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