Radiation Reaction in High-Intense Fields

Transcription

1 Radiation Reaction in High-Intense Fields Keita Seto Extree Light Infrastructure Nuclear Physics (ELI-NP) / Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), 3 Reactorului St., Bucharest-agurele, jud. Ilfov, P.O.B. G-6, RO-7715, Roania *keita.seto@eli-np.ro After the developent of the radiating electron odel by P. A.. Dirac in 1938, any authors have tried to reforulate this odel so-called radiation reaction. Recently, this effects has becoe iportant for ultra-intense laser-electron (plasa) interactions. In our recent research, we found a ethod for the stabilization of radiation reaction in quantu vacuu [PTEP 14, 43A1 (14), PTEP 15, 3A1 (15).]. In the other hand, the field odification by high-intense fields should be required under 1PW lasers, like ELI- NP facility. In this paper, I propose the cobined ethod how to adopt the high-intense field correction with the stabilization by quantu vacuu as the extension fro the odel by Dirac Introduction With the rapid progress of ultra-short pulse laser technology, the axiu intensities of these lasers has reached the order of 1 W/c [1, ]. If the laser-intense is higher than 1 W/c, strong radiation ay be generated fro an electron. Accopanying this, radiation reaction as the feedback to the electron s otion can have a strong influence on electrons [3]. One of the facilities which can achieve these regie, Extree Light Infrastructure - Nuclear Physics (ELI-NP) will equip two 1PW- 1 ~4 W/c class lasers [4-6]. We can t neglect this effect of radiation reaction in the experients carried out in this level of lasers. The original odel of radiation reaction, the Lorentz-Abraha-Dirac (LAD) equation [7] has a significant atheatical difficulty which is an exponential divergence rapidly naed run-away [7, 8]. In y previous research, I succeeded to perfor the stabilization freeing fro this run-away in quantu vacuu [8-1]. The last for of y equation is dw e [ F g(* F ) ] w. (1) d (1 f ) Here,, e and are the rest ass, the charge and the proper tie of an electron. The vector space denotes the set of the vector in inkowski spacetie. w is the 4-velocity defined by 4 wdx d (, c v). 4 4 * represents the dual space of 4, the Lorentz etric 1 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

2 g * * has the signature of (,,, ), g aa aa aa aa 1 1 aa aa 3 3. The, F is an arbitrary external field like lasers. The field F Fex FLAD field FLAD ex is the radiation LAD field, dw dw FLAD w w xx, () ec d d 3 and ( ), therefore, the liit derive the equation of otion dw d ef w, so-called the LAD equation. f and g are certain Lorentz invariant functions depending on the odel of quantu vacuu. In the case of the Heisenberg-Euler vacuu, f F F FF and g 74 F * F 74 F (* F ). These works suggest us (i) quantu vacuu stabilizes the LAD field and (ii) there is the good behavior that Eq.(1) agreed one of the ajor references proposed by Landau and Lifshitz [11] via y analysis and nuerical siulations. In the other hand, the dynaics of an electron is considered that it should be corrected by the high intense fields like 1PW lasers. In these physics regie, it is often discussed with the paraeter representing the field strength. For exaple, I. Sokolov, et al. [1] proposed the radiation reaction odel with this dependence. dp dx q( ) ef g f f p (3) d d c ex ex ex dx 1 p q( ) f d (4) ex For instance, I nae this as QED-Sokolov equation/odel since the function q( ) depends on the QED cross-section: q( ) drr drk5 ( r) rrk ( r) 8 r 3 3. (5) Where the definition C c, r r (1 r) and is the QED-field strength paraeter, 3 C g f ex fex. (6) c In the low field-intense regie 1, then q( ) 1. This liit converges to the result of Landau- Lifshitz equation [11]. So, the function q( ) odified the dynaics fro in classical to in quantu high-field dynaics. However QED-Sokolov equation breaks the invariance ( dx d)( dx d) c which should be satisfied in the particle description under classical dynaics, we should recover this acceptable relation when we consider the classical-relativistic equation of otion. oreover, we ay discover a new factor different fro q( ) in the experients. I considered we should ake a degree of freedo of the high-intense field correction for this case. In this paper, I discuss the general way how to describe radiation reaction in the high-intense fields under quantu vacuu with Radiation Reaction in High-Intense Fields, Febuary 6, 15.

3 this degree of freedo, the extension fro Ref.[9] and Ref.[1]. For the deonstration of it at first, I bring the new function and for the odification of the LAD field in the high-intense external field. Actually, we can t find the stability of the equation of otion by using only this correction. Fro this point of view, I will perfor the stabilization by the field propagation in quantu vacuu. Unfortunately the functions and can t be defined fro y ethod. Therefore, I perfor the nuerical calculations with the Sokolov s q( ) function as the saple of. We can find the result that the new equation agree well the QED-Sokolov equation with the relation ( dx d)( dx d) c.. Ultra-high field-intense correction.1. Idea of the correction ethod In ultra-high intense fields, the coupling of an electron ay be corrected. This effect has been discussed by I. Sokolov, et al. [1], they proposed the equation of otion for the radiation reaction odel upgraded fro this classic to QED radiation via the cross-section. Finally, they forulated the following interesting relation, e q( ) q( ). (7) QED Classic 3 6c 3 is the constant in Eq.(), e 6c and c describes the order of the classical electron radius. Equation (7) represents the paraeter replaceent fro e to q( ) e in QED- Sokolov ethod. Here taking the new non-zero functions, C ( ) satisfying with q( ), we can obtain the separated replaceents of e e and. These two function and should be the Lorentz invariance. Fro here, we try to re-derive the equation of radiation reaction with the new ass and the charge e. Before correction by quantu vacuu, we consider the odification of the LAD field. The equation of otion of an electron and the axwell equation becoes as follows: dw ( ) e( ) F w (8) d [ ( ) ( ) 4 ( ( ))] F ec d w x x (9) F is the hoogenous solution of Eq.(9). The solutions of Eq.(9), F are the retarded and the advance field (with hoogenous incoing/outgoing field). 3 d( w ) d( w ) ( ) Fret w w d xx 4 ec d d d ( w ) d ( w ) w w ec d d (1) 3 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

4 3 d( w ) d( w ) ( ) Fadv w w d xx 4 ec d d d ( w ) d ( w ) w w ec d d (11) Following Dirac s ideas the radiated field ( Fret( x) Fadv ( x))/, we can obtain the odified LAD field, d ( w ) d ( w ) Fod-LAD w w xx ec d d d dw dw FLAD w w xx. (1) ec d d d We can find that this field avoids the singularity of d ( ). Defining the hoogenous field F F F, the equation of an electron otion Eq.(8) becoes as follows: ex od-lad dw e F ( ) w. (13) d d d Here F F F, we choose the relation between functions as ( ) ex LAD. (14) For the basic charge to ass ratio ( e) ( ) e, dw 1 e F ( ) w. (15) d d 1 d We assue the ripple of is very slow, d d 1. This equation can t be solved by the sae reason of the LAD equation because of the ter of the second order derivative d d d w d (1 ), we use the perturbation like the ethod by Landau-Lifshitz with 4 p w with q( ), the definition dp e q( ) F [ p q ( ) g f ] g f f p d c ex ex ex ex e df e dq( ) ( ) ( ) d d. (16) ex q p Fex p The ter e [ q( ) df ex d] p e [ dq( ) d] Fex p are very sall value, so this Eq.(16) converge to the QED-Sokolov equation (3-4). When we know the style of the function, 4 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

5 we can take in the correction by high-intense fields. The key is the introduction of the two invariant function and which we can arrange freely under the condition of Eq.(15) 1. In the next section, we try to stabilize Eq.(15) in quantu vacuu... Quantu vacuu correction In the Sect..1, I perfored the odification of the LAD field via which arranges the coupling between an electron and radiation. It derived the odified LAD field Fod-LAD. In this section, we consider how to stabilize Eq.(15). At first what we should consider is F od-lad also satisfies with the source free axwell s equation Fod-LAD. By shifting fro F LAD to F od-lad, we can observe the hoogenous field F ho Fex Fod-LAD at the point far fro an electron. The field dresses the vacuu polarization during the field propagation, we take a stand that this field F ho represents the dressed field [Ref.8-1]. Here we need to derive the undress field F acting on an electron. That general dynaics of the propagating field is described by, L 1 F F F F F F Quantu Vacuu F F F F. (17) 4, * L, * Here, L Quantu Vacuu is an undefined Lagrangian density for quantu vacuu. The iportant reark is we can only apply this Lagrangian density for the field propagation without the field source. By solving this, we can obtain the axwell s equation like as follows: 1 F c (18) The RHS in Eq.(18) eans source-free, The field F c should be also the source-free field. eans the polarization of vacuu, 1 f F g* F (19) c f g L F F, F * F 4 F F Quantu Vacuu L F F, F * F 4 F F Quantu Vacuu () (1) Here, 4 /45 c. F c refers to the dressed field set of ( D, H ). In addition, the following axwell s equation is also held F ho. Thus, the solution of Eq.(18), F c connects to ( D, H) F ho Fex Fod-LAD with the continuity and soothness with C on all points in the inkowski spacetie, 1 Of cause, I can t deonstrate the independence between and. I stand the assuption of Eq.(15), but the independence is an unsolved proble in this paper. 5 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

6 F f F g* F F. () ho Via the algebraic treatents, we can solve Eq.() for F, (1 f) Fho g(* Fho ) F (3) (1 f) ( g) (see Ref.[1]). We consider the for of the equation corresponding to axwell equation (18) as follows: dw ( ) e( ) F w (4) d Transforing this Eq.(4) with the relation, we can get the for, dw e d [(1 f ) F( ) g (* F( ) ) ] w d (1 f) ( g) (1 f) d (5) (see Appendices 1). Here, f f( F( ) F( ), F( ) * F ( ) ) and g g( F ( ) F ( ), F ( ) * F ( ) ). Introducing the new tensor 1 (1 f) g g g K! ( ) d, (6) (1 f) ( g) (1 f) d The equation of an electron otion becoes briefly, dw e K( ) F ( ) w. (7) d For the iic of the Sokolov s odel, the function should have the dependence 1 in the low-intense liit, converging to Eq.(1). 3. Ultra-high field-intense correction under the first order Heisenberg-Euler vacuu 3.1 Equation of otion The failiar odel of quantu vacuu was represented by Heisenberg and Euler [13, 14]. L 4 7 * c 3 Quantu Vacuu Lthelowestorderof 4 Heisenberg-Euler 36 F F F F (8) 6 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

7 The Heisenberg-Euler Lagrangian is the dynaics only for the constant field. If ore general Lagrangian which we can apply any general field exists, that general Lagrangian includes the coponent of Eq.(8) since the constant field should be also applied for the general Lagrangian. In this section, I assue we can apply Eq.(8) for the field propagation. In this case, the functions f and g are f F F F F F F (9) ( ) ( ) LAD LAD LAD ex g 7 F( ) * F( ) 7 FLAD * Fex 4 (3) and we transfor Eq.(5) like, dw e d [(1 f ) F g (* F ) ] w ( ) ( ) d (1 f) 1 f ( g) d. (31) We can find the singular points when g and 1 f d d or 1 f in Eq.(31). Fro the condition of the low-intense liit, 1 f d d ust be required for avoiding the singular point: ex rest 1 d E d f ( v rest eeex rest ) 1 d e c c d physical requireents ex rest d E 1 (3) c d Where I eployed the relation F F ec F F e c g f f ( ec) v and LAD LAD LAD LAD rest 5 LAD ex v Eex rest. 1 f d d 1 (5. 1 ) ( Eex rest ESchwinger) 3 d d with the Schwinger liit field ESchwinger c e E ex Schwinger is satisfied. Considering the extree condition like E ex ( ESchwinger ) ( E ex rest ESchwinger ) is saller than unity, 1 f d d ~ d d E coefficient of. Norally, the relation, the ust be held, it is the requireent for avoid the instability and for taking in the high-intense fields at least. 3. Run-away avoidance In the original odel of radiation reaction, the LAD equation has an instability naed the run-away solution diverging exponential even if the external field doesn t exist. Therefore, this atheatical proble is also called self-acceleration. The new equation ust be required the stability and we need to understand how large dynaical range we can apply it. At first, we assue the condition of Eq.(3). For instance to write, I re-write the equation of an electron s otion Eq.(31) like The subscript of rest eans the value in the rest frae. 7 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

8 dw (1 f) fex (1 f) flad g(* fex), (33) d d (1 f) ( g) (1 f) d with the definition of the forces fex efex w and (* fex ) e(* Fex ) w. Here, we follow the two-stage analysis which I used in Ref.[1]. At first, we check the finiteness of the radiation energy because there is a possibility that run-away coes fro infinite radiated energy. In the second stage, we proceed to the asyptotic analysis proposed by F. Röhrlich for investigation after freeing fro the external field [15]. Eq.(33): In the first stage, we ake the Laror s forula dw d g ( dw d)( dw d) fro ec ec (1 f) f (1 f)( g) dw dw 7 g d d [(1 f) ( g) (1 f) ( )] g [(1 f ) f g * f ][(1 f ) f g * f ] f g f ex ex ex ex [(1 ) ( ) (1 ) ( )] (34) Here, I put the sybol ( ) d d. We research the finiteness of this equation. Considering invariances in the rest frae, f ( ec) v 4 ec ve ( v ) and rest ex rest rest g 7 F * F 7 ec vb ( v ). When the dynaics becoes run-away, LAD ex ex rest rest v rest, the self-acceleration of an electron. In the run-away case, g f ( ) ( ) is obviously satisfied. We use this relation under the condition of Eq.(3), Eq.(34) is evaluated as follows (detail is in Appendices 3): g run-away dw dw d d ec f ec 1 ( f ) f f f [1 ( )] 7 (1 )[1 ( )] g f f g f (* f ) f f f ex ex f ex ex [1 ( )] 1 [1 ( )] g f (* f ) f f ( g) ex ex [1 ( )] 1 (35) Now we only consider the case that the external field is weaker than the Schwinger liit. If (1), the functions 1 1 x, 1 1 x, x 1 x and x 1 x are finite in the 5 doain x (,1 ), Since xf E ex rest c (1 ), x is included in this doain. In these conditions, 8 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

9 dw dw dw run-away g dt d d. (36) Though we can consider the infinite energy eission of light as one of the reason of run-away, but now that possibility was vanished due to Eq.(36). Next, we proceed the asyptotic analysis. For this analysis, we need to ake the pre-acceleration for. The for of Eq.(33) is as follows: (1 f) fex g(* fex) dw dw (1 f ) g w dw d c d d d (1 f ) 1 d a ( ) e ( ) 1 f ( g) d d d a [ln ( ) ln ( )] (1 f ) e e e e (37) Here, I eployed the paraeter a. Now we want to consider the acceleration dw d at the infinite future,. Following the Röhrlich s ethod, the velocity converges to a constant by the assuption of external field vanishing at,. In this liit, the dynaics becoes classical liit 1 like li q( ) 1 in the QED-Sokolov odel because of the field absence ( ). Therefore, the liit of Eq.(37) becoes dw g dw dw ( ) fex ( ) (* fex) ( ) g w ( ) d 1f c d d dw dw ( ) c d d g w, (38) by using the l'hôpital's rule with the relation. The finiteness of Eq.(36) is iportant for the constant velocity, otherwise dw d ( ). After this procedure, we can follow the sae way of Ref.[1]. Finally, we can obtain the liit of the acceleration of an electron; dw li (39) d This is the requireent for the avoidance of run-away proposed by Rörhlich [15]. y odel also converge to the constant velocity after the release fro the external field. Above here, we could deonstrate that run-away doesn t essentially exist under the condition of Eq.(3) as the iic ethod of Ref.[1]. 3.3 Calculations Finally, I present the nuerical calculation results with other radiation reaction odels. As the saple of, I eploy the relation q( ), Eq.(33) becoes 9 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

10 dw e 7 1 F F F F * F (* F ) 4 q( ) q( ) q( ) q( ) q( ) q( ) d 7 dq 1 Fq( ) Fq( ) Fq( ) * Fq( ) 1 Fq( ) Fq( ) 4 w, ( ) d (4) here F q( ) Fexq( ) FLAD. I perfored the calculation of Eq.(4) and the following odels: Seto I odel which is Eq.(1) [9], the Landau-Lifshitz odel [11], Classical Sokolov [16] and QED-Sokolov [1]. And I nae Eq.(4) Seto II. I assued the case of the head-on collision between the laser and electron for the initial configuration of the siulations (Fig.1). I used the paraeters of Extree Light Infrastructure - Nuclear Physics (ELI-NP) [5-6]. The peak-intense of the laser is 11 W c in a Gaussian shaped plane-wave like Eq.(8,9) in Ref.[1]. The pulse width is fsec and the laser wavelength is.8μ. The electric field is set in the y direction, the agnetic field is in the z direction. The electron travels in the negative x direction, with the energy of 6eV initially. The nuerical calculations were carried out in the laboratory frae. The tie evolution of an electron s energy shows the typical behavior of radiation reaction. I show it in Fig.. The energies of an electron drop fro the initial energy of 6eV to the final energies. Depending on the odels, the final energy of the electron converges to two separated levels. The first group includes Seto I, Landau-Lifshitz, and Classical Sokolov odels around 165eV. The second group is QED-Sokolov and Seto II odels, around 6eV. The difference between these two group depends on the function q( ), obviously. In this laser regie and the energy of an electron, runs fro to.3 in this case of 1PW-1 1 W c. Figure 3 is the graph of q( ). Fig.1 Setup of the laser - electron head-on collision. The laser propagates along the x axis. An electron travels in the negative x direction. 1 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

11 Fig. The tie evolution of an electron s energy. The final electron s energies are, Seto I, Eq.(1): 166.5eV, Landau-Lifshitz: 165.3eV, Classical Sokolov: 165.3eV, QED-Sokolov: 59.eV and and Seto II, Eq.(4): 6.5eV. The insets are close-up of the figures. Fig.3 The function of q( ). In this calculations, the doain of this calculation is [,1]. 11 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

12 Fig.4 The function of d d. 5 The following is satisfied d d (1 ) (see Fig.4) and also f Fq( ) Fq( ) 8 (1 ) Therefore, 1 f d d ~1 in this case. In the head-on collision between the laser and the electron, g 7 4 Fq( ) * F q( ) 7 ec vbex rest since the initial condition liit the electron s otion on the x-y plane. Rounding the invariance f into 1, Eq.(4) is reduced like as follows: dw 1 1 e Fq( ) w ~ e Fq( ) w. (41) d dq( ) dq( ) 1 Fq( ) Fq( ) 1 d d Since it is Eq.(15), we can derive Eq.(16), the quasi-qed-sokolov equation by using perturbation fro Eq.(4). The convergence between the odel of Seto II: Eq.(4) and QED-Sokolov Eq.(3-4) appeared because of the reason why I entioned above, the difference of the final energies between tow groups depends on the invariant function. In the theoretical point of view, y new equation (4) can satisfies both of the relation ( dx d )( dx d ) c other hand, in the Classical/QED-Sokolov odel, and Eq.(4)]. This is an algebraic difference laying on two odels. p p ( dx d)( dx d) c and c at any tie. In the p p c [see 4. Conclusion In suary, I updated y previous equation of a radiating electron s otion considering in the high- 1 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

13 intense fields, in quantu vacuu. The correction by high-intense fields odified the eitted field fro an electron and quantu vacuu stabilized the instability of run-away. The atheatical treatents in the derivation of the new equation was based on our previous paper [9, 1]. I assued the paraeter replaceents e e and at first. It eans the source ter of the axwell s equation was defored, therefore, the LAD field was odified fro F LAD to F od-lad (Sect..1). The field acts on an electron in quantu vacuu F ust be satisfied F f F g* F F F, (4) ex od-lad the new equation of an electron s otion with radiation reaction is dw e K( ) F ( ) w (43) d or dw e d [(1 f ) F( ) g (* F( ) ) ] w d (1 f) ( g) (1 f) d. (44) Here, F ( ) F ex FLAD (Sect..). Fro y analysis, the following relation ust be satisfied for the stability of this equation in Heisenberg-Euler vacuu; 1 E ex rest c d d (Sect.3.1). Under this condition, I could deonstrate the run-away avoidance by using the Röhrlich ethod [15] (Sect.3.), and could perfor the nuerical calculation for checking the difference between the proposed odels. I eployed the relation q( ) for the nuerical siulations. The results of the showed us the dependence of q( ), it eans the correction in high-intense fields, is essential between each odels (Sect.3.3). I introduce the easure of the electron s ass ( x) ( ) and the anisotropic charge E ( x) ( ) following Ref.[1]. dw d( ) x de( ) xf ( ) w (45) d The Radon-Nikody derivative [17] should be d E( ) d( ) e K ( ), depending on the invariance. The anisotropy of charge coes fro the polarization of quantu vacuu. Before the taking the relation of Eq.(14), the Radon-Nikody derivative is, 1 (1 f) g g g de e! d d [(1 f) ( g) ] (1 f) d. (47) I couldn t give the evidence whether we can use the relation at any tie, we ay find this 13 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

14 relation breakings of this in the experients with the extree light eission in high-intense fields. Finally, I discuss about the utility of this new radiation reaction odel. In the siulations, I used the q( ) following the Sokolov s suggestion [1], but it ay not be true in nature. The Eq.(43/44/45) is required d d 1 under the stabilization, we can observe the radiated field fro an electron F od-lad ~ F. (48) LAD When we find the new behavior of by significant trials of experients, the invariant function ay be the upgraded fro q( ). Since represents a coupling between an electron and the radiated field, the investigation of will becoe ore iportant for radiation reaction / an electron s odel in high-intense fields. Acknowledgeents This work is supported by Extree Light Infrastructure Nuclear Physics (ELI-NP) Phase I, a project co-financed by the Roania Governent and European Union through the European Regional Developent Fund, and also partly supported under the auspices of the Japanese inistry of Education, Culture, Sports, Science and Technology (EXT) project on Prootion of relativistic nuclear physics with ultra-intense laser. 14 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

15 Appendices 1: Detail of the derivation of equation of otion Eq.(5) Here I derive the undressed field F via perturbation fro Eq.(). (1 f) F g(* F ) F (A-1.1) ho ho 1 f g The definition of the invariant function f and g are Eq.(19) and Eq.(). We can expect the for of F F ( ) F ( )(* F ) G, (A-1.) ( ) ( ) * ( ) We assue that the paraeter satisfies with the relation 1. The functions and * depend on F. This relation leads the expansion of the invariant functions: f ( F F, F * F ) f( F( ) F( ), F( ) * F ( ) ) f f (A-1.3) g( F F, F * F ) g( F( ) F( ), F( ) * F ( ) ) g g (A-1.4) For instance, I introduce f f( F ( ) F ( ), F ( ) * F ( ) ) and g g( F( ) F( ), F( ) * F ( ) ). The ters f and g coe fro G in Eq.(A-1.). By using these equations, Eq.(A-1.1) becoes (1 f f) F ( g g)(* F ) F f ho g ho (1 f) ( g) n f ff gg f f g g n (1 f) ( g) ( ) ( ) [( ) ( ) ] n (A-1.5) By coparing Eq.(A.1-) and Eq.(A.1-5), we can estiate the order of and * with the condition 1. ~ f f (A.1-6) ~ g g (A.1-7) * We can choose here, then ~ f f, * ~ g g. By neglecting the ters of ( f, g, f, g), 3 (1 f )( F F ) g (* F * F ) F (A.1-8) ( ) ( ) (1 f) ( g) with the definition 3 This order cut-off is iportant for the stability of the new equation. See Sect. 3.1 and Radiation Reaction in High-Intense Fields, Febuary 6, 15.

16 F F F, (A.1-9) ho ( ) d dw dw F w w. (A.1-1) ec d d d The equation of otion dw ef w (A-1.11) d becoes dw e d [(1 f ) F( ) g (* F( ) ) ] w. (A-1.1) d [(1 f) ( g) ] (1 f) d When we assue ( ) ( ), dw e d [(1 f ) F( ) g (* F( ) ) ] w d [(1 f) ( g) ] (1 f) d. (A-1.13) 16 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

17 Appendices : Errata of K. Seto, PTEP 15 [1] Strictly speaking, Eq.(1) should be dw [(1 f ) g (* ) ] w d f g 1 e F F (1 ) ( ), (A.-1) for connecting fro Eq.(5) or Eq. (A.1-13). Here, F F F F. Norally, ( ) 1 ex LAD 1 f g is satisfied, under this condition the Eq.(A.-1) is transfored as follow: g F (* F ) w [ F g(* F ) ] w (1 ) (1 ) dw 1 e 1 f e d f ( g) f 1 (1 f ) (A.-) This is the Eq.(1) derived in Ref.[1]. However in the strict order treatent, it should be 1 e g w (1 ) 1 dw F. (A.-3) d f f In this for, the analysis of run-away avoidance in Ref.[1] becoes ore easier 4, the calculation was all ost agree since F g(* F ) is satisfied. I suggested the anisotropy of the quantu vacuu. We can confir the anisotropic field *F in the for of Eq.(A.-1), but it doesn t exist in (A.-3). The higher orders of g (1 f) describe the degree of the anisotropy of quantu vacuu. 4 For considering Eq.(A.-), We only put the relation g at Ch.3 in Ref.[1] and any anisotropy is vanished. 17 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

18 Appendices 3: the detail of Eq.(35) Fro Eq(34), ec ec (1 f) f (1 f)( g) dw dw 7 g d d (1 f)[1 f ( )] ( g) g [(1 f ) f g * f ][(1 f ) f g * f ]. f f g ex ex ex ex (1 )[1 ( )] ( ) (A.3-1) We consider this equation under the condition of Eq.(3); physical requireents 1 f ( ). (A.3-) This condition supports the relation proceed to consider the absolute value of Eq.(A.3-1). f f g. Above here, we can (1 )[1 ( )] ( ) ec (1 f) f (1 f)( g) dw dw 7 g d d (1 f)[1 f ( )] ( g) ec ex ex ex ex (1 )[1 ( )] ( ) (1 f) ec g [(1 f ) f g (* f ) ][(1 f ) f g (* f ) ] f f g ex ex ex ex (1 )[1 ( )] f ( g) 7 [1 ( )] (1 )[1 ( )] ec f (1 f)( g) 7 (1 f )[1 f ( )] g [(1 f ) f g (* f ) ][(1 f ) f g (* f ) ] f f ec ec f f f g f f g f (* f ) [1 ( )] 1 [1 ( )] ex ex g ex ex f f f g g fex (* fex ) 1 f [1 f ( )] (A.3-3) Finally, ( g ) ( f ) is satisfied in the case of run-away, we can obtain the relation of Eq.(35). 18 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

19 g run-away dw dw d d ec f ec 1 ( f ) f f f [1 ( )] 7 (1 )[1 ( )] g f f g f (* f ) f f f ex ex f ex ex [1 ( )] 1 [1 ( )] g f (* f ) f f ( g) ex ex [1 ( )] 1 (A.3-4) 19 Radiation Reaction in High-Intense Fields, Febuary 6, 15.

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