how to fall into a black holeqpt.physics.harvard.edu/simons/verlinde.pdf · de grisogono delacour...

Simons Symposium

February, 2013

Quantum Entanglement: From Quantum Matter to String Theory

How to Fall into a Black Hole

Princeton University

PCTS

Herman Verlinde

Tuesday, February 19, 2013

?

Can a finite temperature CFTdescribe an object falling through the horizon of an AdS black hole?

The AdS/CFT correspondence has convinced usthat black hole physics follows the rules of QM.


?

Can a finite temperature CFTdescribe an object falling through the horizon of an AdS black hole?

The AdS/CFT correspondence has convinced usthat black hole physics follows the rules of QM.

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Hawking pair

horizon

b

a

r

e

Thursday, December 13, 2012

Figure 1: The firewall argument in a nut shell: quanta r and e can escape the black hole thanks to

entanglement with Hawking partners b and a. Monogamy prevents the partners to get entangled

with other interior quanta. So after the Page time, the black hole is in a maximally mixed state,

and a and b can not escape via the normal Hawking process.

bFIREWALL

ea

a

rb


Figure 2: When the Hawking partner b leaves the black hole, it can not be entangled with any

mode a right behind the horizon: the horizon turns into a firewall. To avoid this conclusion, one

needs a mechanism for allowing entanglement to re-establish inside the black hole.

Hawking pair

horizon

b

a

r

e






bFIREWALL

ea

a

rb





Rn =1

√Wn

i

i

iC †

n , R j

0a=

n

Rn

jn

a

RU i

0a

0b=

i 0U

if

i∈ Hcode

0 if i

/∈ Hcode

0U=

n

√Wn

na

nb.

Rn =1

√Wn

i

i

iC †

n , R j

0a=

n

Rn

jn

a

RU i

0a

0b=

i 0U

if

i∈ Hcode

0 if i

/∈ Hcode

0U=

n

√Wn

na

nb.


Hawking pair

horizon

b

a

r

e






bFIREWALL

ea

a

rb





Hawking pair

horizon

b

a

r

e






bFIREWALL

ea

a

rb






“What are the characteristics of a black hole?”


A black hole is always in a mixed state!

i∈ HM N = dimHM = eSBH , S

BH= 4πM2.

Ψ=

i∈Hcode

a i

i Φ i

↔ ρ

BH=

i∈Hcode

p i

i

i


BH= 4πM2.

Ψ=

i∈Hcode

a i

i Φ i

↔ ρ

BH=

i∈Hcode

p i

i

i


BH= 4πM2.

Ψ=

i∈Hcode

a i

i Φ i

↔ ρ

BH=

i∈Hcode

p i

i

i


BH= 4πM2.

Ψ=

i∈Hcode

a i

i Φ i

↔ ρ

BH=

i∈Hcode

p i

i

i


Hamiltonian evolution in Interaction Picture

Hilbert space splits up into three sectors:

with

H = HA⊗H

B⊗H

earlyR

HA= H

BH, H

B= H

lateR .

H = HBH

+HR+Hint,

H = HA⊗H

B⊗H

earlyR

HA= H

BH, H

B= H

lateR .

H = HBH

+HR+Hint,


ρAB(0) =

i

p i

i

i ⊗

0b

0,

ρAB(τ) = U ρ(0)U † =

n,m

ρnmn

b

m

ρB =

n

Wn

nb

n ; Wn =

e−βEn

ZSB = logZ + βE

Initial mixed state

evolves into

B = Hawking atmosphere

ρAB(0) =

i

p i

i

i ⊗

0b

0,

ρAB(τ) = U ρ(0)U † =

n,m

ρnmn

b

m

ρB =

n

Wn

nb

n ; Wn =

e−βEn

ZSB = logZ + βE

ρAB(0) =

i

p i

i

i ⊗

0b

0,

ρAB(τ) = U ρ(0)U † =

n,m

ρnmn

b

m

ρB =

n

Wn

nb

n ; Wn =

e−βEn

ZSB = logZ + βE


Time evolution in terms of Kraus Operators C

C’s are ergodic `random’ matrices with known statistical properties!

cf. loss of coherence

of a quantum computer

U (τ) i

0b=

n

Cn

i n

.

n

C †nCn = M .

CnC†m=

1

Zδnm M−En

iC †

nCm

k= Wn δnm δ ik .

U (τ) i

0b=

n

Cn

i n

.

n

C †nCn = M .

CnC†m=

1

Zδnm M−En

iC †

nCm

k= Wn δnm δ ik .

“BAD” “GOOD”

b

Unitary:

U (τ) i

0b=

n

Cn

i n

.

n

C †nCn = M .

CnC†m=

1

Zδnm M−En

iC †

nCm

k= Wn δnm δ ik .

U (τ) i

0b=

n

Cn

i n

.

n

C †nCn = M .

CnC†m=

1

Zδnm M−En

iC †

nCm

k= Wn δnm δ ik .


ρ(E)2=

wE

Nc+ 2

dB(E)

NZ

ρ(E)−

dB(E)

NZ

2

M−E

S(E) = wEdB(E)

log

NZ

dB(E)

−

y

2 + ylog

1 + y +

2y + y2

y =N

Nc

e−βE

2dB(E),

young BH : SAB = logNc SA = logNc + logZ + βE

old BH : SAB = logN, SA = logN − βE

We can compute the Page curve of our model!

Two limiting cases:

ρ(E)2=

wE

Nc+ 2

dB(E)

NZ

ρ(E)−

dB(E)

NZ

2

M−E

S(E) = wEdB(E)

log

NZ

dB(E)

−

y

2 + ylog

1 + y +

2y + y2

y =N

Nc

e−βE

2dB(E),



ρ(E)2=

wE

Nc+ 2

dB(E)

NZ

ρ(E)−

dB(E)

NZ

2

M−E

S(E) = wEdB(E)

log

NZ

dB(E)

−

y

2 + ylog

1 + y +

2y + y2

y =N

Nc

e−βE

2dB(E),



ρ(E)2=

wE

Nc+ 2

dB(E)

NZ

ρ(E)−

dB(E)

NZ

2

M−E

S(E) = wEdB(E)

log

NZ

dB(E)

−

y

2 + ylog

1 + y +

2y + y2

y =N

Nc

e−βE

2dB(E),



ρ(E)2=

wE

Nc+ 2

dB(E)

NZ

ρ(E)−

dB(E)

NZ

2

M−E

S(E) = wEdB(E)

log

NZ

dB(E)

−

y

2 + ylog

1 + y +

2y + y2

y =N

Nc

e−βE

2dB(E),




Key Tool: Recovery Super Operator R

Produces the Unruh Vacuum!

Rn =1

√Wn

i

i

iC †

n , R j

0a=

n

Rn

jn

a

RU i

0a

0b=

i 0U

if

i∈ Hcode

0 if i

/∈ Hcode

0U=

n

√Wn

na

nb.

Rn =1

√Wn

i

i

iC †

n , R j

0a=

n

Rn

jn

a

RU i

0a

0b=

i 0U

if

i∈ Hcode

0 if i

/∈ Hcode

0U=

n

√Wn

na

nb.

Rn =1

√Wn

i

i

iC †

n , R j

0a=

n

Rn

jn

a

RU i

0a

0b=

i 0U

if

i∈ Hcode

0 if i

/∈ Hcode

0U=

n

√Wn

na

nb.


Reconstruction of Interior Operators:

A acts on black hole Hilbert spacesatisfies QFT operator algebra

Π = a

0R†R

0a, Π2 = Π.

ΠU i

0b=

U

i|0b

if i

∈ Hcode

0 if i

/∈ Hcode

A = a

0R†AR

0a

φ(y) = a

0R†φ(y)R

0a.


U+|

R+|

A B 0U=

n

√Wn|na|nb . (79)

Acknowledgements:

References

[1]

[2]

34

2 Setting the Stage

In this section and the next, we will describe the basic setup and main assumptions. Our

guiding principle is that a black hole is an ordinary quantum mechanical system, – much

like an atom, albeit a very large one and with an enormous level density of states – that

evolves and interacts with its environment via standard hamiltonian evolution. We start

by setting up some notation that will help us model the evaporation process in a way that

keeps the rules of quantum mechanics manifestly intact. Along the way, we will state a

version of the firewal argument put forward in [1], that is most directly addressed by our

subsequent study.

2.1 Black Holes States: Old and Young

Consider a macrosopic black hole. Suppose that we have measured its mass M with some

high accuracy. For now, we will ignore the uncertainty in M , and any other macroscopic

properties such as angular momentum and charge. We associate a Hilbert space HM

to the

black hole with mass M , whose states

i∈ HM (1)

are indistinguishable from a macroscopic perspective. Each state | i describes the interior

of a black hole, with microscopic properties that are hidden from the outside observer by the

stretched horizon. The number of black holes micro-states is determined by the Bekenstein

Hawking entropy

N = dimHM = eSBH , (2)

with SBH

= 4πM2. We will assume that the time evolution of every micro state describes

a black hole that slowly evaporates via the Hawking process. So as time progresses, every

state i

∈ HM will evolve into a new state given by the product of a micro state of a black

hole with smaller mass M = M − E and a radiation state with energy E.

To describe this evaporation process we need to enlarge our Hilbert space, so that it

incorporates all possible intermediate situations. So we introduce the total Hilbert space

H, given by the tensor product

H = HBH

⊗HR (3)

5

We read off that

RU | i |0a|0b =

| i | 0U if | i ∈ Hcode

0 if | i /∈ Hcode

(42)

where | 0U is the following familiar looking state, defined on the tensor product of the

Hilbert space of the outside radiation region with the ancillary Hilbert space

| 0U =

n

√Wn |na |n b . (43)

Given that Wn are the Boltzmann weights, we recognize this state as the Unruh vacuum

state, with the ancilla states playing the role of the interior radiation region. We see explic-

itly how the R operation works: it sweeps all entanglement between the time evolved black

hole and the emitted radiation, encoded in U |i|0b, into entanglement of the radiation

with the ancilla. Thus the recovery operation restores the purity of the black hole state,

provided it started out within the code space.

We see from (42) that the recovery operator R is not a unitary map, since it projects out

all states that do not map back into the code space. This means that the operation R as

defined in (40)-(39) in fact is not really a valid superoperator. Preferrably, one would want

the recovery operation to be a unitary operation. To achieve this, consider the operator

Π = a0|R†R|0a (44)

We may also write Π =

nR†nRn =

i ,n

1Wn

Cn | i i |C †n. If R had been a valid su-

peroperator, then Π would have been equal to the identity operator. Instead it defines a

projection operator on the space of all states that can be reached by acting with the Cn

operators on the code subspace. In other words, it is the projection of all states that the

code states can evolve into as a result of the evaporation process.

A straightforward calculation, similar to the one done in eqn (41), shows that Π indeed

has the property of a projection operator:

Π2= Π (45)

To complete the recovery super operator, one thus adds one extra term to the right-hand

21

Acknowledgements

We thank Bartek Chech, Jan de Boer, Daniel Harlow, Steve Jackson,

helpful discussions. The work of HV is supported by NSF grant PHY-0756966.

Appendix A: Some calculations

b

0U †ABU |0b =

0b

U † 0a|R†AR|0aBU |0b (69)

=0b

a0|U †R†ABRU |0a|0b (70)

= 0UAB| 0U code (71)

trρ(τ)AB

=

i ,k

ρ ik i0b

U †ABU |0b|k

(72)

=

i ,k

ρ ik i0U

AB| 0U|k= 0U

AB| 0U (73)

πnm =1

√WnWm

i

Cn | i i |C †m (74)

πns πtm =1

√WnWsWtWm

i ,k

Cn| i i |C †sCt |kk|C †

m (75)

=δst√WnWm

i

Cn | i i |C †m = δst πnm (76)

32

Acknowledgements




0b

U †ABU |0b =0b

U † 0a|R†AR|0aBU |0b (69)

=0b

a0|U †R†ABRU |0a|0b (70)


trρ(τ)AB

=

i ,k

ρ ik i0b

U †ABU |0b|k

(72)

=

i ,k

ρ ik i0U

AB| 0U|k= 0U

AB| 0U (73)

πnm =1

√WnWm

i

Cn | i i |C †m (74)

πns πtm =1

√WnWsWtWm

i ,k


m (75)

=δst√WnWm

i


32

We read off that

RU | i |0a|0b =


0 if | i /∈ Hcode

(42)



| 0U =

n

√Wn |na |n b . (43)











Π = a0|R†R|0a (44)


nR†nRn =

i ,n

1Wn








Π2= Π (45)


21

the outgoing field φout = φ. The corresponding interaction operator V(t) can be expanded

in modes Vω and V †ω, that map black hole states with mass M to states with mass M ±ω.

The matrix elements Vij(ω) = jVω

i determine the emission and absorption rates of

particles of frequency ω. Semi-classically this process is described by the creation of a

Hawking pair. Hence it is natural to conjecture that the operators V(ω) and V †(ω) have

something to do with the creating or annihilating particles behind the horizon.

In principle one can express the Cn operators in terms of theV operators as follows: one

inserts the Fock states of the radiation and commutes the annihilation operators through

the evolution operator. One obtains an expression involving products of a number of V

operators, one for every emitted particle. It is suggestive that the number of V insertions

matches the number of infalling partner modes of the Hawking pairs. This suggests that

it is natural to view the transition amplitudes Cn as creating a state with a collection of

particles behind the horizon. Our aim is to make this intuition more precise.

There exists of course a close parallel between the above discussion and the GKPW

dictionary that underlies the AdS/CFT correspondence. Indeed, we may compare the

process of falling into a black hole with entering the near-horizon geometry of a stack of

D-branes. In this comparison, the AdS space replaces the black hole interior geometry, the

CFT represents the interior Hilbert space, the value of the scalar field φ at the stretched

horizon plays the role of the asymptotic boundary value of the bulk field in AdS, and the

V operators represent the CFT operator dual to φ.

2.4 Old and Young Black Holes Revisited

As a first application of our formal set up, let us revisit the time evolution of the young

and old black hole states (6) and (8).

First consider the young black hole. At time τ it has evolved into a state of the form

ρyoungAB

= U i

| 0

b

0 i

U † (26)

which we may expand in terms of the radiation basis as in eqn (12), with

ρyoungnm

= Cn

i

iC †

m (27)

We now make the assumption that the density matrix on the radiation states takes the

15

2 Setting the Stage








subsequent study.





to the


i∈ HM (1)




Hawking entropy


with SBH



state i






H = HBH

⊗HR (3)

5

R

form depends on unknown details of Planck scale physics. It looks like cheating to assume

that Alice can access these observables, without actually having such detailed knowledge.

The situation is of course similar to other holographic dualities, like AdS/CFT. From the

dual perspective, the interior space-time emerges in a mysterious way. But for Alice, the

smooth local space-time is a classical reality, relative to which the operators (52) behave

like ordinary observables. So she has no trouble using them. Note, however, that Alice

does not organize the modes in terms of their Schwarschild energy ω, but reassembles them

into local Minkowski modes, that make no reference to the precise location of the horizon.

To make this a bit more concrete, consider the two point function of local field operators,

say one behind and one in front of the horizon. The general result (58) gives that

trρφa(y)φb(x)

= 0U |φa(y)φb(x)| 0U (61)

where the subscript indicates that φa is made up from a-modes, and φb from b-modes.

What is the meaning of this expectation value? Strictly speaking, it does not yet describe

what we want. In our discussion so far, we have only included the outgoing modes that

are emitted by the black hole. The ancillary a modes play their guest role as the Hawking

partner of the outgoing b modes: they initially live on the left region in fig 1, but can be

analytically continued to live just behind the future horizon. To really get the complete

set of modes behind the horizon, we would need to extend our framework a bit, by adding

the infalling modes. Since the horizon state looks smooth, the infalling modes can be

analytically continued to the region behind the horizon. So we have indeed explained how

a QFT mode, and by extension any object, can fall into a black hole.

Note that we can also try to go beyond the past horizon, by switching time direction.

The outgoing modes behind the past horizon are obtained by analytic continuation of

outgoing modes on the right region in fig 1. The time-flipped Hawking partners of the

incoming modes live behind the past horizon, and are obtain by analytic continuation of

modes on the left region in fig 1. To define them, we would need to introduce transition

amplitudes for the ingoing modes involved in the black hole formation process.

3.3 Ancilla versus Black Hole Final State

Our use of the ancillary Hilbert space Ha is somewhat reminiscent of the black hole final

state proposal of [2]. However, there is a key difference: unlike the final state proposal, our

procedure is completely in accord with the conventional rules of quantum mechanics. We

have stated explicitly in what Hilbert space we are working, and time evolution is unitary.

Our internal operators involve an expectation value inside an auxilary Hilbert space Ha,

26

U

horizon

trρφa(y)φb(x)

=

0U

φa(y)φb(x) 0U

,

U+|

R+|

A B 0U=

n

√Wn|na|nb . (79)

Acknowledgements:

References

[1]

[2]

34

2 Setting the Stage








subsequent study.





to the


i∈ HM (1)




Hawking entropy


with SBH



state i






H = HBH

⊗HR (3)

5

We read off that

RU | i |0a|0b =


0 if | i /∈ Hcode

(42)



| 0U =

n

√Wn |na |n b . (43)











Π = a0|R†R|0a (44)


nR†nRn =

i ,n

1Wn








Π2= Π (45)


21

Acknowledgements




b

0U †ABU |0b =

0b

U † 0a|R†AR|0aBU |0b (69)

=0b

a0|U †R†ABRU |0a|0b (70)


trρ(τ)AB

=

i ,k

ρ ik i0b

U †ABU |0b|k

(72)

=

i ,k

ρ ik i0U

AB| 0U|k= 0U

AB| 0U (73)

πnm =1

√WnWm

i

Cn | i i |C †m (74)

πns πtm =1

√WnWsWtWm

i ,k


m (75)

=δst√WnWm

i


32

Acknowledgements




0b

U †ABU |0b =0b

U † 0a|R†AR|0aBU |0b (69)

=0b

a0|U †R†ABRU |0a|0b (70)


trρ(τ)AB

=

i ,k

ρ ik i0b

U †ABU |0b|k

(72)

=

i ,k

ρ ik i0U

AB| 0U|k= 0U

AB| 0U (73)

πnm =1

√WnWm

i

Cn | i i |C †m (74)

πns πtm =1

√WnWsWtWm

i ,k


m (75)

=δst√WnWm

i


32

We read off that

RU | i |0a|0b =


0 if | i /∈ Hcode

(42)



| 0U =

n

√Wn |na |n b . (43)











Π = a0|R†R|0a (44)


nR†nRn =

i ,n

1Wn








Π2= Π (45)


21

the outgoing field φout = φ. The corresponding interaction operator V(t) can be expanded

in modes Vω and V †ω, that map black hole states with mass M to states with mass M ±ω.

The matrix elements Vij(ω) = jVω

i determine the emission and absorption rates of

particles of frequency ω. Semi-classically this process is described by the creation of a

Hawking pair. Hence it is natural to conjecture that the operators V(ω) and V †(ω) have

something to do with the creating or annihilating particles behind the horizon.

In principle one can express the Cn operators in terms of theV operators as follows: one

inserts the Fock states of the radiation and commutes the annihilation operators through

the evolution operator. One obtains an expression involving products of a number of V

operators, one for every emitted particle. It is suggestive that the number of V insertions

matches the number of infalling partner modes of the Hawking pairs. This suggests that

it is natural to view the transition amplitudes Cn as creating a state with a collection of

particles behind the horizon. Our aim is to make this intuition more precise.

There exists of course a close parallel between the above discussion and the GKPW

dictionary that underlies the AdS/CFT correspondence. Indeed, we may compare the

process of falling into a black hole with entering the near-horizon geometry of a stack of

D-branes. In this comparison, the AdS space replaces the black hole interior geometry, the

CFT represents the interior Hilbert space, the value of the scalar field φ at the stretched

horizon plays the role of the asymptotic boundary value of the bulk field in AdS, and the

V operators represent the CFT operator dual to φ.

2.4 Old and Young Black Holes Revisited

As a first application of our formal set up, let us revisit the time evolution of the young

and old black hole states (6) and (8).

First consider the young black hole. At time τ it has evolved into a state of the form

ρyoungAB

= U i

| 0

b

0 i

U † (26)

which we may expand in terms of the radiation basis as in eqn (12), with

ρyoungnm

= Cn

i

iC †

m (27)

We now make the assumption that the density matrix on the radiation states takes the

15

2 Setting the Stage








subsequent study.





to the


i∈ HM (1)




Hawking entropy


with SBH



state i






H = HBH

⊗HR (3)

5

R

form depends on unknown details of Planck scale physics. It looks like cheating to assume

that Alice can access these observables, without actually having such detailed knowledge.

The situation is of course similar to other holographic dualities, like AdS/CFT. From the

dual perspective, the interior space-time emerges in a mysterious way. But for Alice, the

smooth local space-time is a classical reality, relative to which the operators (52) behave

like ordinary observables. So she has no trouble using them. Note, however, that Alice

does not organize the modes in terms of their Schwarschild energy ω, but reassembles them

into local Minkowski modes, that make no reference to the precise location of the horizon.

To make this a bit more concrete, consider the two point function of local field operators,

say one behind and one in front of the horizon. The general result (58) gives that

trρφa(y)φb(x)

= 0U |φa(y)φb(x)| 0U (61)

where the subscript indicates that φa is made up from a-modes, and φb from b-modes.

What is the meaning of this expectation value? Strictly speaking, it does not yet describe

what we want. In our discussion so far, we have only included the outgoing modes that

are emitted by the black hole. The ancillary a modes play their guest role as the Hawking

partner of the outgoing b modes: they initially live on the left region in fig 1, but can be

analytically continued to live just behind the future horizon. To really get the complete

set of modes behind the horizon, we would need to extend our framework a bit, by adding

the infalling modes. Since the horizon state looks smooth, the infalling modes can be

analytically continued to the region behind the horizon. So we have indeed explained how

a QFT mode, and by extension any object, can fall into a black hole.

Note that we can also try to go beyond the past horizon, by switching time direction.

The outgoing modes behind the past horizon are obtained by analytic continuation of

outgoing modes on the right region in fig 1. The time-flipped Hawking partners of the

incoming modes live behind the past horizon, and are obtain by analytic continuation of

modes on the left region in fig 1. To define them, we would need to introduce transition

amplitudes for the ingoing modes involved in the black hole formation process.

3.3 Ancilla versus Black Hole Final State

Our use of the ancillary Hilbert space Ha is somewhat reminiscent of the black hole final

state proposal of [2]. However, there is a key difference: unlike the final state proposal, our

procedure is completely in accord with the conventional rules of quantum mechanics. We

have stated explicitly in what Hilbert space we are working, and time evolution is unitary.

Our internal operators involve an expectation value inside an auxilary Hilbert space Ha,

26

U

horizon

trρφa(y)φb(x)

=

0U

φa(y)φb(x) 0U

,


ΠU i

0b= U

i0

b+ error.

error = EU i

0b, with E =

n

Ncode

NnM−En.

Error of the Recovery Operation:

Reconstruction breaksdown for maximally

mixed states!

Firewall ParadoxAMPS

ΠU i

0b= U

i0

b+ error.

error = EU i

0b, with E =

n

Ncode

NnM−En.


ΠU i

0b= U

i0

b+ error.

error = EU i

0b, with E =

n

Ncode

NnM−En.

Error of the Recovery Operation:

Reconstruction breaksdown for maximally

mixed states!

Firewall ParadoxAMPS

ΠU i

0b= U

i0

b+ error.

error = EU i

0b, with E =

n

Ncode

NnM−En.

Reconstruction worksfor all states |Psi) that fit inside a code space!

?


Decoherence

In practice this means that not all of the quantum information in the early state can be

extracted in real time, since it is unsuitable for delicate quantum measurements involving

interference experiments, etc. In other words, in it is unavoidable that part of the informa-

tion contained decoheres and becomes, as far as the infalling observer is concerned, classical

information. An essential difference between classical and quantum information is that the

latter is delicate and exclusive – it can not be cloned, and while it can be highly non-local,

it can not be simultaneously accessed at different location first can be duplicated – once

it is known in some location it can be broadcast throughout space and become knowledge

that is shared by different semi-classical observers.

OBH

=

c

O(c)BH

⊗Ωc

Ωc

(10)

6

Solving the problem:

Does this preserve linearity and locality?


Using QECC, we’ve given a general construction of localQFT operators relative to which the horizon is smooth.*

Generic BH states describe a semi-classical space-time with an interior geometry and a smooth event horizon. *

Some conclusions:

Maximally mixed BH state = sum of semi-classical states Einstein locality + general logic => old BHs have no firewall*

Formulate of the rules of observer complementarity

Some outstanding challenges:

Trace the flow of quantum information and entanglement

Quantify violations of locality and limitations of effective QFT.


Hawking pair

horizon

b

a

r

e


bFIREWALL

ea

a

rb



Code bits

Measurement:entangles qubits with

environment, produces classical information. r

a

b

e

horizon



ba

Code bits

horizon

Measurement:enables the quantum

teleportation of Hawkingquanta back into the black hole interior

assists QT protocol

r

a

b

e

Tuesday, January 22, 2013


Via the Einstein equations, this will result in correspondingly large quantum fluctu-ations in the local background geometry on the Cauchy surface Σ. For a given cut-offand Cauchy surface Σ, one can in principle calculate these by integrating the cumulativegravitational effect of the local stress-energy fluctuations. It is clear that when thesegeometry fluctuations become too large, relative to the cut-off scale, the semi-classical de-scription of the Hilbert space as the space of states on a given Cauchy slice breaks down.We will call a cut-off violating this bound super-critical, and we will call it sub-critical orsemi-classical if the fluctuations of the geometry are controlled in this sense. It is clearfrom the above discussion that semi-classical cut-offs have a minimal size.

The question arises, however, how such a semi-classical truncation of the Hilbert spacemust be interpreted at a fundamental level? On the one hand, any short distance cut-offwould appear to constitute an unacceptable violation of Lorentz invariance (= frame-independence), since different observers will in general be inclined to truncate the Hilbertspace in different ways. On the other hand, there is no correspondence principle that tellsus that the Hilbert space must extend into this super-critical regime. On the contrary: ifwe would assume it does extend into the super-critical regime, we would need to explainwhy there are no large space-time fluctuations at scales much larger than the Planck scale.

Thus it seems we are faced with a dilemma: apparently we must either give up strictLorentz invariance, or find a way to deal with a Hilbert space supported on field config-urations with arbitrarily high frequencies. A way out of this dilemma is provided by theblack hole complementarity principle proposed in [3, 6, 5].

We now propose a new formulation of this principle, which we name space-time com-plementarity, because its formulation and possible consequences are in principle not re-stricted to the black hole context. The spirit in which this principle is meant is as aproposal for a reasonable effective description of some underlying, consistent theory ofquantum gravity, such as the one provided by string theory.† Clearly, however, a muchmore detailed knowledge of this underlying fundamental theory is necessary to verify andquantify this effective description.

Space-time complementarity: A variable cut-off scale (x) on a Cauchy surface

Σ provides a permissible semi-classical description of the second quantized Hilbert

space, only when the quantum fluctuations of the local background geometry induced

by the corresponding stress-energy fluctuations do not exceed the cut-off scale itself.

All critical cut-off scales that saturate this requirement provide complete, comple-

mentary descriptions of the Hilbert space.

The key step here is that, while different observers may under certain circumstances be

† L. Susskind has also advocated that a complementarity principle of the type formulatedin this section may be realized in string theory. There indeed exist several indications that,compared to local field theory, string theory has drastically fewer degrees of freedom at shortdistances.

5

Space-Time Complementarity:

Kiem, EV & HV, 1995

Via the Einstein equations, this will result in correspondingly large quantum fluctu-ations in the local background geometry on the Cauchy surface Σ. For a given cut-offand Cauchy surface Σ, one can in principle calculate these by integrating the cumulativegravitational effect of the local stress-energy fluctuations. It is clear that when thesegeometry fluctuations become too large, relative to the cut-off scale, the semi-classical de-scription of the Hilbert space as the space of states on a given Cauchy slice breaks down.We will call a cut-off violating this bound super-critical, and we will call it sub-critical orsemi-classical if the fluctuations of the geometry are controlled in this sense. It is clearfrom the above discussion that semi-classical cut-offs have a minimal size.

The question arises, however, how such a semi-classical truncation of the Hilbert spacemust be interpreted at a fundamental level? On the one hand, any short distance cut-offwould appear to constitute an unacceptable violation of Lorentz invariance (= frame-independence), since different observers will in general be inclined to truncate the Hilbertspace in different ways. On the other hand, there is no correspondence principle that tellsus that the Hilbert space must extend into this super-critical regime. On the contrary: ifwe would assume it does extend into the super-critical regime, we would need to explainwhy there are no large space-time fluctuations at scales much larger than the Planck scale.

Thus it seems we are faced with a dilemma: apparently we must either give up strictLorentz invariance, or find a way to deal with a Hilbert space supported on field config-urations with arbitrarily high frequencies. A way out of this dilemma is provided by theblack hole complementarity principle proposed in [3, 6, 5].

We now propose a new formulation of this principle, which we name space-time com-plementarity, because its formulation and possible consequences are in principle not re-stricted to the black hole context. The spirit in which this principle is meant is as aproposal for a reasonable effective description of some underlying, consistent theory ofquantum gravity, such as the one provided by string theory.† Clearly, however, a muchmore detailed knowledge of this underlying fundamental theory is necessary to verify andquantify this effective description.

Space-time complementarity: A variable cut-off scale (x) on a Cauchy surface

Σ provides a permissible semi-classical description of the second quantized Hilbert

space, only when the quantum fluctuations of the local background geometry induced

by the corresponding stress-energy fluctuations do not exceed the cut-off scale itself.

All critical cut-off scales that saturate this requirement provide complete, comple-

mentary descriptions of the Hilbert space.

The key step here is that, while different observers may under certain circumstances be

† L. Susskind has also advocated that a complementarity principle of the type formulatedin this section may be realized in string theory. There indeed exist several indications that,compared to local field theory, string theory has drastically fewer degrees of freedom at shortdistances.

5


inclined to use very di!erent cut-o!s, and thus very di!erent bases of observables for doingmeasurements, the Hilbert spaces spanned by these di!erent bases is assumed to be thesame! The consequences of this assumption are particularly striking in a situation withtwo di!erent observers whose reference frames are related by very large red- or blue-shifts,such as the infalling and outside observer on a black hole background.

To illustrate this, let us consider the simultaneous measurement of an outside observ-able O!out

, supported on free field configurations of typical wavelength !out, and an insideobservable O!in

near or just behind the horizon, of typical wavelength !in. (See fig. 1b.)One would expect that such a simultaneous measurement should indeed be possible, sincethe two observables on "final are space-like separated.

! out"

2e

out"

in" cm

in"

t

/4Mt#E ~

#

HorizonEvent

final

Fig 1b. This figure shows two space-like separated observables defined on !final of typical wave-

length !in and !out. The field modes associated with these observables have collided in the past

with a center of mass energy that grows as exp("t/8M). The proposed complementarity prin-

ciple states that observables for which this collision energy exceeds some (possibly macroscopic)

critical value do not simultaneously exist as mutually commuting operators.

To accurately compute transition amplitudes involving these operators, however, wemust consider their past history. In first instance, we can try to compute this pasthistory using the free field propagation of the modes contained in these observables.We then discover that the past history of these observables in fact contains an ultra-high energy collision very close to the horizon, with a center of mass energy that growsexponentially in the out-going time! While it is true that this interaction takes placebetween virtual instead of real particle excitations, the absurd magnitude of the collisionenergy nevertheless indicates that the classical geometry is no longer the appropriatesetting for considering the simultaneous measurement by these two observables.

6

inclined to use very di!erent cut-o!s, and thus very di!erent bases of observables for doingmeasurements, the Hilbert spaces spanned by these di!erent bases is assumed to be thesame! The consequences of this assumption are particularly striking in a situation withtwo di!erent observers whose reference frames are related by very large red- or blue-shifts,such as the infalling and outside observer on a black hole background.

To illustrate this, let us consider the simultaneous measurement of an outside observ-able O!out

, supported on free field configurations of typical wavelength !out, and an insideobservable O!in

near or just behind the horizon, of typical wavelength !in. (See fig. 1b.)One would expect that such a simultaneous measurement should indeed be possible, sincethe two observables on "final are space-like separated.

! out"

2e

out"

in" cm

in"

t

/4Mt#E ~

#

HorizonEvent

final

Fig 1b. This figure shows two space-like separated observables defined on !final of typical wave-

length !in and !out. The field modes associated with these observables have collided in the past

with a center of mass energy that grows as exp("t/8M). The proposed complementarity prin-

ciple states that observables for which this collision energy exceeds some (possibly macroscopic)

critical value do not simultaneously exist as mutually commuting operators.

To accurately compute transition amplitudes involving these operators, however, wemust consider their past history. In first instance, we can try to compute this pasthistory using the free field propagation of the modes contained in these observables.We then discover that the past history of these observables in fact contains an ultra-high energy collision very close to the horizon, with a center of mass energy that growsexponentially in the out-going time! While it is true that this interaction takes placebetween virtual instead of real particle excitations, the absurd magnitude of the collisionenergy nevertheless indicates that the classical geometry is no longer the appropriatesetting for considering the simultaneous measurement by these two observables.

6


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