Gerard ’t HooftGerard ’t Hooft

Erice Opening Lecture

August 2008

Utrecht University

Remembering

June 30 - July 6, 1971: Amsterdam International EPS Conference.

This was the first occasion for me to present my theories about the renormalizabilityof non-Abelian gauge theories with spontaneous symmetry breakdown. My then

advisor, Martinus Veltman, had given me the opportunity to explain this in 10 minutesto the audience. “Let me introduce you to two American gangsters”, he said to me. I was puzzled by the remark at that time, but later realized that, in Veltman’s view,

anybody who ignored his work, and did physics differently, quickly turned into a gangster.Anyway, talking to these two people, Shelly Glashow and Sidney Coleman, made me realize that these must be the smartest gangsters on the planet. Sidney immediately

saw the most essential ingredients of the theory. He later explained that Quantum FieldTheory had been an ugly monster, until it received a kiss by our work, at which it turned

into a beautiful and wealthy prince.

July 11 - August 1, 1975: Erice“The Whys of Subnuclear Physics”

I met Sidney many times since, and I vividly remember the lectures he gave in the 1975Erice School. At that school, not only the ‘best student’ and ‘best secretary’ were chosen,but also ‘best lecturer’. This was Sidney Coleman, who, at the farewell party, also cameas the ‘most fancy dressed’ participant, in his fluorescent pink jacket. His lectures hadbeen about topology in QFT. Previously, when renormalizing the theory, we had always

limited ourselves to perturbation expansion, where the topology is trivial.

Sidney would illustrate vividly what topology is, using the cord of his microphone, by which he was connected to the loudspeaker system. Winding the cord around

his neck he explained what winding number was. The students actually worried thathe might strangle himself, so they never forgot the definition of winding number.

October 1988, HarvardDuring one of my later encounters at Harvard, I visited him at his home, with my two

little children. There, we would play the game of “Charades”;he was very good at it. My children loved it.

As emphasized by others here, Sidney was a great story teller:

One of his stories was about how he rescued mankind from a pending planetary disaster,

while sitting at a bar, with his friend Carl Sagan. With Sagan, Sidney shared his love forscience-fiction. Behind a good glass of whiskey, or two, Sagan showed a problem to

Sidney: “Suppose that there exist alien, totally unknown, life forms on Mars, or, for thatmatter, on the Moon as well. The Moon would soon be visited by astronauts.

What should space agencies do to avoid the danger thatthese life forms would infect the Earth, and destroy mankind? How do we avoidcontamination of one planet by another in general, when space flight becomes

routine? This became a vivid discussion.

But a month later, Sidney was highly surprised when a manuscript arrived in

his mail: Spacecraft Sterilization Standards and Contamination of Mars

by S. Coleman and C. Sagan, Journal of Astronautics and Aeronautics 3, 22 (1965).

The norm for the possibility of cross contamination from one celestial body to another should be less than 0.1%. Further,

astronauts should be put in quarantine, and the chances of survival for every individual organism of a life form from one planet to the next should be less than

one in 10,000.

Now, this was the only existing publication in this field, so, NASA had to followthe advice.

Committee of Space Research COSPARdecided that astronauts returning from the Moon would have to go into quarantine for

a certain amount of time.

→ the Lunar Receiving Laboratory (LRL) in Houston, Texas Dangerous life forms were fortunately never found.

Gerard ’t HooftGerard ’t Hooft

Erice Interlude

August 2008

Utrecht University

Some effects of

in QCD

boundary ofUniverse

3(2)SU S

x

x

x

x

x

x

x

x

x

x

x

x

xx

x

We can map the gauge transformations on theboundary of space-time, but this mapping cannot

continuously be extended to the interior

Applying this gauge transformation to the vacuum configuration, gives a vacuum at infinity, but continuity

in the fields demands a region in space-time thatdiffers from the vacuum: the instanton.

instanton

Instantons can exist in Minkowski space-time as well

as in Euclidean space-time, since they are topological.Instantons are time-dependent, therefore fields

such asfermion (quark) fields, can have modes where

theenergy before and after an instanton is different.

time0

+

_

Energy

This fermion cameout of thevacuum !

There is one mode (for each chiral flavor), that hops

from the antiparticle sea to the particle states:

time

Now, consider Euclidean timet i

| | | |i E t Ee e

EiEte e

0igA

Therefore, this effect happens if and only if thereis a bounded solution of the Dirac equation inEuclidean space:

instanton

Under parity, instanton goes to anti-instanton, andleft goes to right. Therefore left-helicity fermionsdo the opposite of right helicity fermions:

If a left-helicity appears, then a right-helicity disappears,and vice versa.

One instanton causes one flip left → right for everyflavor:

up

down

strange

RL

L

R

LR

Effective instanton action for fermions:

inst

,

13!

det( ) . .R La p

a p

R R R L L Labc pqr a b c p q r

h c

L Note: alsosinglet in color

space

In QCD, this has important effects for― pseudoscalar mesons

― scalar mesons

Pseudoscalars:

s

L Rsm s s

L

R

u

u R

L

d

d

Exactly the quantum numbers of theη mass term; it violates chiral symmetry because

left ↔ right

Scalars: why does the scalar meson appear to be lighter than the non-strangemesons? Is it a “tetra-quark”?

s s

No!But it mixes with the tetraquark:

s

s

u

ud

d

s s d d

uu d d

s s

uu

GtH, Isidori, Maiani, Polosa and Riquer, arXiv:0801.2288, To appear in Phys Lett B.

Utrecht University

Gerard ‘t HooftErice LecturesAugust 2008

What will space-time look like at scalesmuch smaller than the Planck length?No physical degrees of freedom anymore

Gravity may become topological

It may well make sense to describe space-timethere as if made of locally flat pieces gluedtogether (as in “dynamical triangulation” or “Regge calculus”)

The dynamical degrees of freedom are then line-like defectsThe dynamical degrees of freedom are thenpointlike defects[??]

This theory will have a clear vacuum state:flat Minkowski space-time

“Matter fields” are identified with the defects.

There are no gravitons: all curvature comes fromthe defects, therefore:

Gravity = matter. Furthermore:

The vacuum has 0

What are the dynamical rules? What is the “matter Lagrangian”?

First do this in

A gravitating particle in 2 + 1 dimensions:

A moving particle in 2 + 1 dimensions:

xx

xx

A′A

A′

A

xx v t

'

'

x x

t t

A many-particle universe in 2+1 dimensions can now be constructed by tesselation

(Staruszkiewicz, G.’t H, Deser, Jackiw, Kadar)

There is no localcurvature; the onlyphysical variablesare N particle coordinates, withN finite.

2 + 1 dimensional cosmology is finite and interesting

BANG

CRUNCH

Quantization is difficult.

How to generalize this to 3 +1 (or more) dimensions ?

straight strings are linelike grav. defects

0

0

Calculation deficit angle:

2

33 00

2

2

11 22

21

( )

( ) ( )

( ) ,

8

T t x

t t

x r

R r x

r r G

Deficit angle →22 (1 cos ) 8 G

In units such that :11 22 1R R

A static universe would contain a large number of such strings

But what happens when we makethem move ?

One might have thought:

But this cannot be right !

New s

tring

??

Holonomies on curves around strings: Q : member of Poincaré group:

Lorentz trf. plus translationC

(3,1) (2, )SO SL C

Static string: pure rotation,

Lorentz boost:

Moving string:

(2)Q U SU †Q Q V

1Q V U V

Tr real , Tr 2U U

Tr real , Tr 2Q Q All strings must have holonomies that are constrained by

C1Q1

2Q

1Q1

1 2Q Q 12Q

12 1Q Q

1 1 1 1 12 1 1 2 1 2 1 2( )C Q Q QQ Q Q QQ

In general, Tr (C) = a + ib can be anything. Only if the angle is exactly 90° can the newly formed object be a string.What if the angle is not 90°?

12 1

2 12

Can this happen ?

a

c

1 11 2 1 2 a cC Q Q QQ Q Q

This is a delicate exercise in mathematical physicsAnswer: sometimes yes, but sometimes no !

bd

1Q 2Q

4Q 3Q

aQ

bQcQ

dQ

1 1 1 13 1 2 1 4 3 1 3

13

2 3 4

2 1

1 ,,

, .,b c b d aa

Q QQ Q Q Q Q Q

Q Q Q Q Q

QQ Q

Q Q Q

Q

Q

1

Notation and conventions:

1Q 2Q

4Q 3Q

aQ

bQcQdQ

The Lorentz group elements have 6degrees of freedom. Im(Q) = 0 is one constraint for each of the 4 internal lines →

We have a 2 dimensional space ofsolutions.

Strategy: write in SL(2,C) 1 1 2 2

3 3 4 4a

a ib a ibQ

a ib a ib

Im(Tr(Q)) = 0 are four linear constraints on the coefficients a, b .

|Re(Tr(Q))| < 2 gives a 4 - dimensional hypercube.

Two quadratic conditions remain: Re (det( )) = 1Im (det( )) = 0

aQaQ

In some cases, the overlap is found to be empty !- relativistic velocities- sharp angles,- and others ...

In those cases, one might expect

1

2

N

where we expect free parameters.

We were unable to verify wether such solutions always exist

6( 1) (3 2) 3 4N N N

Assuming that some solutions always exist, we arriveat a dynamical, Lorentz-invariant model.

1. When two strings collide, new string segments form.2. When a string segment shrinks to zero length, a similar rule will give newly formed string segments.3. The choice of a solution out of a 2- or more dimensional manyfold, constitutes the

matter equations of motion.4. However, there are no independent gravity degrees of freedom. “Gravitons” are composed of matter.5. it has not been checked whether the string constants can always be chosen positive. This is probably not the case. hence there may be negative energy.

6. The model cannot be quantized in the usualfashion: a) because the matter - strings tend to generate a continuous spectrum of string constants; b) because there is no time reversal (or PCT) symmetry; c) because it appears that strings break up, but do not often rejoin.

7. However, there may be interesting ways to arrive at more interesting schemes. For instance:

crystalline gravity

❖ Replace 4d space-time by a discrete, rectangular lattice:

This means that the Poincaré group is replaced byone of its discrete subgroups

Actually, there are infinitely many discrete subgroups of thePoincaré group

One discrete subgroup: (3,1, )SO

Compose SO(3) rotations over 90° and powers of

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 2

B

But, there are (probably infinitely) many more scarcediscrete subgroups. Take in SL(2,C)

with

and

are rotated by 90° along z and y axis.Then, generically,

diverge as stronger and stronger boosts.

xB

, , ,

1 ; | |, | |, | |, | | 1

,y zB B

1 2 3 4...n n n nx y z xB B B B

❖ add defect lines :

A

A′

identify

❖ postulate how the defect lines evolve:

1

2

N

Besides defect angles, surplus angles will probably beinevitable.

❖ quantize... ❖ pre - quantize...

Define the basis of a Hilbert space as spanned byontological states / equivalence classes ..

Diagonalize the Hamiltonian (evolution operator) tofind the energy eigenstates

... and do the renormalization group transformationsto obtain and effective field theory with gravity

At this point, this theory is still in its infancy.

Utrecht University

Gerard ’t HooftMeeting on Integrability in Gauge and String Theory

Utrecht, August 11-15, 2008

Utrecht University

Gerard ’t HooftPerimeter Institute,

1/5/08

Utrecht University

Gerard ’t HooftStueckelberg Lectures

Pescara, July 7-19, 2008