Transcript Geen diatitel

Remembering
Utrecht University
Gerard ’t Hooft
Erice
Opening Lecture
August 2008
June 30 - July 6, 1971: Amsterdam
International EPS Conference.
This was the first occasion for me to present my theories about the renormalizability
of non-Abelian gauge theories with spontaneous symmetry breakdown. My then
advisor, Martinus Veltman, had given me the opportunity to explain this in 10 minutes
to 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 Field
Theory 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 1975
Erice 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 came
as the ‘most fancy dressed’ participant, in his fluorescent pink jacket. His lectures had
been 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 that
he might strangle himself, so they never forgot the definition of winding number.
October 1988, Harvard
During 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 for
science-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 that
matter, on the Moon as well. The Moon would soon be visited by astronauts.
What should space agencies do to avoid the danger that
these life forms would infect the Earth, and destroy mankind? How do we avoid
contamination 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 follow
the advice.
Committee of Space Research COSPAR
decided 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.
Some effects of
Utrecht University
in QCD
Gerard ’t Hooft
Erice
Interlude
August 2008
x
x
We can map the gauge transformations on the
boundary of space-timex, but this mapping cannot
continuously be extended to the interior
x
x
x
SU (2)  S3
x
x
x
x
x
x Applying this gauge transformation to the vacuum
x
configuration, gives a vacuum
at infinity, but continuity x
in the fields demands a region in space-time that
differs from the vacuum: the instanton.
x
boundary of
Universe
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 as
fermion (quark) fields, can have modes where the
energy before and after an instanton is different.
There is one mode (for each chiral flavor), that hops
from the antiparticle sea to the particle states:
This fermion came
out of the
+
Energy
vacuum !
0
_
instanton
time
Now, consider Euclidean time t  i
e
e
i|E|t
e
|E|
iEt
e
 E
time

Therefore, this effect happens if and only if there
is a bounded solution of the Dirac equation in
Euclidean space:
      igA   0


Under parity, instanton goes to anti-instanton, and
left goes to right. Therefore left-helicity fermions
do 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 every
L
flavor:
up
R
L
instanton
down
R
R
L
strange
Effective instanton action for fermions:
L inst   det( aR pL )  h.c. 
a, p
1
3!
  abc pqr     
R
a
R
b
R
c
L
p
L
q
L
r
Note: also
singlet in color
space
In QCD, this has important effects for
― pseudoscalar mesons
― scalar mesons
Pseudoscalars:
L R
ms s s
uL
s
uR
dR
dL
Exactly the quantum numbers of the
η mass term; it violates chiral symmetry because
left ↔ right
Scalars:
why does the scalar meson s s
appear to be lighter than the non-strange
mesons?
Is it a “tetra-quark”?
No!
But it mixes with the tetraquark:
s
s
u
u
d
d
ss d d
uu d d
ss
uu
GtH, Isidori, Maiani, Polosa and Riquer, arXiv:0801.2288,
To appear in Phys Lett B.
Gerard ‘t Hooft
Erice Lectures
August 2008
Utrecht University
What will space-time look like at scales
much smaller than the Planck length?
No physical degrees of freedom anymore
Gravity may become topological
It may well make sense to describe space-time
there as if made of locally flat pieces glued
together (as in “dynamical triangulation” or “Regge calculus”)
The dynamical degrees of freedom are then
pointlike
defects
line-like [??]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 from
the defects, therefore:
Gravity = matter.
The vacuum has
Furthermore:
0
What are the dynamical rules?
What is the “matter Lagrangian”?
First do this in
A gravitating particle in
2 + 1 dimensions:
x
A
x
A moving particle in
2 + 1 dimensions:
A′
x  vx t
x  x'
 t t'
A
A′
A many-particle universe in 2+1 dimensions
can now be constructed by tesselation
(Staruszkiewicz, G.’t H, Deser, Jackiw, Kadar)
There is no local
curvature; the only
physical variables
are N particle
coordinates, with
N finite.
2 + 1 dimensional cosmology is finite and interesting
Quantization is difficult.
CRUNCH
BANG
How to generalize this to 3 +1 (or more) dimensions ?
 0
straight strings are linelike grav. defects
0
Calculation deficit angle:

T  t  2 ( x)
t33  t00  
 2 ( x) 
1
 2
 (  r )
R  r  2 ( x) ,
r11  r22  8 G 
In units such that R11  R22  1 :
2
Deficit angle →   2 (1  cos  )    8 G
A static universe would contain
a large number of such strings
But what happens when we make
them move ?
One might have thought:
But this cannot be right !
Holonomies on curves around strings:
Q : member of Poincaré group:
Lorentz trf. plus translation
C
SO(3,1)  SL(2, C )
Static string: pure rotation, Q  U  SU (2)
Lorentz boost:
Q  Q†  V
Moving string:
Tr U  real ,
Q  V U V 1
Tr U  2
All strings must have holonomies that are
constrained by
Tr Q  real ,
Tr Q  2
1
2
1
2
2
Q1 Q21
Q1
Q1
1
C
Q21
Q21 Q1
1
1
2
1
1
2
1
1
1
2
C  (Q Q ) QQ
 Q Q2 QQ
1
1
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°?
Q21
Can this happen ?
a
d
c
1
1
b
1
2
C  Q Q2 Q1Q  Qa Qc
This is a delicate exercise in mathematical physics
Answer: sometimes yes, but sometimes no !
Q2
Q1
Qa
Qd
Qc
Q4
Qb
Q3
Notation and conventions:
Q1Q2Q3Q4  1 ,
Qb  QaQ2 ,
Q3  Q1Q21Q11 ,
Qc  QbQ3 ,
Q4  Q31Q11Q3
1
1
Qd  QaQ
.
Q2
Q1
Qa
Qd
Q4
Qc
Qb
Q3
The Lorentz group elements have 6
degrees of freedom.
Im(Q) = 0 is one constraint for each
of the 4 internal lines →
We have a 2 dimensional space of
solutions.
Strategy: write in SL(2,C)
 a1  ib1
Qa  
 a3  ib3
a2  ib2 
a4  ib4 
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(Qa )) = 1
Im (det(Qa )) = 0
In some cases, the overlap is found to be empty !
- relativistic velocities
- sharp angles,
- and others ...
In those cases, one might expect
N
2
1
where we expect
free parameters.
6( N  1)  (3N  2)  3N  4
We were unable to verify wether such solutions
always exist
Assuming that some solutions always exist, we arrive
at 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 usual
fashion:
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 by
one of its discrete subgroups
Actually, there are infinitely many discrete subgroups of the
Poincaré group
One discrete subgroup:
SO(3,1, )
Compose SO(3) rotations over 90° and powers of
0
1
B
1

1
1
0
1
1
1
1
0
1
1
1 
1

2
But, there are (probably infinitely) many more scarce
discrete subgroups. Take in SL(2,C)

Bx  

with
and

 
, ,  ,  
    1 ; |  |, |  |, |  |, |  |
By , Bz
are rotated by 90° along z and y axis.
Then, generically,
Bxn1By n2 Bz n3 Bxn4 ...
diverge as stronger and stronger boosts.
1
❖ add defect lines :
A′
identify
A
❖ postulate how the defect lines evolve:
N
2
1
Besides defect angles, surplus angles will probably be
inevitable.
❖ quantize...
pre - quantize...
Define the basis of a Hilbert space as spanned by
ontological states / equivalence classes ..
Diagonalize the Hamiltonian (evolution operator) to
find the energy eigenstates
... and do the renormalization group transformations
to obtain and effective field theory with gravity
At this point, this theory is still in its infancy.
Utrecht University
Gerard ’t Hooft
Meeting on Integrability in Gauge and String Theory
Utrecht, August 11-15, 2008
Utrecht University
Gerard ’t Hooft
Perimeter Institute,
1/5/08
Utrecht University
Gerard ’t Hooft
Stueckelberg Lectures
Pescara, July 7-19, 2008