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Fundamental principles of particle physics
Our description of the fundamental interactions and
particles rests on two fundamental structures :
 Quantum Mechanics
 Symmetries
Symmetries
Central to our description of the fundamental forces :
Relativity - translations and Lorentz transformations
Lie symmetries -
SU (3)  SU (2)  U (1)
Copernican principle : “Your system of co-ordinates and units is nothing special”
Physics independent of system choice
Fundamental principles of particle physics
Quantum Mechanics
+
Relativity
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Quantum Field theory
Relativistic quantum field theory
Fundamental division of physicist’s world :
speed
slow
A
c
large
fast
Classical
Newton
Classical
relativity
(QM amplitude  e
t
i
o
n
S
i( )
small
Classical
Quantum
mechanics
Quantum
Field
theory
(S )
c
)
Quantum Mechanics : Quantization of dynamical system of particles
Quantum Field Theory : Application of QM to dynamical system of fields
Why fields?


No right to assume that any relativistic process can be explained by
single particle since E=mc2 allows pair creation
(Relativistic) QM has physical problems. For example it violates causality
U (t )  x | e  i ( p
Amplitude for free
propagation from x0 to x

1
2
3
/ 2 m )t
| x0   d 3 p  x | e i ( p
3
i ( p
d p e
 m 


 2 it 
Relativistic case :
2
2
/ 2 m )t
2
/ 2 m )t
| p  p | x0 
eip ( x  x0 )
3/ 2
eim ( x  x0 )
U(t)  e
2
/ 2t
-m x 2 t 2
... nonzero for all x, t
... nonzero for all x, t

Relativistic QM - The Klein Gordon equation (1926)
 (x)
Scalar particle (field) (J=0) :
E p m
2
2
2
Energy eigenvalues
2
  t 2  2  m2
E  (p2  m2 )1/ 2 ???
1927 Dirac tried to eliminate negative solutions by writing a relativistic
equation linear in E (a theory of fermions)
1934 Pauli and Weisskopf revived KG equation with E<0 solutions as E>0
solutions for particles of opposite charge (antiparticles). Unlike Dirac’s hole
theory this interpretation is applicable to bosons (integer spin) as well as to
fermions (half integer spin).
As we shall see the antiparticle states make the field theory causal
Special relativity
a   (ct , x, y, z )

Space time point

Space-time vector
not invariant under translations
(a  a)   a   a   (ct , x, y, z )
Invariant under translations …but not invariant under rotations or boosts

Einstein postulate : the real invariant distance is
 a

0 2
  a

1 2
  a

2 2
  a

3 2

3


, 0
g  a  a  a  a   a 
2
g   diag (1, 1, 1, 1)

Physics invariant under all transformations that leave all such distances invariant :
Translations and Lorentz transformations
Lorentz transformations :
3
x   x   x  x

 

 0
 



 
g x x  g x  x
 g     g
(Summation assumed)
Solutions :

0
1

 0 cos 
0
0

 0  sin 


3 rotations R
3 boosts B
 cosh 

 sinh 
 0

 0
0
0 

0 sin  
1
0 

0 cos  
Space reflection – parity P
1 0 0 0 


0

1
0
0


 0 0 1 0 


0
0
0

1



sinh 
cosh 
0
0
0
0
1
0
0

0
0

1 
Time reflection, time reversal T
 1

0
0

0
0
1
0
0
0
0
1
0
0

0
0

1 

The Lorentz transformations form a group, G
( g1 g2  G
if
g1 , g 2  G)
Rotations
R( )  e
 iJ . /
[ Ji , J j ]  i
 ijk
J z  i ( x y  y x )..
,
(c. f . J  r  p)
3

k 1
Angular momentum
operator
ijk
Jk
totally antisymmetric Levi-Civita symbol,
123   231   312  1;  213  132   321  1
Demonstration that
RZ ( )  e iJ z
 x   cos 
RZ ( ) ( x, y)  RZ ( )    
 y    sin 
For small  = ,
sin   x   x ' 
       ( x ', y ')
cos   y   y ' 
 x'  x  y
 

y
'
y


x
  



RZ ( ) ( x, y)   ( x   y, y   x)  ( x, y)   ( y
x
)
x
y
 (1  i ( xp y  ypx )) ( x, y)
i.e. RZ ( )  (1  i ( xp y  ypx ))  1  i J z
Hence
iJ z
RZ (  n ) (1  i J Z )n 

e
n

The Lorentz transformations form a group, G
( g1 g2  G
if
g1 , g 2  G)
Rotations
R( )  e
 iJ . /
[ Ji , J j ]  i
 ijk
J z  i ( x y  y x )..
,
(c. f . J  r  p)
3

k 1
Angular momentum
operator
ijk
Jk
totally antisymmetric Levi-Civita symbol,
123   231   312  1;  213  132   321  1
Derivation of the commutation relations of SO(3) (SU(2))
Rx ( ) Ry ( ) Rx ( ) Ry ( )
1 0

 0 1
 0 

 ,  small (infinitesimal)
0  1 0   1 0 0  1 0  



 
0
1
0
0
1


0
1
0







1   0 1  0  1   0 1 
 1  0 




1
0

  Rz ( ) (1  i J z )
 0
0 1 

Rx ( ) Ry ( ) Rx ( ) Ry ( )
 (1  i J x )(1  i J y )(1  i J x )(1  i J y )   ( J x J y  J y J x )
Equating the two equations implies
[ J x , J y ]  iJ z
QED

The Lorentz transformations form a group, G
( g1 g2  G
if
g1 , g 2  G)
Rotations
R( )  e
 iJ . /
[ Ji , J j ]  i
J z  i ( x y  y x )..
,
(c. f . J  r  p)
3

k 1
ijk
Jk
J  0, 12 ,1,...
Representations
0 1
 Spin  J 1/ 2
Pauli spin matrices
0
i
1
0
 
 2  
 3  

1
0

i
0
0

1






e.g. J  1/ 2 J i   i / 2
z
Angular momentum
operator
1
0
   ...
1
(SU(2))
The Klein Gordon equation (1926)
Scalar field (J=0) :
E p m
2
2
2
Energy eigenvalues
 (x)
2
  t 2  2  m2
E  (p2  m2 )1/ 2 ???
Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
j   2im ( *   * )
2
“probability density”
“probability current”
2
 t 2  2  m2
Klein Gordon equation
  i( * t   t )
*
  Neip. x ,   2 E N
  dV    d
V
3
x  2E
j  i( *   * )
j   (  , j)
2
f p  e
ip. x
1
2 p0V
Normalised free particle solutions
Physical interpretation of Quantum Mechanics
i t  21m  2  0
Schrödinger equation (S.E.)

t
i ( S .E.)  i (S .E.)
*
*
= 
 .j  0 continuity eq.
j   2im ( *   * )
2
“probability density”
“probability current”
2
 t 2  2  m2
Klein Gordon equation
  i( * t   t )
*
  Neip. x ,   2 E N
Negative probability?
j  i( *   * )
j   (  , j)
2
f p  e
ip. x
1
2 p0V
Normalised free particle solutions
Field theory of


Scalar particle – satisfies KG equation
   ( 1c t , ),    ( 1c t , )
(    m )  0


2
Classical electrodynamics, motion of charge –e in EM potential
is obtained by the substitution :

Quantum mechanics :
A   (A0 , A)
p   p   eA
i   i   eA
The Klein Gordon equation becomes:
(     m2 )  V
where V  ie(  A  A   )  e2 A2
em  4e
2
The smallness of the EM coupling,
1
137 ,
Make a “perturbation” expansion of V in powers of
means that it is sensible to
 em