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					               infinity

Gerard ’t Hooft, Nobel     Lecture 1999
                k
  q           
              
                    k q
 d k 2
             1
 (k  m2 )((k  q)2  m2 )
  4
                            
                                
   What does
Renormalizability
   Mean ???

Understanding Small
Distance Behavior !!
The Differential Equation
         x   dx
                = velocity
         t   dt
Discretized Space   and   Time
Mass and Charge Renormalization
            -
                                
                  +


                                

      0
                      +
 Bare  Observed           Bare Observed
Charge Charge         -   Mass Mass
 Keeping the Observed Properties
         -
            Fixed
                 +

  

                       
                      

 Bare Observed          Bare Observed
Charge Charge           Mass Mass
All problems with renormalizing
          infinities can
   be resolved by considering


    The Small Distance Limit


      of our theory(ies)
The scale transformation
when particles are quantized ...




                        g´
           Scaling and Dimensions
            Examples:  theory
                         4

                                      2
          and: Electro-magnetism, e


                , e      2




               distance scale
     4              2                        2
10              10            1           10
          Negative screening:
        Yang-Mills gauge theory


                             2
                         g

            distance scale
10 4       10 2       1         102
     Chiral theories:
These are theories in which a field
       has a fixed length:




                     Field strength
                Compare large distance with
                     small distance:

  The quantum fluctuations at small distance
 in such a theory undermine its own structure.
Its small-distance behaviour is ILL-DEFINED

                                     At small distances, strong
At large distance scales, the        curvature  strong interactions
curvature is weak 
near linearity = weak interactions
Some theories have BAD short
    distance behaviour:



             


  10 2      1        102
  Spontaneous symmetry breaking
              ( left - right symmetry )
                                 At large distance
 At short distance             scales, the situation
scales, our particle           is as described here
   theory looks
      like this                   This degree of
                             freedom corresponds to
                                the Higgs particle

                                              
                                              
      Breaking Rotational Symmetry


Now THIS becomes
an essential degree       And THIS is the
    of freedom            Higgs degree of

                           
                             Freedom
    If there were no HIGGS particle in our
     theory, then the “Mexican Hat” would
              be infinitely steep, or:

                M Higgs  
      This is exactly like the situation in a
             “chiral field theory”:

                    F
                      2       2


   Such a theory is ill-defined, since its
small-distance structure runs out of control...
          How does force depend on distance ?
                                Force
Weak:



Electro-magnetic:


                       

                                                  Strong


                                             EM
                                      Weak
Strong:
            q               q     0                    x
     The Standard Model
          Generation I      Generation II   Generation III

          e    e                              
Leptons    e  e
                        R
                                
                                       R
                                                     
                                                           R
            L                  L                  L
          u u u              c c c          t t t
          d d dL             s s sL         b b b  L
Quarks
          u u u               c c c          t t t
                R                  R                  R
          d d d               s s s           b b b
                                                         Higgs
Gauge
           Z0
                W
                    
                                      g
Bosons          W                                     Graviton
CERN



SpS
 &     **

LEP
  Linear
Accelerator


    Fermilab
 linear booster
  A symmetric object can be
slightly out of equilibrium …
An asymmetric equilibrium is unnatural ...
          Running Coupling Strengths
                     *  g strong
1
      *
                               *    g Weak
0.5                            *    eElect-Magn
          *
              *
                  *
      *   *   *   *
      *   *   *   *
              3       6    9        12     15      18
  1 GeV 10 10             10       10    10     10
              Super symmetric theories
                        *  g strong
1
      *
                                 *    g Weak
0.5       *                      *    eElect-Magn
               *
                    *
      *   *    *    *
      *   *    *    *
                3       6    9        12     15      18
  1 GeV 10 10               10       10    10     10
  Super String Theory

Are strings continuous or
     are they discrete
 at tiny distance scales ?
A theory can only be successful
 if we understand completely
  how its dynamical variables
 behave at the tiniest possible
   time- and distance scales

      Otherwise, it is likely to
            explode ….
        With thanks to:

         M. Veltman (teaching)
        C.T. de Laat (animation)
my wife and the rest of my family (support)
         many other physicists


     Otherwise, it is likely to
            Royal Swedish
  and the explode ….
   Academy of Sciences

				
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