Schwinger Effect in Curved Spacetimes
Transcript of Schwinger Effect in Curved Spacetimes
Schwinger Effect in Curved
Spacetimes
Sang Pyo KimKunsan National University
Black Holesâ 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
ĎT,
T
mN S
S
S2
1
2exp
Schwinger Effect in Charged Black HolesZaumen (â74)
Carter (â74)
Gibbons (â75)
Damour, Ruffini (â76)
âŽ
Khriplovich (â99)
Gabriel (â01)
SPK, Page (â04), (â05), (â08)
Ruffini, Vereshchagin, Xue (â10)
Chen, SPK, Lin, Sun, Wu (â12); Chen, Sun, Tang, Tsai (â15)
Ruffini, Wu, Xue (â13)
SPK (â13)
Cai, SPK (â14)
SPK, Lee, Yoon (â15); SPK (â15)
Chen, SPK, Tang, Kerr-Newman BH, in preparation (â16)
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. tâHooft & A. Strominger, âconformal symmetry near the
horizon of BH,â MG14, July 2015.
magnification
Schwinger Effect in (A)dS[Cai, SPK (â14)]
Vacuum
Fluctuations
QEDSchwinger
Effect/ Unruh Effect
de SitterGibbons-Hawking Radiation
Effective Temperature for
Unruh Effect in (A)dS[Narnhofer, Peter, Thirring (â96); Deser, Levin (â97)]
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 (â90)]
⢠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 (â99);
two-loop by Dunne, Schubert (â00)]
⢠Borel summation of real effective action for dS (AdS) and
vacuum persistence for Gibbons-Hawking radiation
[Dunne, Das (â06)]
2
2424226
24)1( ,
3
1
2
11
)22(
41
m
eEg
ngma
nnn
n
n
eE
Ďkm
km
eE
Ď
m(E)L
k
eff
2
12
2
23
4
exp1
4Im2
Perturbative QED Action in Curved Spacetime[Davila, Schubert (â10)]
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 (â51); DeWitt (â75), (â03)]
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 (â08),
(â10); SPK (â11)]; gamma-function & zeta-function
regularization [SPK, Lee (â14)]; quantum kinematic
approach [Bastianelli, SPK, Schubert, in preparation (â16)]
⢠finite-temperature effective action [SPK, Lee, Yoon (â09),
(â10)]
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 (â08), (â10); SPK
(10), (â11)]
⢠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 (â94); SPK, Page (â08)] and in AdS2 [Pioline, Troost
(â05); SPK, Page (â08)]
)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 (â96); Deser, Levin (â97)]
⢠Effective temperature for Schwinger formula in (A)dS
[Cai, SPK (â14)]
)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 (â14)]
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 (â15)]
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 đĄ and đĽ due to the symmetry of
spacetime and the field, and the integration of đ 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 (â15)]
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 (â15); SPK (â15)]
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: đ,đ, đ =đ˝
đ; đ0
2 ⥠đ2 + đ2
⢠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.