text: Increasing the intensity of the electromagnetic field and the particle velocity, the emission of photons with energy comparable to the electron one becomes more probable and non-linear inverse Compton scattering  starts to progressively differ from the classical limit because of quantum effects such as photon recoil. A dimensionless parameter, called electron quantum parameter, can be introduced to describe how far the physical condition are from the classical limit and how much non-linear and quantum effects matter. This parameter is given by the following expression:where  V/m is the Schwinger field. In scientific literature,  is also called . The Schwinger field , appearing in this definition, is a critical field capable of performing on electrons a work of  over a reduced Compton length , where  is the reduced Planck constant. The presence of such a strong field implies the instability of vacuum and it is necessary to explore non-linear QED effects, such as the production of pairs from vacuum. The Schwinger field corresponds to an intensity of nearly  W/cm. Consequently,  represents the work, in units of , performed by the field over the Compton length  and in this way it also measures the importance of quantum non-linear effects since it compares the field strength in the rest frame of the electron with that of the critical field. Non-linear quantum effects, like the production of an electron-positron pair in vacuum, occur above the critical field , however, they can be observed also well below this limit since ultra-relativistic particles with Lorentz factor equal to  see fields of the order of  in their rest frame.  is called also non-linear quantum parameter whereas it is a measure of the magnitude of non-linear quantum effects. The electron quantum parameter is linked to the magnitude of the Lorentz four-force acting on the particle due to the electromagnetic field and it is a Lorentz-invariant:The four-force acting on the particle is equal to the derivative of the four-momentum with respect to proper time. Using this fact in the classical limit, the radiated power according to the relativistic generalization of the Larmor formula becomes:<math display="block">
  P=\dfrac{2}{3}\dfrac{e^2m^2c^3}{\hbar^2}\chi^2
   and, therefore, some considerations can be done on which are the conditions for prolific emission, further evaluating the definition (). The electron quantum parameter increases with the energy of the electron (direct proportionality to ) and it is larger when the force exerted by the field perpendicularly to the particle velocity increases.