Can an electron borrow momentum from its field? Let us consider a charged particle moving with uniform velocity $v$. We know that the EM field due to it has some momentum too. If the mass of the particle is $M$,
then the momentum of the particle is $Mv$. Why can't the particle borrow some of its momentum from the field or lend some of its momentum to the field, thereby 
increasing or reducing its velocity? 
This shouldn't be a case of violation of law of conservation of momentum. But now let's take a look at law of conservation of 
energy. Here we can see the kinetic energy of the particle is changing, but it's only its field with which the exchange of momentum is going on. Should we 
consider the field of the particle as another system and can be fair by law of conservation of energy. I guess this kind of events doesn't happen. But can we have a
reason that puts restriction on it? Why is this not spontaneous, is there any kind of entropy for EM fields?
 A: It's acutally possible. The phenomenon is called Bremsstrahlung (its direct translations would be something like "stopping radiation"). It can be described purely with classical theories like special relativty and electromagnetism.
If a charged particle is accelerating, it "borrows" some of his momentum and energy to the EM-field which is then radiated as EM-waves carrying this momentum and energy.
A: Why can't the particle borrow some of its momentum from the field or lend some of its momentum to the field thereby increasing or reducing its velocity?
Because the particle is the field. We make an electron (and a positron) out of electromagnetic waves in gamma-gamma pair production. In atomic orbitals electrons "exist as standing waves". We can diffract electrons. In quantum field theory the electron is described as a field excitation. It isn't some billiard-ball particle that has a field. You can't take the electron's field away leaving some little point-particle thing. If you take away the field, there is no electron. For an analogy, think of a cyclone, and ask a similar question: 
Why can't the cyclone borrow some of its momentum from the rotational airflow or lend some of its momentum to the rotational airflow thereby increasing or reducing its velocity?
Because the cyclone is rotational airflow. If you took away the rotational airflow, there is no cyclone. Having said all that, take a look at Compton scattering. The photon gives some of its momentum to the electron's field thereby increasing its velocity.
Edit: see what Marcel said above. Bremsstrahlung is also known as the inverse Compton. 
