Three-dimensional loop quantum gravity: physical scalar product and spin-foam models

By K. Noui, Alejandro Perez. 2004

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In this paper, we address the problem of the dynamics in three-dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space—corresponding to the quantization of the infinite-dimensional kinematical configuration space of the theory—to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three-dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard ‘lattice’ methods), and second that no ill-defined sum over spins (‘bubble’ divergences) are present in the spin-foam representation.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/179fcf0528b4f53b7a9a785a29096a655576f666

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Citations in the corpus

On the semiclassical limit of 4d spin foam models (2008, background): “More tangible results have been obtained in 3 dimensions, where the classical and quantum theory are far simpler [15, 16, 17, 18].”
Group Field Theory: An Overview (2005, background): “this construction has been exemplified in 2+1 gravity [15] It was soon realized that spin foam models can naturally incorporate causality [16], Lorentzian signature [17] and coupling to gauge field theory [18].”
The continuum limit of loop quantum gravity - a framework for solving the theory (2014, background): “that is the path integral over the configuration variables X with (diffeomorphism invariant) action S(X) serves as projector onto states satisfying the constraints [10].”
Introduction to loop quantum gravity and spin foams (2004, background): “G[g] the operator generating a local g(x) ∈ SU(2) transformation then its action can be deﬁned directly on the elements of the basis of Hkin deﬁned above, thus UG[g]φj mn [he] = φ j mn [gsheg −1 t ], (89) where gs is the value of g(x) at the source point of the edge eand gt the value of g(x) at the ”
Spin-foams for all loop quantum gravity (2009, methodology): “An alternative derivation of the spin-foam model from 2 + 1 LQG was proposed by [5, 7].”
The Spin-Foam Approach to Quantum Gravity (2012, background): “We will see in Part IV that natural regularizations are available and that the model is well defined [61, 275, 180].”
Flipped spinfoam vertex and loop gravity (2007, methodology): “Recall that in the LQG approach the action that is quantized is the Holst action [33], obtained adding to the action a topological term that doesn’t change the equations of motion SHolst = ∫ ∗(e ∧ e) ∧ F + 1 γ ∫ (e ∧ e) ∧ F. (77) where γ is called the Immirzi parameter.”
Loop Quantum Gravity (2003, background): “The dynamics defined in the two versions of the theory ‘resemble’ one another, but it is has not yet been possible to clearly derive the relation in the 4D theory (in the 3D theory, the corresponding relation has been nicely proven [167].”
LQG vertex with finite Immirzi parameter (2007, background): “It would be be of great interest if a direct relation between these two nonperturbative quantizations of GR could be completely established, as it was done in three dimensions by Perez and Noui [29].”