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

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 [7–9], 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 lP = √ Gh̄/c3. 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 E–mail: [email protected] E–mail: [email protected]