Solutions of the Klein-Gordon and Dirac equations for a particle in a constant electric field and a plane electromagnetic wave propagating along the field

Pair production: The view from the lightfront

“…This is reaffirmed by integrating over momenta to obtain, for example, the total probability of pair production or the vacuum persistence amplitude: one finds that all dependence on the plane wave fields vanishes because π ⊥ (−p − ) can be absorbed into p ⊥ by a change of variable. Thus the plane wave fields, and in particular any effective mass they may generate, do not influence the probability of vacuum decay [20,21,28].…”

“…It was found in [28] that the addition of a plane wave to a constant electric field (Schwinger's case) does not change the vacuum persistence amplitude. We will see for our fields that this remains true: the plane wave has no impact on the creation of particles, nor the properties they are created with.…”

“…The models for which most analytic progress can be made (in terms of calculating scattering amplitudes) are plane waves depending on x + [26][27][28]. The combination of a constant longitudinal electric field and periodic plane waves was described in [28]. The case of a purely longitudinal electric field E(x + ) depending arbitrarily on x + was covered by [20,21] (and includes Schwinger's result as a particular case).…”

“…A variety of models for the laser field can be found in the literature: the case of a constant, longitudinal electric field is of course covered by Schwinger's classic result [29]. The models for which most analytic progress can be made (in terms of calculating scattering amplitudes) are plane waves depending on x + [26][27][28]. The combination of a constant longitudinal electric field and periodic plane waves was described in [28].…”

“…where M 2 is clearly an effective mass (see [10,25] for details) which extends the intensity-dependent mass shift from purely periodic plane waves [26][27][28], to arbitrary plane wave backgrounds. Integrating out p + , one recovers, by definition, our W : this is precisely the same as in the free theory (4).…”