24P166-B
The Theoretical study of Radiation Reaction under Ultrarelativistic laser
超相対論レーザーの下での放射の反作用の理論研究
Keita Seto, Hideo Nagatomo, Kunioki Mima瀬戸慧大，長友英夫，三間圀興Institute of Laser Engineering, Osaka University, 2-6 Yamada-oka, Suita, Osaka 565-0871, Japan
大阪大学・レーザーエネルギー学研究センター 〒565-0871 大阪府吹田市山田丘2-6
In Ultrarelativistic laser-plasma interaction over the intensity of 1022W/cm2, it is known the ‘radiation
reaction’ effect becomes large because of the significant bremsstrahlung. If the laser intensity can become
higher and higher, we consider, for example, the electron-proton pair creation appears. Therefore, the nextof laser-plasma interaction field is the quantum domain. In this presentation, We show the quantum shift ofthe radiation reaction dynamics from the classical description, without electron-proton pair creations as thebasic study of the next generation laser-plasma interactions.

1. Introduction
With the advent of ultra–relativistic laser, it
will reach to the Intensity 1022 W/cm2. If an electron
is under this laser when laser intensity is larger than
1018W/cm2, an electron will carry out relativistic
behavior. The most important phenomenon in thisregime is an effect of ponderomotive force, and anelectrons are pushed out in the propagation
direction of laser. If we can get the laser intensity
1022 W/cm2, the strong bremsstrahlung might becaused. Moreover, simultaneously with it,“radiation reaction force (or damping force)” works
to charge particles. We could reach to describe thisreaction force working on an electron in Minkowski
spacetime, it is as follows [1]:d 2 wµ m0τ 0  dw dw  µ+ 2 g,wdτ 2c dτ dτ w0 wµ − c 2δ 0 µ d 2 w0−m0τ 0(1)22w0 − c 2 dτf reaction µ = m0τ 0( )Where τ 0 = Q 6πε 0 m0 c3 , w, τ and g is therelativistic 4-velocity, proper time and Lorentz
metric. Before this equation appears, theLandau-Lifshitz (L-L) equation [2] as the method of
perturbation of Lorentz-Abraham-Dirac (LAD)
equation [3] is used. But, the L-L equation breaks
the relation of relativistic covariance. Our Eq.(1) issatisfied with the covariance.
22. Quantum model of radiation reaction
In the next laser intensity generation (over2410 W/cm2), we consider the electron-positron paircreations and something effects appear [4]. Thisintensity level requires us to treat phenomena as
QED (Quantum Electro Dynamics). However, we
can consider an electron as quantum description

under 1024W/cm2, without the pair creations. In thispresentation, we suggest the quantum model of theradiation reaction without the electron-positron paircreation using the Dirac equation.

 µ
ie i γ  ∂ µ − Aµ  − mc ψ = 0(2)Here, γ µ =0,1,2,3 are Dirac matrix and Aµ means
the electronic field, Aµ = (φ c , −A ) . In QED, weconsider this field contains not only laser, but thebremsstrahlung from the electron. Under the theoryof quantum field, the wave function ψ and theelectromagnetic field A are described the fields
operator. When the free electron’s propagator isSfree ( x, x′ ) and the solution of the Dirac equationof free electron is ψ 0 , the full solution of Eq.(2)becomesψ ( x ) = ψ 0 ( x ) − ∫ dx′Sfree ( x, x′) eγ µ Aµ ( x′) . (3)Therefore, the second term means photon-electroninteraction. R. P. Feynman suggests that this iscalculated by the Feynman diagram (see Fig.1)[5,6]. This diagram leads the probability (called theS-matrix) which the electron p and the laser kFig.1 The Feynman diagram oflaser-electron interaction

become p + k − k ′ and the radiation k ′ (in theeasiest perturbation). But, in QED, this calculation
requires us to decide the condition of the initial andthe final. It is described in the Fock space,S pp,+k k − k ′, k ′ = p + k − k ′ ⊗ k ′− dx′Sfree ( x, x′ ) eγ µ Aµ ( x′) ∫p ⊗k .

(4)

Here, p ⊗ k is the initial state which theelectron has the momentum p and the photon has

k , the Fock vector.

p + k − k ′ ⊗ k ′ is thefinal state. Moreover, the red line in Fig.1 is the“scattered photon” which behaves as the radiation.
It requires us to decide the final value of the
momentum of the bremsstrahlung. This means it isnecessary to integrate of the 4-momentum of the′.
bremsstrahlung kbremsstrahlung∫′kbremsstrahlung= d 4 k ′ k ′ S pp,+k k − k ′, k ′2(5)This method is difficult for the simulation of thelaser-electron interaction.
In classical theory of Dirac [3], he suggested
that the reaction force should be described by theretarded field Aret and advanced field Aadv . Thesefields are satisfied with the equation;∂ µ ∂ µ Afullν = µ0 jν .

1
( Aret + Aadv )2
1
Areaction = ( Aret − Aadv )2

Afull =(6)(7)

Equation (4) means we can consider Aret 2 is thephoton field and Aadv 2 is the antiparticle field in
QED method. The reaction force of the LAD theory
is described byf reaction µ =−e ( ∂ µ Areactionν − ∂ν Areaction µ ) wν .

(8)This means the reaction force is defined by thepotential Areaction like an electromagnetic field.
The potential in the Dirac equation is replaced as(9)A Alaser + Areaction
if the radiation reaction is considered without theelectron-positron pair creation. Therefore, the Diracequation with the radiation reaction is Alaser   µ 
ie  i γ ∂ µ −  +   − mc ψ = 0 . (10)A  reaction  µ We will show the detail of the radiation reaction in

the classical and the quantum description in ourpresentation.
References[1] K. Seto, H. Nagatomo and K. Mima, Plasma Fusion
Res. 6, 2404099 (2011).
[2] L. D. Landau and E. M. Lifshitz, The Classicaltheory of fields (Pergamon, New York, 1994).
[3] P. A. M. Dirac, Proc. Roy Soc. A 167, 148 (1938).
[4] G.A.Mourou, C.P.J.Barry and M.D.Perry, Phys.
Today 51,22 (1998).
[5] R. P. Feynman, Phys. Rev. 76, 749 (1949)[6] R. P. Feynman, Phys. Rev. 76, 769 (1949)