Hawking Radiation and Vacuum Polarization
Sang Pyo Kim (金相杓 )Kunsan Nat’l Univ. & YITP, Kyoto Univ.
竹原理論物理硏究會 , June 06-08, 2011[AS & NCU & NTHU & Kinki seminars]
Outline
• Introduction• Spin Statistics Inversion in QED• Vacuum Polarization and Hawking Radia-
tion• Conclusion
QED Vacuum Polarization
• Scalar QED: Weisskopf/Schwinger effective action per volume and per time in a constant E-field
• Spinor QED: Heisenberg-Euler/Schwinger effective action per volume and per time in a constant E-field
62
)2/sin(
1
)(2
k
)2(2)(
0
2
k
2
22sceff
22
s
se
s
dsP
dqEEL
sqE
m
122
)2/sin(
)2/cos(
)(2
k
)2(2
)(2)(
0
2
k
2
22speff
22
s
s
se
s
dsP
dqEEL
sqE
m
QED Vacuum Persistence• Spinor QED: Schwinger pair production in a con-
stant E-field
• Scalar QED: Schwinger pair production
)/(
2,
,1ln)2(
k
2exp
)1(
8
)()Im(2
)22
k(
)k(
k
k2
2
1
2
2
1
3
2sceff
222
mqEeeN
NdqE
qE
nm
n
qEL
m
mqE
m
n
n
k2
2
1
2
23
2speff 1ln
)2(
k
2
)(2exp
1
4
)()Im(2 N
dqE
qE
nm
n
qEL
n
Nonperturbative Aspect of QED & sQG
[SPK, JHEP11(2007)048]
Davies-Unruh Effect &
Pair Production
Schwinger Mechanism
QED
QCD
Hawking Radia-tion
Black Holes
de Sitter/ Ex-panding Uni-
verse
One-Loop Effective Actions
• The in-out formalism via the Schwinger variational principle [Schwinger, PNAS(‘51); DeWitt, Phys. Rep. (‘75), The Global Approach to Quantum Field Theory (‘03)]
• The vacuum persistence (twice of the imaginary part ) and the mean number of produced pairs
in0,|out0,
in0,out0,in0,|out0,
eff
xLdgiiWD
ee
Si
k
Im22
)1ln(Im2
in0,|out0,
k
W
NVTW
e
Bogoliubov Transformation & In-Out Formalism
• The Bogoliubov transformation between the in-state and the out-state, equivalent to the S-ma-trix,
• Commutation relations from quantization rule (CTP):
• Particle (pair) production
kink,kink,*
ink,ink,ink,outk,
kink,kink,*
ink,ink,ink,outk,
UbUabb
UaUbaa
p)(k,),pk(,
);pk(,),pk(,
outp,outk,outp,outk,
outp,outk,outk, outp,
bbaa
bbaa
1;2
k
2
k
2
kk N
Out-Vacuum from In-Vacuum
• For bosons, the out-vacuum is the multi-particle states of but unitary inequivalent to the in-vacuum:
• The out-vacuum for fermions:
kkk
ink,
*ink,
ink,kk in;,
1in;0out;0
k
k
n
n
nnU
k
kkink,kk*
ink,k
k in;0,0in;1,1in;0out;0 U
0in;0|out;0
Out-Vacuum from S-Matrix
• The out-vacuum in terms of the S-matrix (evolution operator)
• The diagrammatic representation for pair production
)sinh(,cosh
)](exp[
)]1(exp[,
2*
2ink,ink,
2ink,ink,kk
ink,ink,ink,ink,kkkkk kk
kii
kki
k
ii
reere
ebaebarS
bbaaiPPSU
kkk
in)](exp[out kk 2ink,ink,
2ink,ink,k
ii
k
i ebaebare k
Effective Actions at T=0 & T
• Zero-temperature effective action for scalar and spinor [SKP, Lee, Yoon, PRD 78, 105013 (‘08); 82, 025016 (‘10)]
• finite-temperature effective action for scalar and spinor [SKP, Lee, Yoon, PRD 82, 025016 (‘10)]
k
*klnin0,|out0,ln iiW
)(Tr
)(Trin,0,in,0,]exp[
in
ineff
3
UUxdtLdi
Spin Statistics in QFT• The spin statistics theorem
-Bosons: Bose-Einstein distribution and commutator-Fermions: Fermi-Dirac distribution and anticommutator
• The vacuum persistence for boson production takes the form of spinor QED [Stephens, AP193 (’89)]
• The vacuum persistence for fermion production takes the form of scalar QED
KK
K
Ke
eW
1ln1
11ln)Im(2 bos
KK
K
Ke
eW
1ln1
11ln)Im(2 fer
Spin-Statistics Inversion• The vacuum polarization of scalar QED can be
written as a spectral function times the Fermi-Dirac distribution and that for spinor QED as a spectral function times the Bose-Einstein distribu-tion [Muller, Greiner, Rafelski, PLA 63 (‘77)]
)/(2
2,
1
1ln)1ln()(
)1(
)(
)2(4
)1||2()Re(2
02
022
4
eff0
mqEis
isisssF
e
sFds
mL
s
Spin-Statistics Inversion• The vacuum persistence in terms of the trans-
verse energy [Hwang, SPK, PRD 80, 065004(‘09)]
• The vacuum persistence in terms of the instanton action or the worldline instanton
)/(
2,
2
k
)1(2
)1||2()Im(2
2
02)2/(
2
eff0
mqEm
ed
mL
m
02)(eff
)1(
/)(
22)Im(2
Sz
e
ddSd
dkmL
Vacuum Polarization and Hawking Radiation
SPK & W-Y. Pauchy Hwang, “Vacuum Polariza-tion and Persistence on the Black Hole Hori-
zon,” [arXiv:1103.5264]
Vacuum Persistence for BH
• Hawking radiation of bosons and fermions in a charged rotating black hole
• Vacuum persistence for zero amplification (RJ = 0)
• 2Im(W) as ln(Z), the logarithm of the partition func-tion, plus the vacuum energy implies that 2Im(W)/V is the pressure of boson and fermion gas [Ritus, (‘84), (‘98)]
,2
,1
,1
||1)(
)(
2
H
HBqmJ
J TTke
RN
HH
)(1ln)1ln()Im(2 HH qm
JJ
J
eNW
Effective Action for Black Hole
• Schwarzschild black hole in d-dimensions
• Bogoliubov coefficients for a massless boson [De-Witt, Phys.Rep.19 (‘75)]
• Effective action
ieAieA JJJJ 1,1 2/2/
2
,2
)(',
)()( 2
22
222
Hd
rfdr
rf
drdtrfds
)1(ln
2
horizon) of (area0
,,
i
diW
pml
Vacuum Polarization for BH
• Hawking radiation in a Schwarzschild black hole is effectively two-dimensional one, in the (t-r) space-time
• Reduced effective action for bosons and fermions
• Vacuum polarization as a thermal effective action
• Renormalized vacuum polarization for massive emission
ei
s
se
s
dsdL s
pml
1ln)2/sin(
)}2/{cos(
22
10
2/
0,,
red
0 0 220,,
red'
'
122
1)Re(
s
d
e
sds
dL
spml
0
02
,,2red )2/sin(
)}2/{cos(
2
1)Re( sa
s
a
s
se
s
dsL s
s
pml
Vacuum Persistence and Gravi-tational Anomalies
• Vacuum persistence = decay rate of vacuum due to Hawking radiation or Schwinger mechanism
• Trace anomalies = Hawking radiation and Schwinger mechanism [Christensen, Fulling, PRD 15 (’77); Dit-trich, Sieber, JPA 21 (‘88)]
• Vacuum persistence for bosons and fermions
equals to the total flux of Hawking radiation from gravitational anomalies [Robinson, Wilczek, PRL 95 (‘05); Iso, Umetsu, Wilczek, PRL 96 (‘06)].
pmlpml
LL,,
2ferred
,,2
bosred
1
24)Im(2,
1
12)Im(2
Duality between BH and QED
)2
sin(
)]2
[cos()(
)2
sin(
)}2
{cos()(
2
1onPolarizati Vac.
)1ln()()1ln()(ePersistenc Vac.
)2(
k
2
1 States of#
2
)/(QED
2
BH ildSchwarzsch1Notation
2)
22(
02
)22
(
2
2
,,
B
2
2
s
s
es
dss
s
es
ds
ee
dmd
mqETk
sm
m
k
J
s
J
m
m
k
JJ
pmlJ
Conclusion
• The nonperturbative effective actions for QED and gravity in the in-/out formalism al a Schwinger varia-tional principle.
• The effective action for pair production are character-ized by two prominent aspects: the vacuum polariza-tion (real part) and persistence (imaginary part)
• The effective action for Schwarzschild black hole, modulo number of states, is dual (equivalent) to QED effective action in constant E.
• Duality of gravity coupling and gauge coupling? (Davies-Unruh temperature)
m
e
c
ETk
GM
cTk EH
B
3
B
1
8