  In this article we perform the quantization of graphene plasmons using both a
macroscopic approach based on the classical average electromagnetic energy and
a quantum hydrodynamic model, in which graphene charge carriers are modeled as
a charged fluid. Both models allow to take into account the dispersion of
graphenes optical response, with the hydrodynamic model also allowing for the
inclusion of non-local effects. Using both methods, the electromagnetic field
mode-functions, and the respective frequencies, are determined for two
different graphene structures. we show how to quantize graphene plasmons,
considering that graphene is a dispersive medium, and taking into account both
local and nonlocal descriptions. It is found that the dispersion of graphene's
optical response leads to a non-trivial normalization condition for the
mode-functions. The obtained mode-functions are then used to calculate the
decay of an emitter, represented by a dipole, via the excitation of graphene
surface plasmon-polaritons. The obtained results are compared with the total
spontaneous decay rate of the emitter and a near perfect match is found in the
relevant spectral range. It is found that non-local effects in graphene's
conductivity, become relevant for the emission rate for small Fermi energies
and small distances between the dipole and the graphene sheet.
