Post on 25-Feb-2022
Schwinger Effect in Curved
Spacetimes
Sang Pyo KimKunsan National University
Black Holes’ New Horizons
CMO, Oaxaca, May 15-19, 2016
Outline
• Why Schwinger Effect in Curved Spacetimes?
• Perturbation Theory & Borel Summation &
Vacuum Persistence Amplitude
• Effective Actions in In-Out Formalism
• Reconstructing Effective Action
• QED in (anti-) de Sitter Space
• Schwinger Effect in Near-extremal RN BHs
• Schwinger Effect in Near-extremal KN BHs
• Conclusion
What is Schwinger Effect?
• Constant E-field changes energy
spectra in Minkowski spacetime:
• Spontaneous creation of a particle-
antiparticle pair from the Dirac sea
(quantum mechanical tunneling)
• Critical (Schwinger) field to
energetically separate the pair
22 mpxeE
2mcmc
eEc
m
qE
πT,
T
mN S
S
S2
1
2exp
Schwinger Effect in Charged Black HolesZaumen (‘74)
Carter (‘74)
Gibbons (‘75)
Damour, Ruffini (‘76)
⋮
Khriplovich (‘99)
Gabriel (‘01)
SPK, Page (‘04), (‘05), (‘08)
Ruffini, Vereshchagin, Xue (‘10)
Chen, SPK, Lin, Sun, Wu (‘12); Chen, Sun, Tang, Tsai (‘15)
Ruffini, Wu, Xue (‘13)
SPK (‘13)
Cai, SPK (‘14)
SPK, Lee, Yoon (‘15); SPK (‘15)
Chen, SPK, Tang, Kerr-Newman BH, in preparation (‘16)
Why Schwinger Effect in (A)dS2?Near-Horizon Geometry of RN BHs
RN Black Holes
Near-extremal BH
AdS2S2
NonextremalBH
Rindler2 S2
Rotating BH in dS
S-scalar wave
QED in dS2
Near-horizon Geometry of Near-extremal
RN BH
Inner Outer Horizons
AdS2S2
G. t’Hooft & A. Strominger, “conformal symmetry near the
horizon of BH,” MG14, July 2015.
magnification
Schwinger Effect in (A)dS[Cai, SPK (‘14)]
Vacuum
Fluctuations
QEDSchwinger
Effect/ Unruh Effect
de SitterGibbons-Hawking Radiation
Effective Temperature for
Unruh Effect in (A)dS[Narnhofer, Peter, Thirring (‘96); Deser, Levin (‘97)]
Effective Temperature
Unruh Effect
TU = a/2
(A)dSR = 2H2
or -2K2
2
2
Ueff8
RTT
Borel Summation
• Large-order perturbation theory may have a divergent
power series with the asymptotic form with three real
constant , >0 and [Le Guillou, Zinn-Justin (‘90)]
• Leading Borel approximation for alternating case (+ sign)/
nonalternating case ( sign) and vacuum persistence
)(,)()1()(00
ngngagf n
n
nn
n
n
n
/1/
/1/
0
1exp
1)(Im
exp1
11)(
gggf
g
s
g
s
ss
dsgf
Borel Summation
• Heisenberg-Euler-Schwinger QED action in a constant
electric field E
• Borel summation leads to the vacuum persistence in a
constant electric field E [one-loop by Dunne, Hall (‘99);
two-loop by Dunne, Schubert (‘00)]
• Borel summation of real effective action for dS (AdS) and
vacuum persistence for Gibbons-Hawking radiation
[Dunne, Das (’06)]
2
2424226
24)1( ,
3
1
2
11
)22(
41
m
eEg
ngma
nnn
n
n
eE
πkm
km
eE
π
m(E)L
k
eff
2
12
2
23
4
exp1
4Im2
Perturbative QED Action in Curved Spacetime[Davila, Schubert (‘10)]
Borel summation? Find large-order perturbation for vacuum persistence!
In-Out Formalism for QED Actions
• In the in-out formalism, the vacuum persistence amplitude
gives the effective action [Schwinger (‘51); DeWitt (‘75), (‘03)]
and is equivalent to the Feynman integral
=
• The complex effective action and the vacuum persistence for
particle production
in0,|out0,eff
2/1)(
xLdgDiiW ee
k
Im22
)1ln(Im2,in0,|out0, k
W NVTWe
Effective Actions at T=0 & T
• Zero-temperature effective actions in proper-time integral
via the gamma-function regularization [SPK, Lee, Yoon (‘08),
(‘10); SPK (‘11)]; gamma-function & zeta-function
regularization [SPK, Lee (‘14)]; quantum kinematic
approach [Bastianelli, SPK, Schubert, in preparation (’16)]
• finite-temperature effective action [SPK, Lee, Yoon (‘09),
(‘10)]
kk
*
k )k(lnln ll
l
ibaiiW
)(Tr
)(Trin,0,in,0,exp
in
ineff
3
UUxdtLdi
-Regularization
• Assumption: Bogoliubov coefficients of the form
• The effective action from Schwinger variational
principle
• The integral representation for gamma-function
)(
)(;
)(
)(
ikh
igf
idc
ibakk
away regulated be totermsedrenormaliz be toterm
)(
0)1(
11)(ln z
z
z
z
zibaz
eibae
e
e
e
z
dziba
k
*
keff lnin0,|out0,ln iiW
-Regularization• -regularization [SPK, Lee, Yoon (‘08), (‘10); SPK
(10), (‘11)]
• The Cauchy residue theorem
epersistenc vacuum
1
)2)((
onpolarizati vacuum
0
))((
0
)(
211
n
iniba
is
isiba
z
ziba
in
ei
e
e
s
dsP
e
e
z
dz
Conjecture
• Can one find the effective action from the pair-production
rate? inverse procedure of Borel summation (Gies, SPK,
Schubert)
• If the imaginary part (vacuum persistence) of the effective
action can be factorized into a product of one plus or one
minus exponential factors, then the structure of simple poles
and their residues of these factors uniquely determine the
analytical structure of the proper-time integrand of the
effective action (vacuum polarization) (modulo entire
function independent of renormalization via Mittag-Leffler
theorem):
states }{stateseff )1ln()()1ln()Im(2
}{
I
S I
eNL
Reconstructing Effective Action
from Pair-Production Rate
• Scalar/Spinor effective action (Real part) vs Imaginary part
(Cauchy theorem vs Mittag-Leffler theorem/Borel
summation)
• Most of imaginary parts from the pair-production rate
(Schwinger formula in E & B, Bose-Einstein or Fermi-Dirac
distribution) can be written as a sum of the form
10 2
1
1
0 2
11ln
3
1
sin
cos
)1(1ln
6
1
sin
1
n
SnSSs
n
Snn
SSs
en
ieis
ss
s
s
edsP
en
ieis
sss
edsP
}{}{states
eff )1ln()1ln()Im(2}{}{
II
R
I
S III
eeL
Schwinger formula in (A)dS
• (A)dS metric and the gauge potential for E
• Schwinger formula (mean number) for scalars in dS2
[Garriga (‘94); SPK, Page (‘08)] and in AdS2 [Pioline, Troost
(‘05); SPK, Page (‘08)]
)1)(/(,
)1)(/(,
0
2222
1
2222
KxKx
HtHt
eKEAdxdteds
eHEAdxedtds
2
2
2
2
22
16211
42
,
qE
R
qE
Rm
m
R
qE
mSeN S
Effective Temperature for
Schwinger formula• Effective temperature for accelerating observer in (A)dS
[Narnhofer, Peter, Thirring (‘96); Deser, Levin (‘97)]
• Effective temperature for Schwinger formula in (A)dS
[Cai, SPK (‘14)]
)2(,2,8
, 22
2
2
Ueff
/ eff KHRR
TTeNTm
U2
2
UAdSU
2
GH
2
UdS
GHU
2/
8;
2,
2
/,
8,eff
TR
TTTTTT
HT
mqET
RmmeN
Tm
Scalar QED Action in dS2• Mean number for pair production and vacuum polarization
from the in-out formalism [Cai, SPK (‘14)]
2
22
2
/
0
nsubtractio Schwinger
2/)(2
(1)
dS
dS
)1(
dS2
2)(
dS
2,4
12
6
2
)2/sin(
)2/cos(
12
2
)2/sin(
1
22
1lnIm2,1
H
qES
H
m
H
qES
s
ss
se
s
sse
s
dsP
SHL
NWe
eeN
sS
sSS
S
SSS
Scalar QED Action in AdS2
• Mean number for pair production and vacuum polarization
2
22
2
0
2/2
(1)
AdS
AdS
)1(
AdS)(
)()(
AdS
2,4
12
12
2
)2/sin(
1)2/cosh(
22
1lnIm2,1
K
qES
K
m
K
qES
s
sssSe
s
dsP
SKL
NWe
eeN
sS
SS
SSSS
Spinor QED Action in dS2
• Mean number for pairs and vacuum polarization [SPK (‘15)]
2
22
2
0
/2/)(2
sp
dS
sp
ds
)1(
dS2
2)(
sp
ds
2,2
6
2)
2cot(
2
1lnIm2,1
H
qES
H
m
H
qES
s
s
see
s
dsP
SHL
NWe
eeN
sSsSS
S
SSS
Spinor QED Action in AdS2
• Mean number for pairs and vacuum polarization
2
22
2
0
2/)(2/)(2
sp
AdS
sp
AdS
sp
AdS)(
)()(sp
AdS
2,2
6
2)
2cot(
2
1lnIm2,1
K
qES
K
m
K
qES
s
s
see
s
dsP
SKL
NWe
eeN
sSSsSS
SS
SSSS
Schwinger Effect in D-dimensional dS
• The Schwinger effect in a constant E in a D-dimensional dS
should be independent of 𝑡 and 𝑥 due to the symmetry of
spacetime and the field, and the integration of 𝑘 gives the
density of states D.
• dS radiation in E=0 limit and Schwinger effect in H=0 limit
222
222
2
2
2)(
2
22
1
dS
/
/2,
2
12
1)2(22
12
kHqE
HqE
H
qES
D
H
m
H
qES
e
eekdSH
xdtd
NdS
SSS
D
D
D
D
Interpretation of Schwinger Effect
• Thermal interpretation of Schwinger formula for
charged scalars (upper signs) and fermions (lower signs)
in spherical harmonics [SPK, Lee, Yoon (‘15)]
Qm
qmqET
QTTT
QTTT
e
e
e
eeN
HU
UURNUURN
T
m
T
qA
T
qA
T
m
T
m
T
m
NBH
RNH
H
RN
RNRN
22
/
)2
1(,)
2
1(
,
1
1
1
2222
)( 0
0
Schwinger Effect and Hawking
Radiation• Thermal interpretation of Schwinger formula for charged
scalars and fermions [SPK, Lee, Yoon (‘15); SPK (‘15)]
charges ofRadiation Hawking('00) AP Spindel & Gabriel
SpaceRindler in Effect Schwinger ('14) JHEPSPK & Cai
AdSin Effect Schwinger
0
0
2
1
)1(
1
RNH
HRN
RN
RNRN
RN
T
m
T
qA
T
qA
T
m
T
m
T
m
T
m
T
m
NBH
ee
ee
e
eeeN
Near-Horizon Geometry
• Kerr-Newman (KN) black hole: 𝑀,𝑄, 𝑎 =𝐽
𝑀; 𝑟0
2 ≡ 𝑄2 + 𝑎2
• Near-horizon geometry warped AdS2 of near-extremal KN
BH
dar
arrdt
ar
arQA
rdtar
ard
ar
ar
dBr
drdtBrards
r
BrMt
artt
ar
arrr
222
0
2
0
222
0
222
0
2
22
0
0
222
0
222
0
2
22
2222222
0
2
0
2
0
22
0
22
0
0
cos
sin
cos
cos
,2
cos
sin
cos
2,,,
Schwinger Effect for Charged Scalars
• Schwinger formula for charged scalars in spheroidal
harmonics in near-extremal KN BH
BS
armS
Sar
narqQS
e
e
e
eeN
c
aba
SS
SS
SS
SSSS
NKNbc
ac
ba
baba
2
,4
1
22,
22
,1
1
1
22
0
2
2
22
0
0
3
)(
)(
)(
)()(
Interpretation of Schwinger Effect• Thermal interpretation of Schwinger formula for charged
scalars in spheroidal harmonics in near-extremal KN BH
BH KN extremal-Nearfor Factor BH KN Extremal
1
1
1
MH
t
MH
t
KN
KNKN
T
m
T
T
m
T
T
m
T
m
T
m
NKN
e
e
e
eeN
22
0
222
0
222
0
0
3
2
2
2
2
4/11,
2,
2
2
112,
112
8,
8
armm
arR
arm
narqQT
TTTTTT
RTTT
RTTT
U
KNKNMKNKNM
UUKNUUKN
Conclusion• The in-out formalism is consistent and systematic QFT
method for vacuum polarization and vacuum persistence in
backgrounds (gauge and/or curved spacetimes) [cf. worldline
formalism and instanton in progress with Schubert.]
• The vacuum polarization of QED in (A)dS and near-extremal
RN black hole exhibits the gravity-gauge relation (or
AdS/CFT).
• The production of charged particles from an near-extremal
RN and Kerr-Newman black hole shows a strong interplay of
the Schwinger effect and the Hawking radiation and may
have a thermal interpretation.