Abstract
We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett–Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett–Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO ( 3 ) loop quantum gravity—the two can be identified as eigenstates of the same physical quantities—providing a solution to the problem of connecting the covariant SO ( 4 ) spinfoam formalism with the canonical SO ( 3 ) spin-network one. The vertex amplitude is SO ( 3 ) and SO ( 4 ) -covariant. It rectifies the triviality of the intertwiner dependence of the Barrett–Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.
Highlights
While the kinematics of loop quantum gravity (LQG) [1] is rather well understood [2, 3], its dynamics is not understood as cleanly
The spinfoam formalism [5, 6, 7, 8, 9] can be viewed as a tool for answering this question: the spinfoam vertex plays a role similar to the vertices of Feynman’s covariant quantum field theory. This picture is nicely implemented in three dimensions (3d) by the Ponzano-Regge model [10], whose boundary states match those of LQG [11] and whose vertex amplitude can be obtained as a matrix element of the hamiltonian of 3d LQG [12]
(i) There is a relation of the SO(4) states determined in this model to the projected spin network states studied by Livine in [42]. (A similar approach is developed by Alexandrov in [44].) The constrained SO(4) states that form the physical Hilbert space of the model presented here can be constructed from these projected spin networks
Summary
While the kinematics of loop quantum gravity (LQG) [1] is rather well understood [2, 3], its dynamics is not understood as cleanly. We express Regge calculus in terms of the elementary fields used in the loop and spinfoam approach, namely holonomies and the Plebanski two-form, and we study the quantization of the resulting discrete theory (on lattice derivations of loop gravity, see [6, 27, 28]). The identification is not arbitrary: the boundary states of the model are precisely eigenstates of the same quantities as the corresponding LQG states This last result provides a solution to the long-standing difficulty of connecting the covariant SO(4) spinfoam formalism with the SO(3) canonical LQG one. It provides a novel independent derivation of the LQG kinematics, and, in particular, of the quantization of area and volume. The issues raised by recovering triangulation independence and the relation with the Lorentzian–signature theory will be discussed elsewhere
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