Radiation Reaction in Quantum Vacuum

From the development of the electron theory by H. A. Lorentz in 1906, many authors have tried to reformulate this model named"radiation reaction". P. A. M. Dirac derived the relativistic-classical electron model in 1938, which is now called the Lorentz-Abraham-Dirac model. But this model has the big difficulty of the run-away solution. Recently, this equation has become important for ultra-intense laser-electron (plasma) interactions. Therefore, it is desirable to stabilize this model of the radiation reaction for estimations. Via my recent research, I found a stabilized model of radiation reaction in quantum vacuum. This leads us to an updated Fletcher-Millikan's charge to mass ratio including radiation, de/dm, derived as the 4th order tensor measure. In this paper, I will discuss the latest update of the model and the ability of the equation of motion with radiation reaction in quantum vacuum via photon-photon scatterings.

Abraham [3]. 2 4 00 LAD LAD 0 0 M 22 [ , ] , This is called the LAD radiation reaction force. J. Schwinger [5,6]. One laser facility, which can achieve such ultra-high intensity is LFEX (Laser for fast ignition experiment) at the Institute of Laser Engineering (ILE), Osaka University [7] and another is the next laser generation project, the Extreme Light Infrastructure (ELI) project [8] in Europe. If the laser intensity is higher than 10 22 W/cm 2 , strong bremsstrahlung might occur.
Accompanying this, the radiation reaction force (or damping force) can have a strong influence on the charged particle [9]. This is in spite of the fact that the LAD equation has a very significant mathematical problem. The solution of the LAD equation has the factor of an exponential. Let the vector function be , the the solution of the LAD equation is This solution is derived by integration of the LAD equation, but this solution goes rapidly to infinity, is a very small value [10,11]. We call this run-away depending on the first term in Eq.(3) named the Schott term. It should be avoided to solve the equation stably.
For avoidance of this run-away problem, we've considered a radiating electron with a dress of field, in our previous paper [12].
I named here this equation the Seto-Zhang-Koga (SZK) equation for instance. This dressed electron was described by vacuum polarization via the Heisenberg-Euler Lagrangian density [13,14]. The dress stabilizes run-away by changing the coupling constant However, the previous model considered only the correction of radiation from an electron. Moreover, the introduction of the external field was artificial (Eq.(24) in [12]).
To address these points, I will introduce a new model of radiation reaction which incorporates a smooth installation of the external fields, including the radiation-external field interaction in this paper.
To achieve this, we first consider a more general equation of motion with radiation reaction in quantum vacuum in Section 2. But, we will not investigate a more concrete dynamics of quantum vacuum beyond the Heisenberg-Euler's vacuum. In this phase, we only assume the Lagrangian density is a function of | FF  and |* FF  . Next, I will proceed to a concrete model by using the lowest order Heisenberg-Euler Lagrangian density as the model of quantum vacuum in Section 3. I will present the stability of the new equation via analysis and numerical calculations. Finally, this will lead us to an anisotropic correction for the charge to mass ratio by R. Fletcher and H. Millikan [15,16]. Rev., Oct 24, 2014.

Derivation of a new method of radiation reaction
The Heisenberg-Euler Lagrangian density includes the dynamics of the quantum vacuum correction.
Of course, this Lagrangian density Quantum Vacuum L needs to converge to the Heisenberg-Euler Lagrangian density when the field F is a constant field. For instance, we assume that L and Quantum Vacuum L are functions of C  . From this equation, the Maxwell equation is derived as follows: In these equations, represents the vacuum "polarization", therefore, Here, it is denoted as for instance. In our previous model [12], we Therefore, we didn't consider the correction of the external field. I could incorporate the external field naturally here. This is the most important difference between the new and old model.     Here, L is the permittivity tensor in Minkowski spacetime. However, we define a new tensor, Here  Rev., Oct 24, 2014.

Equation of motion
In Section 2, the quantum vacuum was assumed to be functions of | FF  and |* FF  without concrete formulations. The Heisenberg-Euler Lagrangian density expresses the dynamics of quantum vacuum, but can be applied only for constant fields. However, its lowest order should be contained in Quantum Vacuum L [12]. Therefore, in this chapter, I assume that, 2 In this case, instead of Eq. (12), by using perturbations, 0 f and 0 g are Here, I used the relation that The stability only depends on the external field in the rest frame of an electron. By using the Schwinger (33)

Run-away avoidance
My previous model could avoid run-away (the effect of the self-acceleration) [12]. In this section, I will show that my new equation can avoid run-away by using a two-stage analysis. The first is the investigation of the radiation upper limit and the second is the asymptotic analysis proposed by F.
Röhrich [17]. The run-away solution deeply depends on the infinite radiation from an electron. The physical meaning of run-away is a time-continuously infinite light emission via stimulations by an electron's self-radiation. In another words, when we can limit the value of the radiation, we can say the model avoids run-away. . In the rest frame,

Calculations
As the final section in this section, I will present numerical calculation results showing the behavior of each model in a laser-electron interaction.  [19,20].  Eq.(28,29) in reference [12]. The pulse width is 22fsec and the laser wavelength is 0.82μm . The electric field is set in the y direction, the magnetic field is in the z direction. The electron travels in the negative z direction, with the energy of 700MeV initially. The numerical calculations were carried out by using the equations in the laboratory frame.
The radiation reaction appears directly in the time evolution of the electron's energy. I show this in  .