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

Introduction

Lectures at YITPMay 24-25, 2011

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 Inversion in QED

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