I know that free electrons can't absorb photons. It is, however, known that electric forces can accelerate electrons. But if electric interactions are merely an exchange of virtual photons, how can the electron even absorb the virtual photon in the first place? I mean, before the interaction, the electron was free. So it shouldn't absorb the photon.
I suppose this has something to do with the nature of virtual photons. But I'm not sure.
Free electrons are unable to absorb photons because there is no way to achieve that while maintaining the simultaneous conservation of energy and momentum.
Virtual photons are called 'virtual' because, in QFT parlance, they are "off-shell", which means that their energy does not correspond to what you'd expect from their momentum, i.e. they represent "temporary" violations of the conservation of energy and momentum. Since that's already baked-in, it's not surprising at all that the formalism allows free electrons to interact with them.
And, for clarity, "virtual photons" do not exist; they are mere calculational tools used to make the QFT Feynman-diagram calculations easier to parse. They are emphatically not imbued with any ontology, and any questions about their behaviour are exclusively questions about the internal workings of the mathematics of QFT.
