Immirzi parameter

The Immirzi parameter (also known as the Barbero–Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity (LQG), a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking, and the counting of microstates in loop quantum gravity. The Immirzi parameter arises in the process of expressing a Lorentz connection with noncompact group SO(3,1) in terms of a complex connection with values in a compact group of rotations, either SO(3) or its double cover SU(2). Although named after Giorgio Immirzi, the possibility of including this parameter was first pointed out by Fernando Barbero. The significance of this parameter remained obscure until the spectrum of the area operator in LQG was calculated. It turns out that the area spectrum is proportional to the Immirzi parameter. In the 1970s Stephen Hawking, motivated by the analogy between the law of increasing area of black hole event horizons and the second law of thermodynamics, performed a semiclassical calculation showing that black holes are in equilibrium with thermal radiation outside them, and that black hole entropy (that is, the entropy of the black hole itself, not the entropy of the radiation in equilibrium with the black hole, which is infinite) equals (in Planck units) In 1997, Ashtekar, Baez, Corichi and Krasnov quantized the classical phase space of the exterior of a black hole in vacuum General Relativity. They showed that the geometry of spacetime outside a black hole is described by spin networks, some of whose edges puncture the event horizon, contributing area to it, and that the quantum geometry of the horizon can be described by a U(1) Chern–Simons theory. The appearance of the group U(1) is explained by the fact that two-dimensional geometry is described in terms of the rotation group SO(2), which is isomorphic to U(1).