Quantum Theory in Strong
Electromagnetic Fields
Sang Pyo Kim
Kunsan Nat’l Univ. & IOA, Nat’l Taiwan Univ.
National Tsing Hua University
February 20, 2012
Outline
Historical Background
Extreme Light Infrastructure
Schwinger Pair Production
Electron-Positron Pair Production/
Condensed Matter Analog
• Heisenberg-Euler and Schwinger Action
• Vacuum Polarization and Persistence
• Conclusion
•
•
•
•
Dirac Vacuum and Paradox
Dirac Theory of Electron
• Dirac, “The Quantum Theory Electron,” PRSL SA 117 (‘28)

e
  e
 p0  A0  1   , p 
c
c




A    3 mc  0


• “A Theory of Electrons and Protons,” PRSL SA 126(‘30)
– The Dirac equation also has negative energy solutions as well as
positive solutions for electrons, as for all relativity theories, for
instance, the relativity Hamiltonian


  2
W / c  eA0 / c  p  eA / c  m2c 2  0
2
– In classical theory the dynamical variables must always vary
continuously and there will be a sharp distinction between positive
solutions and negative solutions (simply ignored).
Dirac Sea or Vacuum
• “A Theory of Electrons and Protons,” PRSL SA 126(‘30)
– All the states of negative energy are occupied except perhaps a few
of small velocity (holes). We expect the exact uniform distribution
to be completely unobservable.
• “Discussion of the infinite distribution of electrons in the
theory of positron,” MPCPS 30 (‘34)
– One can give a precise meaning of a distribution of electrons in
which nearly all of the negative-energy states are occupied and
nearly all of the positive-energy ones unoccupied.
– Further work that remains to be done is to examine the physical
consequences of the foregoing assumption and to see whether it
leads to any physical phenomena of the nature of a polarization of
a vacuum by an electromagnetic field.
Klein Paradox
• O. Klein found [Z. Phys. 53 (’29)]
that the Dirac eq could predict
the reflection probability greater
than
one.
2
4
 1  
R
 , T
1   2
1  
p E  m  



k E  m V 

V  E 2  m 2
k
Em
E  m V




• The Klein paradox is that
fermions can pass through large
repulsive potential without
exponential damping.
Pair Annihilation and Pair
Production
• Annihilation of electron-positron pair by Dirac (‘30)
e  e   1   2
e e
 

mc2
  C 
e

2
  e   2 e  
2

ln

1
,
(


mc
)



e
 mc2  mc2
 
 

• Breit-Wheeler process of electron-positron production
in collision of two photons (‘34) with threshold energy
 1   2  e  e
 

mc

  C 



2
2

 , (   mc2 )


Photon-Photon Scattering
• Classical Maxwell theory is linear and thus prohibits
a self-interaction (direct  scattering) .
• QED permits  and  to interact with virtual e-e+ pair
from the Dirac sea: the cross section in the low energy
limit of the two colliding  in the center of momentum
frame) [Euler (‘36); Akhiezer (‘37); Karplus, Neuman
(‘50)]: a vacuum polarization effect
  
*2
2
973
2


 2 C  3 , (  2 )
81000
m
 66
*
6
 7.4  10 ( [eV ])


• We have NOT seen the photon-photon scattering since
the early universe, but ELI is highly likely to detect it.
Quest for Vacuum and Pair Production
[SPK, JHEP11(‘07)]
Schwinger
Mechanism/
Polarization
Vacuum
QED
Condensed
Matter
Analogues
Fluctuations
Black Holes
Hawking
Radiation
Black Hole
Analogues
Ultrastrong Laser Sources
PW, Multi-PW and EW Lasers
HiPER
ELI
• High Power laser
Energy Research
(HiPER) facility at CLF
in UK for laser-driven
fusion or fast ignition.
• 200 kJ in 40 beams,
several nanoseconds,
photon energy 3 eV.
• 70 kJ in 24 beams, 15 ps
and photon energy 2 eV.
• Extreme Light
Infrastructure (ELI)
[http://www.extremelight-infrastructure.eu]
• The 4th facility for high
intensity physics will
achieve 200 PW (10
beams of 10-20 PW), one
shot per min and
intensity 1025 W/cm2.
Four Pillars of ELI
ELI-Beamlines Facility: Czech Republic
ELI-Nuclear Physics Facility: Romania
ELI-Attosecond Facility: Hungary
• ELI-Ultra High
Energy Field Facility
– location to be selected in
2012 and scheduled in
commissioning in 2017
Statistics of 4 Pillars of ELI
Country
Facility focus
Power
(PW)
Pulse
energy (J)
Pulse
width
(fs)
Rep rate
(Hz)
Romania
Nuclear
physics
10 (x 2)
200
20
0.1
Hungary
Attosecond
physics
1/20
5/400
5/20
1000/0.1
Czech
Republic
Secondary
1/5/10(x2) 10/50/200
beam radiation,
high-energy
particles
To be
High intensity
determined
10-20
(x10)
30-40 kJ
10/10/20 10/10/0.1
15
0.1
Di Piazza, Muller, Hatsagortsyan, and Keitel, “Extremely high-intensity
laser interactions with fundamental systems,” arXiv:1111.3886 [hep-ph]
Physics from Ultra High Field
•
•
•
•
•
•
Particle physics
Nonlinear field theory
Gravitational physics
Astrophysics and cosmology
Nuclear physics
Ultrahigh-pressure physics
ELI & Schwinger Limit
• The Schwinger limit
(critical strength) for
e-e+ pair production
m2
Ec 
 1.3  1016 (V / cm)
|e|
Ec2
Ic 
 2.3  1029 (W / cm 2 )
8
Fundamental Physics with ELI
• Can test strong QED
(Delbruck scattering) (Photon splitting) (Pair production)
• Can test the Hawking-Unruh radiation(a  1024 g )
Schwinger Pair Production
Klein-Gordon or Dirac Equation
• In the space-dependent gauge A(z) for E(z), the
Fourier spin-component equation (tunneling)
2
2
2
2
2
2
 z  kk  ( z) k  ( z)  0, kk  ( z)  (  qA0 ( z))  (m  k )  2iqE( z)





• In the time-dependent gauge A(t) for E(t), the Fourier
spin-component equation for Dirac or Klein-Gordon
equation
2
2
2
2
2
2
 t  k (t ) k (t )  0, k (t )  (k z  qAz (t ))  (m  k )  2iqE(t )


Dirac Theory of Electron &
Positron
• The vacuum is the Dirac sea, fully
occupied with all electrons with
momenta and negative energy.
• Depletion of a negative energy
state means that the sea has a
positive net charge and positive
net energy (like electron-hole in
semiconductor).
• The conservation of energy and
momentum prohibits spontaneous
pair production (jump from a
negative energy to a positive one).
Fig. from Ruffini, Vereshchagin, and Xue, Phys. Rep. 487 (2010)
Tunneling Picture for
Schwinger Pair Production
• Application of a constant Efield changes the energy
spectra:

   eE z  p 2  m2
Fig. from Ruffini, Vereshchagin, and Xue,
Phys. Rep. 487 (2010)
• Quantum mechanically, a
negative charge from the
Dirac sea can tunnel through
the tilted barrier, which leads
to particle-antiparticle pair.
• The tunneling probability is
the Schwinger formula for
pair production:



m 2  p2
P( p )  exp   
| eE |





Scattering Picture & Stokes
Phenomenon
Scattering over the Barrier
Stokes Phenomenon
[Fig. from Dumlu & Dunne, PRL 104
(2010)]
Boson & Fermion Production
• In the phase-integral method, the mean number of pairs
in the gauge field with one pair of turning points [SPK,
2 Im S
Page, PRD 65 (‘02); 73 (‘06); 75 (‘07)]: N  e
• For gauge field with two pairs of turning points, the
mean number of boson pairs [Dumlu & Dunne, PRL 104
(2010)]
N boson  e 2 Im S ( I )  e 2 Im S ( II )  2 cos(ReS ( I , II ))e  Im S ( I )  Im S ( II )
 4 sin2 (Re S )e  2 Im S
• The mean number of fermion pairs
N fermion  e 2 Im S ( I )  e 2 Im S ( II )  2 cos(ReS ( I , II ))e  Im S ( I )  Im S ( II )
 4 cos2 (Re S )e  2 Im S
Quantum Vlasov Equation
• In the time-dependent gauge for electric field, the energy
of the charged particle
takes the form 2
2


(
t
)


2
2
2
2
()
k
k (t 0 )
 k (t )  (k||  qA|| (t ))  k   m ,  k (t ) 
 k (t 0 )
• The pair production rate in the adiabatic approach
[Popov, JETP 34 (‘72); Kluger et al, PRL 67 (‘91)]
t'
 k (t ) t  k (t ' )
d


1  2 N k (t )  


dt
'
1

2
N
(
t
'
)
cos
2
dt
'
'

(
t
'
'
)

k
k
 t 0

dt
 k (t ) t 0  k (t ' )
• The pair production rate in the non-adiabatic approach
[SPK, Schubert, PRD 84 (‘11)]
t
t
d
( )
( )

1  2N k (t )  k (t )t0 dt' k (t ' )1  2N k (t ' )cos t ' dt"(k) (t") 


dt
Electron-Positron Pair
Production
Electron-Positron Pair
Production
• e-e+ production by a high energy photon
propagating in a strong laser field (BreitWheeler pair production).
• e-e+ production by a Coulomb field in the
presence of a strong laser field.
• e-e+ production by two counter-propagating
strong laser beams forming a standing light
wave (spontaneous production in strong
electric field).
Pair Production
Electron-laser collisions
Laser photon energy: 2.4 eV
[Hu et al, PRL 105 (‘10)]
Dynamically Assisted
Schwinger Mechanism
E

E (t ) 

2
cosh (t ) cosh2 (t )
[Dunne et al, PRD 80 (‘09); PRL 101
(’08)]
Condensed Matter Analogue
Condensed Matter Analogue of QED
• The relation between the theory of dielectric breakdown
in condensed matter and nonlinear QED from the view
point of the effective Lagrangian Leff ( A)  i lim1 ln (t) | (0) 
t  t
[Oka, Aoki, Lect. Notes Phys. 762 (‘09)]
Condensed Matter
Mechanism
Excitation
Effective action
QED
Landau-Zener tunneling
Schwinger mechanism
Electron(doublon)-hole pair Electron-positron pair
Nonadiabatic Berry’s phase
Nonlinear polarization Photovoltaic Hall effect
Floquet picture
Heisenberg-Euler/Schwinger
- interaction (birefringence)
Furry picture
Strong Field Physics in
Condensed Matter
• Several Phenomena in condensed matter physics in
strong electric fields in E(field strength)-(photon
energy) space [Oka, Aoki, arXiv:1102.2482; Lect. Notes
Phys. 762 (‘09)]
Carriers (doublons and holes)
created by an external electric field
Graphene Analogue of QED
• Effectively massless Dirac fermions
H 0  ivF
• The Klein paradox
V0
V ( x)  
0
(0  x  D )
(otherwise)
cos2 
T
1  cos2 (q x D) sin2 
• T = 1 for normal incidence or qxD = N.
• The Klein tunneling was experimentally
observed in graphene heterojunctions [A.
F. Young and P. Kim, Nat. Phys. 5 (‘09)].
Fig. Katsnelson, Novoselov and
Geim, Nat. Phys. 2 (‘06)
Heisenberg-Euler &
Schwinger Effective Action
Heisenberg-Euler/Schwinger
Effective Action
• Maxwell theory and Dirac/Klein-Gordon theory are
gauge invariant:


1 
1 2
F  F F  B  E 2 ,
4
2
X 
1  *
G  F F  B  E
4
2( F  iG)  X r  iX i
• The Heisenberg-Euler/Schwinger effective action per
volume and time [J. Schwinger, “On gauge invariance
and vacuum polarization,” PR 82 (‘51) 664; B. DeWitt:
“This is one of the great papers of all time.”]
Leff   F 
1
8

2
0 ds
e
m2 s
s3


Re cosh(qXs)
2
2
2
 1  (qs) F 
(qs) G
Im cosh(qXs)
3


Going Beyond Schwinger
One-Loop Effective Actions
• The in-/out-state formalism via the Schwinger variational
principle [Schwinger, PNAS(‘51); DeWitt, Phys. Rep. (‘75),
The Global Approach to Quantum Field Theory (‘03)]
e
iW
e
i
 g d D xL
e ff
 0, out | 0,in
• The vacuum persistence
0, out | 0,in
2
 e 2 ImW
2 ImW  VT  ln(1  N k )
k
Out-Vacuum from In-Vacuum
• For bosons, the out-vacuum is the multi-particle states of
but unitary inequivalent 0; out | 0; in  0 to the in-vacuum:
0; out  U k 0; in  
k
k
1
 k,in
 
   
n 
k,in
*
k,in
k




nk
nk , nk ; in
• The out-vacuum for fermions:

0; out  U k 0; in     k,* in 1k ,1k ; in   k,in 0 k , 0k ; in
k
k

Effective Actions at T=0 & T
• Zero-temperature effective action for scalar and spinor
from the gamma function-regularization [SKP, Lee, Yoon,
PRD 78, (‘08); 82, 025016 (‘10); SPK, PRD 84 (‘11) ]

W  i ln 0, out | 0,in  i  ln k*  i  ln  c pk  id pk
k

p,k
• finite-temperature effective action for scalar and spinor
[SKP, Lee, Yoon, PRD 82, 025016 (‘10)]

Tr
(
U
 in )
3

exp[i  d xdtLeff ]  0,  , in U 0,  , in 
Tr (  in )
Thermofield Dynamics
• Thermal vacuum [Takahashi, Umezawa (’75)]
1
0,  , in  1/ 2  exp[Enk ,in / 2] nk , in  n~k , in
Zin k , nk
• Thermal expectation value: the expectation value in
the thermal vacuum
O

 Tr(O in )  0,  , in O 0,  , in
• Finite-temperature field theory is equivalent to zerotemperature field theory in the “thermal vacuum”.
Effective Action at T
• Expectation value of U in thermal vacuum

T
r
(
U
in )
3

exp[i  d xdtLeff ]  0,  , in U 0,  , in 
T r( in )
• Effective action per unit volume and time


Leff  i  ln 1  e   (k  z k )
 z k
 ln 1  e  k 




k , 
vacuum effective action zero field subtractio n 

1
 e  z k , z k  z r (k )  izi (k )

k



Vacuum Polarization & Persistence
• Purely thermal part of the effective action
Leff (T , E )  Leff (T , E )  Leff (T  0, E )
 
 
 i  ln 1  e (k  zk )  ln 1  e k
k ,

• Imaginary part of the effective action
( nFD / BE (k )) j zk
1
zk*
j
Im(Leff )   i
(e  1)  (e  1) j
2 k , j 1
j

• Real part of the effective action


sin(zi (k ))
Re(Leff )   arctan  (k  zk )

 cos(zi (k )) 
k ,
e

Vacuum Polarization at T
• Structure of the effective action at T
Re(Leff (T ))


sin(Re Leff (T  0, k ))
   arctan  k  Im Leff ( 0 )

e
e

cos(Re
L
(
T

0
,
k
))
k ,
eff




sin(Re Leff (T  0, k ))
   arctan

 k
2 (1 2| |) / 2
k ,
 cos(Re Leff (T  0, k )) 
 e 1 |  k |


Pair Production at T
• Imaginary part of the effective action (the limit of
small mean number of produced pairs)
2 Im Leff (T )  |  k |2 nFD / BE (k )
k ,
• Consistent with the pair-production rate at T [SPK,
Lee, PRD 76 (‘07); SPK, Lee, Yoon, PRD 79 (‘09)]
 |  k |2 tanh(k / 2)
k
sp/sc
N (T )  
2
|

|
 k coth(k / 2)
k
QED Effective Action in E=const
• The Bogoliubov coefficient for scalar and
spinor in constant E-field [SPK, Lee, Yoon,
PRD78 (‘08)]
2 i ( p 1) / 2
k 
e
,
(  p )
1
m2  k 2
p    i
2
2(qE)
• The effective action for scalar/volume and time
Lsc ( E )  i
2
qE d k   e
ds

2 0
4 (2 )
( p * 1 / 2 ) s
s

1
2 s

 

 sinh(s / 2) s 12 
QED Vacuum Polarization
• Scalar QED: renormalized effective action per volume
and per time for a constant E-field
 m 2 s / qE
2

(qE)
e
L (E)  
P
ds
16 2 0
s2
sc
eff
1 s
 1

 sin s s  6 


• Spinor QED: renormalized effective action per volume
and per time for a constant E-field
2

 m 2 s / qE
(qE)
e
L ( E) 
P  ds
2
0
8
s2
sp
eff
1 s

cot(s)  s  3 
QED Vacuum Persistence
• Spinor QED: Schwinger pair production in a constant
E-field
2
(
qE
)
2 Im( Lspeff ) 
4 3
 m 2 n 
1
2qE dk2
 n 2 exp  qE    2  (2 ) 2 ln1  N k 
n 1



• Scalar QED: Schwinger pair production
(qE) 2
2 Im( L ) 
8 3
sc
eff
Nk  e

 ( m 2  k 2 )
qE
 m 2 n  qE dk2
(1) n 1
 n 2 exp  qE   2  (2 ) 2 ln1  N k ,
n 1



Vacuum Polarization Beyond Schwinger
• Scalar/Spinor QED in a pulsed E-Field [SPK, Lee, Yoon,
PRD 78 (‘08)]
E(t )  E0sech2 t /  
• Imaginary part from the mean number of pairs
2| | S k
2| | S k

1

2
|

|
dk
(
1

(

1
)
e
)(
1

(

1
)
e

Im( Leff )  (1) 2| |
ln
 (2 ) 3 
Rk( )
Rk( )
2
(1  e
)(1  e
)

3
()
S k( )   [ k ()   k ()]  2 (qE0 2 ) 2  (1  2 |  | / 4)
Rk( )  2 k ()
()
) 


Vacuum Polarization Beyond Schwinger
• Scalar/Spinor QED in a localized E-Field [SPK, Lee, Yoon,
PRD 82 (‘10)]
E( z)  E0sech2 z / L
• Imaginary part from the mean number of pairs
2| | Sk
2| | Sk 

1

2
|

|
d

d
k
(
1

(

1
)
e
)(
1

(

1
)
e
)
2| |


Im(Leff )  (1)
ln
3

 (1  (1) 2| | e R( k) )(1  (1) 2| | eS( k) ) 
2
(2 )


2
()
S( k)  L[kk ()  kk ()]  2 (qE0 L2 ) 2  (1  2 |  |) / 4
R( k)  L[kk ()  kk ()]  2 (qE0 L2 ) 2  (1  2 |  |) / 4
()
Conclusion
• The ultra strong lasers from Extreme Light Infrastructure
(ELI) can detect
– Direct photon-photon scattering (vacuum polarization)
– Schwinger pair production (vacuum persistence)
– Other nonperturbative phenomena
• Strong Field Physics (& quantum structure of vacuum)
will be based on experimentation, direct or indirect:
– Strong QED & QCD
– QFT in curved spacetimes