Download Radiation reaction in ultrarelativistic laser

Prog. Theor. Exp. Phys. 2013, O53A01 (10 pages)
DOI: 10.1093/ptep/ptt023
Radiation reaction in ultrarelativistic
laser –spinning electron interactions
Keita Seto1,∗ , Hideo Nagatomo1, James Koga2 and Kunioki Mima1
1
Institute of Laser Engineering, Osaka University, 2-6 Yamada-oka, Suita, Osaka 565-0871, Japan
Advanced Beam Technology Division, Japan Atomic Energy Agency, Kyoto 619-0215, Japan
∗
Email: [email protected]
2
Received December 23, 2012; Accepted February 23, 2013; Published May 1, 2013
........................................................................................................................................................
The intensity of ultra-short pulse lasers has reached 1022 W/cm2 owing to the advancements in
laser technology. The large radiation from the electron behaves something like resistance in this
ultrarelativistic laser – electron interaction. This effect is called the “radiation reaction”. The
equation of motion with the radiation reaction is known as the Lorentz– Abraham – Dirac
equation; however, this equation does not incorporate the spin property. In laser plasmas,
classical physics descriptions are preferred for simulations. This paper discusses how to describe
the radiation reaction of a spinning relativistic electron in classical dynamics.
........................................................................................................................................................
Subject Index
A00, A01, G03, J25
1. Introduction
With the rapid progress of ultra-short pulse laser technology, the maximum intensities of these lasers
have reached the order of 1022 W/cm2 [1,2]. One laser facility that can achieve such ultra-high
intensities is LFEX (laser for fast ignition experiments) at the Institute of Laser Engineering
(ILE), Osaka University [3]; another is the next laser generation project, the Extreme Light
Infrastructure (ELI) project [4] in Europe. If an electron is in the strong fields because of a laser with
an intensity larger than 1018 W/cm2, the dynamics of the electron should be described by relativistic
equations. The most important phenomenon in this regime is the effect of the ponderomotive force,
where an electron is pushed in the propagation direction of the laser. When a charged particle is accelerated, it proceeds in a trajectory accompanied by bremsstrahlung. If the laser intensity is higher than
1022 W/cm2, referred to as ultrarelativistic laser intensities, strong bremsstrahlung might occur.
Accompanying this, the “radiation reaction force” (or “damping force”) acts on the charged particle.
Therefore, it is necessary to study the radiation reaction effects in the ultrarelativistic laser–electron interaction regime. We can consider this effect as a “self-interaction”. In a laser–electron interaction at this
intensity level, one of the important equations is the equation of motion including the reaction force.
However, in classical physics, these equations are based on Lorentz’s work [5]. The purpose of his
study was research into the electron’s characteristics via classical physics. In particular, he was interested in radiation from a moving electron. The radiation field from a point electron is derived by the
Liénard –Wiechert (retarded) potential in classical physics. Quantum mechanically, bremsstrahlung
comes from the integrated momentum of all the photons emitted, meaning that an electron loses its
kinetic energy via bremsstrahlung. However, classically, an electron feels some force, which is
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PTEP 2013, O53A01
K. Seto et al.
related to bremsstrahlung since there is an energy loss mechanism. Lorentz considered the electron
model in which the charge is distributed on a sphere to investigate this force. His model was applied
only in the nonrelativistic regime, the case in which the electron has low velocity. One part of the
electron interacts with another part by the Liénard – Wiechert electromagnetic field. Here, it is
assumed, for analysis, all elements have the same velocity as the center of the electron. The
Liénard –Wiechert field depends on the electron’s motion and this field has directivity. If the electron
is at rest, the Liénard – Wiechert field becomes the Coulomb repulsion. In this case, the electron has a
spherically symmetrical field. Therefore, the electron does not feel a force apart from the Coulomb
repulsion component of the Liénard –Wiechert field. However, in the case with the moving electron,
the radiation symmetry is broken by the directivity. This asymmetry leads to a self-interaction force
from bremsstrahlung. This force is the “radiation reaction force”.
After this work, Dirac updated Lorentz’s theory with the relativistic covariant equation of motion,
known as the Lorentz –Abraham –Dirac (LAD) equation in Minkowski spacetime [6]. Dirac
suggested that the covariant form of the radiation reaction force should be described with not
only the retarded potential but also the advanced potential. When he suggested this theory, the
theory of quantum electrodynamics (QED) had encountered a difficulty, a “mass divergence”.
This was resolved via the “renormalized theory” by Tomonaga [7], Schwinger [8], Feynman [9],
Dyson [10], and so on. However, Dirac pursued physics without the trick of mathematics. His
hope was that the LAD equation would be a classical starting point to develop a quantum theory
without the infinite self-energy of the electron [11]. In the end, he was unable to achieve what he
had hoped for. This method resulted in an equation that is covariant. The starting point of our discussion is the LAD equation, as follows:
m0
d m
mn
m
w = −eFlaser
wn + freaction
dt
m
= m0 t0
freaction
d 2 wm m0 t0
dwr dws m
+
g
w ,
rs
d t2
c2
dt dt
(1)
(2)
where t0 ¼ e 2/6p10m0c 3 and ct0 is the classical radius of the electron. t is the proper time, w is the
4-velocity defined as w ¼ g(c,v). The Lorentz metric g has the signature (+, 2, 2, 2), gmna ma n ¼
a na n ¼ a 0a 0 2 a 1a 1 2 a 2a 2 2 a 3a 3. Since these equations describe the effect of the radiation reaction, the LAD equations, Eqs. (1) and (2), become important in ultrarelativistic laser – electron interactions. This equation has a runaway solution, which quickly becomes infinity (see Refs. [12, 13]).
To avoid this runaway, there are many techniques from physical assumptions to mathematical treatments. Approximate methods have been suggested by Landau –Lifshitz [14], and Ford –O’Connell
[15], Rohrlich [16], Caldirola [17], and Sokolov [18] have suggested other methods, but we need to
wait to decide which is correct in nature. Applications from the fluid dynamics perspective have been
studied by Tam and Kiang [19] and Berezhiani et al. [20, 21].
However, these models do not treat the spin– field interactions. The origin of this theory is the
model of an electron. It is known that the Dirac equation in quantum theory is the equation of an
electron. The most important thing is that the Dirac equation has spin information. The LAD
equation does not consider this spin– electromagnetic field interaction. A recent model of the radiation reaction force based on the self-energy in QED by Galley et al. arrived at finite size corrections
to the LAD equation [22]. This work is the QED model of the Lorentz theory, but it does not include
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spin 1/2. In addition, recent high-power laser proposals have targeted the QED regime [4]. In this
regime, the effect of spin might be important. For the next model of the electron in classical
dynamics, we need to choose a relativistic spinning particle theory that converges to the Dirac
equation in quantum dynamics. This method without radiation was derived by Holten [23]. He
achieved this by way of the Volkov solution [24], which is a strict solution of the Dirac equation
with an external field, via a Klein – Gordon-like equation as follows:
2
h− e mn
s Fmn c = 0.
−h− Dm Dm − m2 c2 I c −
2
(3)
Dm is a minimal substitution defined by Dm ; ∂m − ieAm /h− , Fab is the electromagnetic field tensor,
and sab is the spin information from the Dirac g matrix. Here, taking the classical quantum relations
ih− Dm = ih− ∇m + eAm 7 ! m0 wm , we can obtain the following equation:
h− e mn
m0 wm (m0 wm ) − m20 c2 −
s Fmn = 0.
2
(4)
This does not differ from the normal relation
m0 wm (m0 wm ) − m20 c2 = 0
(5)
since the spin –field interaction is added to Eq. (4). When we consider a spinning relativistic particle,
we need to choose the relation of Eq. (4) to incorporate the spin interaction into theories. For the strict
electron model in classical dynamics (and laser plasmas), we discuss the radiation reaction of a spinning electron in this paper.
2. Radiation reaction with a dipole moment
In this section, we consider the method of taking the spin into account in the radiation reaction.
However, the spin of the electron is treated as a magnetic dipole moment. Therefore, the up and
down spin were differentiated in the Stern– Gerlach experiment [25]. This is known as the interaction
between a dipole moment and an electromagnetic field,
FS−G = ∇ p · E + m · B .
(6)
Here, p, m are the electro- and magnetic dipole moment. In classical theory, this Stern –Gerlach force
acts as a spin – electromagnetic field interaction. Therefore, the first step involves rewriting this
equation for the theory of relativity. This technique to describe a relativistic spinning particle was
first suggested by Holten [23]. The electromagnetic field tensor F mn and the dipole tensor m mn are
defined as follows:
⎡
F mn
0
⎢ Ex /c
=⎢
⎣ Ey /c
Ez /c
−Ex /c
0
Bz
−By
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−Ey /c
−Bz
0
Bx
⎤
−Ez /c
By ⎥
⎥
−Bx ⎦
0
(7)
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K. Seto et al.
⎡
mmn
0
⎢ px
=⎢
⎣ py
pz
−px
0
−mz /c
my /c
−py
mz /c
0
−mx /c
⎤
−pz
−my /c ⎥
⎥.
mx /c ⎦
0
(8)
From these two definitions, we can obtain a Lorentz invariance of
mmn Fmn = −
2
p·E+m·B .
c
(9)
This is then differentiated with respect to spacetime ∂m = (∂t , −∇),
c m
fS−G
= ∂m mab Fab .
2
(10)
Of course, the time component (0-component) refers to the energy change by the Stern –Gerlach
force. The equation of motion with spin becomes
m0
dwm
c m
= Fex
+ ∂m mab Fab ,
dt
2
(11)
and, multiplying wm, the equation becomes
1 ab
m Fab .
wm w = c 1 +
m0 c
m
2
(12)
This is an equation that corresponds to Eq. (4). For simplification, we define
1 ab
m Fab .
a= 1+
m0 c
(13)
The proper time and the 4-velocity are corrected as follows:
1 − v2 /c2
d t = dt
a
a
c
a
v
wm = , .
1 − v2 /c2
1 − v2 /c2
(14)
(15)
When a ¼ 1, the theory goes back to the spinless one. Here, the spinless 4-velocity is
called wm :
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K. Seto et al.
1 + mmn Fmn /m0 c dwm
dwm dt dwm
=
=
dt
dt
d t dt
1 − v2 /c2
m
1 + mmn Fmn /m0 c d 1 + mmn Fmn /m0 cw
=
dt
1 − v2 /c2
m 1 da m
a2
dw
+
= w
a dt
1 − v2 /c2 dt
m d ln a m
a2
dw
+
= w .
dt
1 − v2 /c2 dt
1
1 − v2 /c2
2
2
2
c2 b · ḃ
m
m dw
dw
c2 ḃ
=−
3 − 2 .
dt dt
1 − b2
1 − b2
(16)
(17)
The energy loss of the radiation is
2
2
2
2
ḃ − b × ḃ
ḃ · b
dW
ḃ
= m 0 t0 3 = m0 t0 2 + m 0 t0 3
dt
1 − b2
1 − b2
1 − b2
2
m
m dw
1
dw
.
= −m0 t0 2
2
dt dt
1 − v /c
Therefore, using the description with spin,
dW
m0 t0 dwm d ln a m dwm d ln a
=− 4
−
w
−
wm .
a
dt
dt
dt
dt
dt
(18)
(19)
After this, it is calculated in the same way as the LAD equation (we follow the method in Ref. [14]).
We assume the style of the radiation reaction force to be
m
freaction
=
1 dW m
w + Cm.
c2 dt
(20)
For the Lorentz invariance, C m is inserted into Eq. (20). The invariance which defines C m is
w mf mreaction ¼ 0. Before this calculation, we derive a few relations:
dwn dwn
d
dwn
d 2 wn 1 d 2
d 2 wn
n
=
wn
=
(
w
w
)
−
w
− wn
n
n
dt
2 d t2
dt dt
dt
d t2
d t2
c2 d 2 a 2
d 2 wn c2 a2 d 2 a2
d 2 wn
−
w
=
−
w
n
n
2 d t2
d t2
2a2 d t2
d t2
1 d 2 a2 n d 2 wn
= wn
w −
d t2
2a2 d t2
=
wn
dwn c2 d a2
=
.
dt
2 dt
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(21)
(22)
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K. Seto et al.
The relativistic relation becomes
m
wm freaction
m0 t0 dwm d ln a m dwm d ln a
=− 2
−
w
−
wm
a
dt
dt
dt
dt
+ wm C m
m0 t0 dwm dwm
d ln a dwm
=− 2
−2
wm
a
dt dt
dt dt
d ln a 2 m
w wm + wm C m
+
dt
m0 t0 dwm dwm
d ln a dwm
=− 2
−2
wm
a
dt dt
dt dt
2 d
ln
a
+ c2 a2
+ wm C m
dt
m0 t0 a2 m
1 d 2 a2
d 2 wm
= 2
C + 2 2 wm −
a
m0 t0
d t2
2a d t
d ln a dwm
d ln a 2 m
+
w wm = 0.
−2
dt dt
dt
In the alternative wm, C m needs to satisfy an equation like
2 2 m
m
2 2
m
t
d
w
d
ln
a
dw
1
d
a
d
ln
a
0
0
Cm = 2
+2
− 2 2 wm −
wm .
a
d t2
dt dt
dt
2a d t
Therefore, the radiation reaction force with the spin dipole moment becomes
m0 t0 d 2 wm
d ln a dwm n
m
w
+2
freaction = 2 4
ca
d t2
dt dt
2 n
d w
d ln a dwn m
−
+2
w wn .
d t2
dt dt
(23)
(24)
(25)
n
as
Moreover, we define the tensor F mreaction
mn
Freaction
2 m
m0 t0
d w
d ln a dwm n
w
=
+2
(−e)c2 a4
d t2
dt dt
2 n
d w
d ln a dwn m
w ;
−
+2
d t2
dt dt
(26)
the radiation reaction force is described as an electromagnetic force (Lorentz force):
m
mn
freaction
= −eFreaction
wn .
(27)
In the state of constant a ¼ 1, the spinless model, Eq. (25), converges to the LAD equation.
3. Angular momentum conservation
The equation of motion is the energy and momentum conservation law in relativistic physics.
However, the spin is regarded as that of the angular momentum and magnetic dipole momentum.
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We derived the radiation reaction force with the spin in Sect. 2. The electromagnetic dipole moment
tensor is called m mn. In this method, it is necessary to consider the time evolution of this dipole
moment. In classical physics, the angular momentum satisfies the following time evolution:
ds
e
e
=m×B+p×E=g
s×B+f
s × E.
dt
2m0
2m0
(28)
Here, it is assumed that the electric and magnetic dipoles are linearly related to the angular momentum. To shift to the covariant description, the 4D angular momentum should be defined as
Sm =
1 mabg
1
wa mbg ,
2G
(29)
with the definition of G ¼ ge/2m0. Here, we consider this in the rest frame. From established experimental fact, we assume that the electric dipole of an elementary particle (an electron) does
not exist in the rest frame [26]. The parameters in the rest frame have bars over the characters.
The spin-angular momentum in 4D space is defined by
1 mabg
m
am
bg = 0, G−1 m
= (0, s).
1
w
S =
2G
Using the Lorentz transformation,
S m = s · b, s ,
(30)
(31)
this is the definition in general inertial frames. One important relation is as follows:
wm S m = 0
(32)
n
Now, for an electron in a laser field, Fmlaser
, the equation of motion becomes
m0
mn
d m
mn
wn .
w = −e Flaser
+ Freaction
dt
(33)
n
Of course, Fmreaction
is defined by Eq. (26). From Eq. (32),
d dwm n
dS n
gmn wm S n = gmn
S + gmn wm
dt
dt
dt
e ml
ml
wl S n
= −gmn
Flaser + Freaction
m0
+ gmn wm
dS n
dt
dS l
e ml
ml
= wl
−
F
+ Freaction Sm ; 0,
d t m0 laser
(34)
therefore, the time evolution of the spin-angular momentum is
dS n
e mn
mn
Sm .
=
Flaser + Freaction
dt
m0
(35)
If this particle is a Dirac particle, the fact that g ¼ 2 is derived from Eqs. (28) and (35) [21].
The inverse relation of Eq. (29) is
mmn =
G mnab
1
wa Sb .
2
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(36)
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K. Seto et al.
4. Renormalization
In Sects. 2 and 3, the conservation laws were derived, but the equation of motion will contain the
runaway problem in the LAD equation. This problem is regarded as the reason why the reaction
force is defined by the acceleration (the external force and the radiation reaction force itself).
Here, we will mimic the method of Landau and Lifshitz [14]. Now, we describe the equation of
motion with the radiation reaction as follows:
m0
d m
m
m
m
w = flaser
+ freaction
+ fS−G
dt
(37)
n
Of course, f mlaser ¼ 2eF mreaction
wn. The radiation reaction force, Eq. (25) with Eq. (37), becomes
⎡ m
⎤
m
m d flaser + freaction
+ fS−G
d ln a m
m
m flaser + freaction + fS−G wn ⎥
+2
⎢
d
t
d
t
⎥
t0 ⎢
m
⎥wn ;
− freaction
= 2 4⎢
(38)
⎢
⎥
n
c a ⎣ d fn +fn
⎦
d ln a n
laser
reaction + fS−G
n
n
wm
+2
flaser + freaction
+ fS−G
dt
dt
all of the reaction force transposes from the R.H.S. to the L.H.S.:
⎤
⎡ m
dfreaction
d ln a m
+2
freaction wn
⎢
⎥
d
t
d
t
t
0 ⎢
⎥
m
−
− 2 4⎢
freaction
wn
⎥
⎦
c a ⎣ df n
d
ln
a
reaction
+2
fn
wm
dt
d t reaction
⎧ m
⎫
m d flaser + fS−G
d ln a m
⎪
⎪
m ⎪
⎪
⎪
+2
flaser + fS−G wn ⎪
⎪
⎪
⎨
⎬
d
t
d
t
t0
−
= 2 4 ⎪wn .
n
n
ca ⎪
⎪ d flaser
⎪
+
f
d
ln
a
⎪
⎪
S−G
n
n
⎪
wm ⎪
+2
flaser
+ fS−G
⎩
⎭
dt
dt
(39)
In this equation, the R.H.S. includes the runaway factor, f mreaction 2 t0/c 2a4 × df mreaction/dt. To use
n
.
Eq. (35), the radiation reaction force should be able to extract the electromagnetic tensor F mreaction
The L.H.S. is suitable for this. To avoid the runaway, we use a method in which the L.H.S. is
replaced by f mreaction. In Eq. (38), f mreaction includes the self-interaction via f mreaction itself. Therefore,
it is considered that the radiation reaction force has the runaway:
⎧ m
⎫
m d flaser + fS−G
⎪
⎪
d ln a m
m ⎪
⎪
⎪
+2
flaser + fS−G wn ⎪
⎪
⎪
⎨
⎬
d
t
d
t
t
0
m
−
(40)
freaction = 2 4
n
⎪wn .
n
ca ⎪
⎪ d flaser
⎪
+
f
d
ln
a
⎪
⎪
S−G
n
n
⎪
+2
flaser
+ fS−G
wm ⎪
⎩
⎭
dt
dt
The equation of motion is the renormalized equation, but neither Eq. (39) nor the LAD equation have
yet been renormalized. The replacement of the equation is equal to the renormalization. For these
reasons, the Stern –Gerlach force is defined by only an external laser field:
c m
(41)
= ∂m mab Flaser,ab .
fS−G
2
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This equation set corresponds to the Landau–Lifshitz method, as an approximation of the LAD
equation, and becomes close to the Dirac equation by incorporating spin into the model. Finally,
the equation of the electron’s motion becomes
⎧ m
⎫
m d flaser + fS−G
⎪
⎪
d ln a m
m ⎪
⎪
⎪
+2
flaser + fS−G wn ⎪
⎪
⎪
⎨
⎬
d
t
d
t
d m
t
0
m
m
−
m0
w = flaser + fS−G + 2 4 (42)
⎪wn .
n
n
ca ⎪
dt
⎪ d flaser
⎪
+
f
d
ln
a
⎪
⎪
S−G
n
n
⎪
+2
flaser
+ fS−G
wm ⎪
⎩
⎭
dt
dt
For calculation of Eq. (35), the field of the radiation reaction is defined as
⎧ m
⎫
m d flaser + fS−G
d ln a m
⎪
⎪
m ⎪
⎪
⎪
+2
flaser + fS−G wn ⎪
⎪
⎪
⎨
⎬
dt
dt
t0
mn
−
Freaction = − 2 4 ⎪,
n
n
ec a ⎪
⎪
⎪
d
f
+
f
d
ln
a
⎪
⎪
laser
S−G
n
n
⎪
wm ⎪
+2
flaser
+ fS−G
⎩
⎭
dt
dt
(43)
mimicking LAD theory [6]. The equation of motion, Eq. (40), is the strict equation of an electron,
equal to the Dirac equation in classical physics.
5. Summary
We have derived Eq. (40) with Eq. (35) as the renormalized equation of a spinning particle’s motion
with the radiation reaction. The spin is treated as the electromagnetic dipole moment, and its time
evolution is described by Eq. (35). An electron must follow the Dirac equation in quantum
physics, but it does not fit the relation m0 wm (m0 wm ) − m20 c2 = 0, because the electron’s spin interacts with the electromagnetic fields [see Eqs. (3)–(5)]. Following the Dirac equation (the Volkov
solution), the relativistic relation becomes wm wm = c2 1 + mab Fab /m0 c , and we can incorporate
the spin –electromagnetic interaction into classical physics. From this relation, the Stern –Gerlach
force depending on the spin and the radiation reaction force by the Stern –Gerlach force are
added and all of the formulas are corrected. The R.H.S. of the equation of motion has only the external force; this model can be solved without the runaway problem. This equation is derived from the
Dirac equation, which is more suitable for an electron model in classical physics than the LAD
model. This new equation of motion will predict the additional effects from the radiation reaction
caused by the electron’s spin.
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