Quasilocal energy in loop quantum gravity

Although there is no known meaningful notion of the energy density of the gravitational field in general relativity, a few notions of quasilocal energy of gravity associated to extended but finite domains have been proposed. In this paper, the notions of quasilocal energy are studied in the framework of loop quantum gravity, in order to see whether these notions can be carried out at quantum level. Two basic quasilocal geometric quantities are quantized, which lead to well-defined operators in the kinematical Hilbert space of loop quantum gravity. We then use them as basic building blocks to construct different versions of quasilocal energy operators. The operators corresponding to Brown-York energy, Liu-Yau energy, Hawking energy, and Geroch energy are obtained, respectively. The virtue of the Geroch energy operator is that it is beneficial for us to derive a rather general entropy-area relation and thus a holographic principle from loop quantum gravity.