Static isolated horizons: \(SU(2)\) invariant phase space, quantization, and black hole entropy
Summary: We study the classical field theoretical formulation of static generic isolated horizons in a manifestly \(SU(2)\) invariant formulation. We show that the usual classical description requires revision in the non-static case due to the breaking of diffeomorphism invariance at the horizon leading to the non-conservation of the usual pre-symplectic structure. We argue how this difficulty could be avoided by a simple enlargement of the field content at the horizon that restores diffeomorphism invariance. Restricting our attention to static isolated horizons we study the effective theories describing the boundary degrees of freedom. A quantization of the horizon degrees of freedom is proposed. By defining a statistical mechanical ensemble where only the area \(a_{H}\) of the horizon is fixed macroscopically-states with fluctuations away from spherical symmetry are allowed-we show that it is possible to obtain agreement with the Hawkings area law (\(S = a_{H }\)/(4l \(^{2}_{p}\))) without fixing the Immirzi parameter to any particular value: consistency with the area law only imposes a relationship between the Immirzi parameter and the level of the Chern-Simons theory involved in the effective description of the horizon degrees of freedom.
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