulation, in which the basic variables are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the connection degrees of freedom have been integrated out. We call this a purely geometric Plebanski-Holst path integral. In prior work in which one of the authors was involved, the measure factor for the Plebanski-Holst path integral with both connection and two-form variables was calculated. Before one discretizes this measure and incorporates it into a spin-foam sum, however, one must integrate out the connection in order to obtain the purely geometric version of the path integral. To calculate this purely geometric path integral is the principal task of the present paper, and it is done in two independent ways. Gauge-fixing and the background independence of the resulting path integral are discussed in the appendices.Comment: 21 page

Piecewise linear loop quantum gravity

We define a modification of LQG in which graphs are required to consist in piecewise linear edges, which we call piecewise linear LQG (plLQG). At the diffeomorphism invariant level, we prove that plLQG is equivalent to standard LQG, as long as one chooses the class of diffeomorphisms appropriately. That is, we exhibit a unitary map between the diffeomorphism invariant Hilbert spaces that maps physically equivalent operators into each other. In addition, using the same ideas as in standard LQG, one can define a Hamiltonian and Master constraint in plLQG, and the unitary map between plLQG and LQG then provides an exact isomorphism of dynamics in the two frameworks. Furthermore, loop quantum cosmology (LQC) can be exactly embedded into plLQG. This allows a prior program of the author to embed LQC into LQG at the dynamical level to proceed. In particular, this allows a formal expression for a physically motivated embedding of LQC into LQG at the diffeomorphism invariant level to be given.Comment: 19 page