  Although there is no known meaningful notion of the energy density of the
gravitational field in general relativity, a few notions of quasi-local energy
of gravity associated to extended but finite domains have been proposed. In
this paper, the notions of quasi-local 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 quasi-local 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 quasi-local 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 beneficial
for us to derive a rather general entropy-area relation and thus a holographic
principle from loop quantum gravity.
