In Quantum Electrodynamics (QED), the photon is indeed the gauge boson that mediates the electromagnetic force between charged particles. However, the photon itself does not carry the electric charge associated with the electromagnetic interaction. The reason for this can be understood from the structure of the QED theory and the gauge symmetry it possesses.
In gauge theories like QED, the fundamental interactions are described by gauge bosons, which are the force-carrying particles. These gauge bosons are associated with the symmetries of the theory. In the case of QED, the gauge symmetry is known as U(1) symmetry, which corresponds to the conservation of electric charge.
The gauge symmetry in QED allows for the existence of a conserved electric charge, meaning that the total electric charge of a system remains constant during interactions. The photon is the gauge boson associated with this U(1) symmetry and mediates the electromagnetic interaction between charged particles.
However, the photon itself is considered to be chargeless. This is because the gauge symmetry requires the gauge boson to be neutral under the symmetry it mediates. If the photon carried an electric charge, it would violate the U(1) symmetry of QED.
In QED, the charged particles, such as electrons and positrons, interact with each other by exchanging virtual photons. These virtual photons facilitate the transfer of electric charge and mediate the electromagnetic force between charged particles.
So, while the photon is responsible for mediating the electromagnetic interaction, it does not possess electric charge itself in order to maintain the gauge symmetry of QED.