Download - A Confrontation with
Gerard ’t Hooft, Nobel Lecture 1999
infinity
q
k q
k
42 2 2 2
1d ( )(( ) )
kk m k q m
What does Renormalizability
Mean ???
Understanding SmallDistance Behavior !!
The Differential Equation
= velocityxt
dxdt
Discretized Space and Time
Continuous space and Time
Mass and Charge Renormalization
Bare Charge
BareMass
ObservedCharge
ObservedMass
0
+
-
+
-
Bare Charge
ObservedCharge
BareMass
ObservedMass
Keeping the Observed Properties Fixed
+
-
All problems with renormalizing infinities can
be resolved by considering
of our theory(ies)
The Small Distance Limit
The scale transformation
gg´
when particles are quantized ...
Scaling and Dimensions4Examples: theory
410 210 1 210
distance scale
2, e
2and: Electro-magnetism, e
Negative screening: Yang-Mills gauge theory
410 210 1 210
distance scale
2g
Chiral theories:These are theories in which a field
has a fixed length:
Field strength
Compare large distance with small distance:
At large distance scales, thecurvature is weak near linearity = weak interactions
At small distances, strongcurvature strong interactions
The quantum fluctuations at small distancein such a theory undermine its own structure.
Its small-distance behaviour is ILL-DEFINED
Some theories have BAD short distance behaviour:
210 1 210
Spontaneous symmetry breaking( left - right symmetry )
At short distancescales, our particle
theory lookslike this
At large distancescales, the situationis as described here
This degree offreedom corresponds to
the Higgs particle
Breaking Rotational Symmetry
Now THIS becomes an essential degree
of freedomAnd THIS is theHiggs degree of
Freedom
If there were no HIGGS particle in ourtheory, then the “Mexican Hat” would
be infinitely steep, or:
HiggsM This is exactly like the situation in a
“chiral field theory”:2 2F
Such a theory is ill-defined, since itssmall-distance structure runs out of control...
How does force depend on distance ?
Weak:
Strong:
Strong
EMWeak
x
Force
q q
Electro-magnetic:
0
Leptons
Quarks
Generation I Generation II Generation IIIThe Standard Model
Gauge Bosons g
us s
e
u
c
c
b b
b b
t t
0Z
dt
cs
e
u
c t
b
s
W
du t
W
e
u
c t
d
Graviton
L L L
e
u
d
cs
d
s b
dHiggs
L L L
R R R
RRR
CERN
SpS&
LEP* *
Linear Accelerator
Fermilablinear booster
A symmetric object can be slightly out of equilibrium …
An asymmetric equilibrium is unnatural ...
Running Coupling Strengths
*
**
***
*
**
***
strongg
Elect-MagneWeakg
*
*
*
310 610 910 1210 1510 18101 GeV
1
0.5
Super symmetric theories
strongg
Elect-MagneWeakg
*
*
*
*
**
*
**
*
**
***
310 610 910 1210 1510 18101 GeV
1
0.5
Are strings continuous or are they discrete
at tiny distance scales ?
Super String Theory
Otherwise, it is likely toexplode ….
A theory can only be successfulif we understand completelyhow its dynamical variables
behave at the tiniest possibletime- and distance scales
Otherwise, it is likely toexplode ….
With thanks to:
M. Veltman (teaching)C.T. de Laat (animation)
my wife and the rest of my family (support)many other physicists
and the Royal SwedishAcademy of Sciences