PHYS 5326 – Lecture #14
Monday, Mar. 10, 2003
Dr. Jae Yu
•Completion of U(1) Gauge Invariance
•SU(2) Gauge invariance and Yang-Mills
Lagrangian
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
1
Announcement
• Remember the mid-term exam Friday, Mar. 14,
between 10am-noon in room 200
– Written exam
• Mostly on concepts to gauge the level of your understanding
on the subjects
• Some simple computations might be necessary
– Covers up to SU(2) gauge invariance
– Bring your own pads for the exam
• Review Wednesday, Mar. 12.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
2
U(1) Local Gauge Invariance
The requirement of local gauge invariance forces the
introduction of a massless vector field into the free
Dirac Lagrangian.


 
L  i c      mc 
Free L for
gauge field.


2

  1 



F F   q   A
16

Vector field for
gauge invariance
A is an electromagnetic potential.
And A  A     is a gauge transformation of an
electromagnetic
potential.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
3
U(1) Local Gauge Invariance
The last two terms in Dirac Lagrangian form the
Maxwell Lagrangian
  1 
 1 
LMaxwell  
F F   J A
16
 c
  1 



F F   q   A
16




with the current density J  cq   
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu


4
U(1) Local Gauge Invariance
Local gauge invariance is preserved if all the derivatives
in the lagrangian are replaced by the covariant derivative
q
D    i
A
c
Minimal
Coupling
Rule
The gauge transformation preserves local invariance
iq

 iq c
D      
A e

c


 iq
 iq
iq


A       e cD 
 e c    
c


Since the gauge transformation, transforms the
q
covariant derivative D      i c A     
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
5
U(1) Gauge Invariance
i


e
 is the same
The global gauge transformation
as multiplication of  by a unitary 1x1 matrix
  U


where U U  1 U  e
i

The group of all such matrices as U is U(1).
The symmetry involved in gauge transformation is
called “U(1) gauge invariance”.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
6
Lagrangian for Two Spin ½ fields
Free Lagrangian for two Dirac fields 1 and 2 with
masses m1 and m2 is


L  i c  1   1  m1c  1 1 

ic 
2

2
  2  m2 c  2 2
2

Applying Euler-Lagrange equation to L we obtain
Dirac equations for two fields
 m1c 
i   1  
 1  0
  

Monday, Mar. 10, 2003
 m2 c 
i   2  
 2  0
  

PHYS 5326, Spring 2003
Jae Yu
7
Lagrangian for Two Spin ½ fields
By defining a two-component column vector
 1  Where  1 and  2 are four
   
component Dirac spinors

 2
The Lagrangian can be compactified as
L  ic     c  M

2
 m1 0 

With the mass matrix M  
 0 m2 
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
8
Lagrangian for Two Spin ½ fields
If m1=m2, the Lagrangian looks the same as one particle
free Dirac Lagrangian
L  ic     mc 

2
However,  now is a two component column vector.
Global gauge transformation of  is   U .

Where U is any 2x2 unitary matrix U U  1
Since  U ,   is invariant under the
transformation.

Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
9
SU(2) Gauge Invariance
Any 2x2 unitary matrix can be written, U  e iH , where
H is a hermitian matrix (H+=H).
The matrix H can be generalized by expressing in
terms of four real numbers, a1, a2, a3 and q as;
H  1  τ a
Where 1 is the 2x2 unit matrix and t is the Pauli matrices
Thus, any unitary 2x2 matrix can be expressed as
i 1 iτa
U e e
Monday, Mar. 10, 2003
U(1) gauge
PHYS 5326, Spring 2003
Jae Yu
SU(2) gauge
10
SU(2) Gauge Invariance
The global SU(2) gauge transformation takes the form
 e 
iτa
iτ a
Since the determinant of the matrix e is 1, the
extended Dirac Lagrangian for two spin ½ fields is
invariant under SU(2) global transformations.
Yang and Mills took this global SU(2) invariance to local
invariance.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
11
SU(2) Local Gauge Invariance
The local SU(2) gauge transformation by taking the
parameter a dependent on the position x and defining
c
   a x 
q
is
  S
where S  e
Where q is a coupling
constant analogous to
electric charge
 iqt   x 
c
L is not invariant under this transformation, since the
derivative becomes    S     S 
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
12
SU(2) Local Gauge Invariance
Local gauge invariance can be preserved by replacing
the derivatives with covariant derivative
q
D    i
τ  Aμ
c
where the vector gauge field follows the transformation
rule D   S D  
with a bit more involved manipulation, the resulting L
that is local gauge invariant is
L  i c    D   mc2

 


2

Monday,
i Mar.c10,




mc



q


τ  A 13

2003
PHYS 5326, Spring 2003
Jae Yu
SU(2) Local Gauge Invariance
Since the intermediate L introduced three new vector
fields A=(A1, A2, A3), and the L requires free L for
these vector fields
1 
1 
1 
1 μν
LA  
F1 F1 
F2 F 2 
F3 F 3  
F3  Fμν3
16
16
16
16
1
8
2
 mc  ν

 A  Aν  0
  
The Proca mass terms in L,
to
preserve local gauge invariance, making the vector
bosons massless.
This time F     A   A  also does not make
the L local gauge invariant due to cross terms.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
14
SU(2) Local Gauge Invariance
By redefining


2q μ
ν
F   A  A 
A A
c
The complete Yang-Mills Lagrangian L becomes

μν


ν
μ



1 μν
L  ic      mc  
F  Fμν  q   τ  A 
16

2
This L
•is invariant under SU(2) local gauge transformation.
•describes two equal mass Dirac fields interacting with
three massless vector gauge fields.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
15
Yang-Mills Lagrangian
The Dirac fields generates three currents


J  c q  τ

These act as sources for the gauge fields whose
lagrangian is
Lgauge


1 μν


F  Fμν  q  τ  A 
16
The complication in SU(2) gauge symmetry stems
from the fact that U(2) group is non-Abelian (noncommutative).
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
16
Epilogue
Yang-Mills gauge symmetry did not work due to
the fact that no-two Dirac particles are equal
mass and the requirement of massless iso-triplet
vector particle.
This was solved by the introduction of Higgs
mechanism to give mass to the vector fields,
thereby causing EW symmetry breaking.
Monday, Mar. 10, 2003
PHYS 5326, Spring 2003
Jae Yu
17