Gerard ’t HooftBerlin

November 1, 2007

Utrecht University

of the

How do we reconcile these with LOCALITY?

paradox

Black Holes require new axioms for thequantization of gravity

Unitarity,Causality, ...

paradox

Black Hole Quantum Coherence is realized in String/Membrane Theories !

-- at the expense of locality? -- How does Nature process information ?

The physical description of the difficulty ...

horizon

Here, gravitational interactions become

strong !!

brick wall

interaction

horizon

b

By taking back reaction into account, one can obtain a unitary scattering matrix

Gravitational effect from ingoing objects

particlesout

in

2 2

The coordinate shift can be calculated

to be :

which obeys :

4 log

8 p ( )

x G p x

x G x

in in in, p

2out out outexp ( ) ( )i d x P x y x

2

2 2

( ) d ' ( ') ( ')

( ') ( ') 8 1

y x x G x x p x

G x x x x G

out

2 2out outexp d d ' ( ) ( ') ( ')i x x P x G x x p x

Strategy: infinitesimal modification of in-state

out

2 2out outexp d d ' ( ) ( ') ( ')i x x P x G x x p x

in in

2 2out out in

( ')

exp d d ' ( ) ( ') ( ')

P x

i x x P x G x x P x

a string theory amplitude !!nearly

Repeat procedure:

D Dout in

2

( ) ( )

exp d

X x X x

x i X X i X P i X P

The string world-sheet

Black Hole Formation & Evaporation by Closed Strings

Horizon Algebra

N

N

2 2out in

2in in

d d ' ( ) ( ') ( ')

out in

d ( ) ( )

in in

2in in

2in

so that

therefore

( ) ( ')

( ) ( ') ,

( ), ( ') ( ') ,

( ) ' (

i x x P x G x x P x

i x P x U x

def

P x P x e

U x P x e

P x U x i x x

U x d x G x x

out

2out in

2 2out in

2 2out in in out

') ( ')

( ) ' ( ') ( ')

( ), ( ') ( ') ; ( ')

( ) ( ) ; ( ) ( )

P x

U x d x G x x P x

U x U x iG x x G x x

U x P x U x P x

BLACK HOLE WHITE HOLE

A black hole is a quantum superposition ofwhite holes and vice versa !!

The Difference between

N

Pl

0G

M

®

® ¥

very early

very late

in

out

0z t= =

S-matrix Ansatz:Qu. Gr. gives us aboundary conditionhere

ìïïïïíïïïïî

But, our algebra does not generate the area law. Canwe be more precise? Transverse gravity?

Now, let us look at the contributionsfrom the Standard Model Scale

It all happensat the originof RindlerSpace

S

In Standard Model

units, the -matrix

is generated at the

origin.

out in

2-d surface

The Standard Model Contribution to theHorizon Algebra. I. The U (1) field

in

2in

2right left in

Ingoing charged particles :

Outgoing charged particles undergo thi

( ) ( ) ( ) , 0

( ) ( ) d ' ( ') ( ') , 0

( ) ( ) ,

( ) ( ) d ' ( ') ( ')

J x x x J J

A x x x G x x x A A

A x x

x x x G x x x

2 2

out ind d ' ( ) ( ') ( ')

s gauge rotation :

out outi x x x G x x x

e

2 2out in

2in in

d d ' ( ) ( ') ( ')

d ( ) ( )

in in

2

In wave functions

the operators and obey commutation rules

, so, we

out out

( ) ( ) ,

( ) ( )

( ) ( ') ( ') ,

i x x x G x x x

i x x x

e

x x e

x x

x x i x x

2in out

2out in

in out

2 2in out

have

( ) d ' ( ') ( ')

( ) d ' ( ') ( ')

( ), ( ') ( ')

( ), ( ') ( ')

x x G x x x

x x G x x x

x x iG x x

x x i x x

L scalar 21

2

in out

let ,

then ( ,0,0) ( ,0,0)

this implies

and commutes with everything.

( ) is a fiesp o luri n d

( ) ( )

i i

i

i

i

i i

x x x x

x x

x

D V

The role of the Standard Model’s scalar fields

Scalar field acts as quantum hair

It generates a modification in the vector fieldequations:

L

2

in

2

* *14

2 2 *

Now, the vector field obeys

( ) ( ) d ' ( ') ( ') with

( ) ( ) ( ') ( ')

( ) ( , )

( )

2

i i

A x

A x x x G x x x

x x G x x x xe

F F D D V

( ) *( ')

( ) ( ') *( ") *( '")

i j

i j k

x x

x x x x

are as their(Euclidean)quantum exp.values

The average values of the scalar fields:

Questions:- What is the effect of Standard Model fermions ?- How do we handle non-Abelian vectors ?- What is the effect of the instanton angle ?- What modifications of the algebra are generated by the transverse grav. force ?- Does this allow for a representation of the algebra with discrete states, as suggested by the entropy – area law?

A related question that we can answer:What is the effect of magnetic monopoles ?

in, out in, out

2

1 out in

1 out in

1 out in

out

( ) , ( )

( ) ( )

( ), ( ') (2 ) log '

( ), ( ') (2 ) '

( ), ( ') (2 ) '

( ),

arg

arg

E M

E Mi i in ij j in

E E

E M

M E

M

x x

F x

x x i x x

x x i x x

x x i x x

x 1in( ') (2 ) log 'M x i x x

arg has a Dirac string - ambiguity 2 n

NS

[ , ]

[ , ]

E M

But if we consider exp and exp

then use

(if [ , ] is a c-number)

is unambiguous if

2 Dirac condition !

and must be quantized:

E M

iA iB iB iA A B

A B

ie im

e e e e e

A B

e

e m n

E 2 ( )

M 2 ( )

( ) ( )

( ) ( )

i

i

j

j

x x x

x x x

e

m

2d ( ) ( )

( ) ( )

i x x x

x x

e

A non-Abelian gauge theory may now be treated in its Cartan subalgebra,where the diagonal components of electric and magnetic charges arewell-defined.

Particles and horizons, the hybrid picture

We intend to obtain the complete algebra relating the in-operators to the out-operators from whatever Standard Model Lagrangian.

Once the rules are clear, we should be able once again to add (transversal) gravity. The representation of our algebra should then respect the entropy / area law.

The algebra should generate the details of quantum black hole dynamics.

“The black Hole Horizon as aDynamical System”,Int.J.Mod.Phys. D15 (2006) 1587 earlier version: gr-qc/0504120