The fermionic contribution to the spectrum of the area operator in nonperturbative quantum gravity

Abstract

The role of fermionic matter in the spectrum of the area operator is analysed using the Baez-Krasnov framework for quantum fermions and gravity. The result is that the fermionic contribution to the area of a surface S is equivalent to the contribution of purely gravitational spin network's edges tangent to S. Therefore, the spectrum of the area operator is the same as in the pure gravity case. PACS: 04.60.Ds, 04.20.Cv.Loop quantum gravity [1], the nonperturbative approach to quantum gravity, is nowadays a mathematically well-defined theory with a powerful predictive character (see [2] for a recent review). The theory is based on the Hamiltonian formulation of general relativity due to Ashtekar [3] which, as was shown in [4], is the ADM formulation of the (self-dual sector of the) Plebanski action [5]. At present, the theory is usually formulated in terms of the real SU(2) Ashtekar connection, whose use has been advocated by Barbero [6], and which can be obtained through a canonical transformation from the original complex Ashtekar variables.Amongst the most striking results of loop quantum gravity are the spectra of the area and volume operators [7][8][9], and the computation of the entropy of black holes [10]. These results point to the existence of discrete aspects of spacetime at the Planck length l P = Gh/c 3 .The research in loop quantum gravity is presently developing along three main directions.The first of these focuses on the physics of black holes [10,11]. The second deals with the the Hamiltonian constraint [12,13] and with Feynman-type formulations [14] of the quantum dynamics. The third studies the coupling of matter fields to quantum gravity. For instance, in [15] the contribution of the quantum states to the fermionic mass has been studied, while the possibility of a quantum-gravity induced vanishing of the cosmological constant has *

Spinfoam fermions

“…e dψ e dψ e e −v e ψ e ψ e , (16) which realizes the scalar product at each edge, seen as a boundary between two 4-cells, and where we interpret the field ψ e as an anticommuting variable, in order to take Pauli principle into account. The volume v e in the exponent is needed for dimensional reasons and to keep into account the fact that in the classical theory…”

“…and is therefore a Lorentz-invariant contraction of the normals to the 3-cells 5 . 5 If we do not include the volume factors in (16), then ( 21) is replaced by…”

“…For earlier studies on how to couple fermions to a discretized gravitational field, see [11][12][13][14]. For detailed studies of fermions in the loop quantum gravity canonical formalism, see [15][16][17][18]. On fermions in 3D, spinfoams and quantum cosmology and the effect of fermions on Holst gravity, see also [19][20][21][22][23].…”

Knot theory and a physical state of quantum gravity

“…This is, however, a very non-trivial problem to address. Matter has been coupled to gravity via the self-dual variables and has been quantized [69][70][71][72][73]. The problem has only been solved on the kinematical Hilbert space.…”

Loop Quantum Gravity

“…See Ref. [75] for more discussions.) In the same regard, both the gravitational and Yang-Mills fields are self-sourcing, as they yield colorings for both links and nodes.…”