Monday, Apr. 2, 2007
Dr. Jae Yu
1. Local Gauge Invariance
2. U(1) Gauge Invariance
3. SU(2) Gauge Invariance
4. Yang-Mills Lagrangian
5. Introduction of Massless Vector Gauge
Fields
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
1

For a single, scalar field f , the Lagrangian is
Since
L 
1
2
   

 L
 f i

   f i and
1
2
 mc
 2 f
2
 
 L
 f i
 mc
 2 f i
From the Euler-Largange equation, we obtain
   f 
 mc
 2 f 
0
This equation is the Klein-Gordon equation describing a free, scalar particle (spin 0) of mass m .
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
2

For a spinor field y , the Lagrangian
L  i
  y  
 y 
  yy
Since 
 L
 y
 
0 and
 L
 y
 i
    
 y  mc
2 y
From the Euler-Largange equation for `y , we obtain i
  
 y 
 mc
 y 
0
Dirac equation for a particle of spin ½ and mass m .
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
3

Suppose we take the Lagrangian for a vector field A 
2
L  
1
16


   
A

 
  

A



1
8

 mc


 
Since
1
16

 L

F

F



8

1


 mc
 2
A A

 
1
  
4
 
A
 
A
 and
 L

A


1
4

 mc
 2
From the Euler-Largange equation for A  , we obtain
A


  
A

  mc
 2
A
  

F
 
 mc
 2
A
 
0
Proca equation for a particle of spin 1 and mass m .
For m =0, this equation is for an electromagnetic field.
Jae Yu
4

• Lagrangians we discussed are concocted to produce desired field equations
– L derived (L=T-V) in classical mechanics
– L taken as axiomatic in field theory
• The Lagrangian for a particular system is not unique
– Can always multiply by a constant
– Or add a divergence
– Since these do not affect field equations due to cancellations
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
5

Dirac Lagrangian for free particle of spin ½ and mass m ;
L  i
  y  
 y 
  yy is invariant under a global phase transformation (global e i
 y i

Why?  is a constant and is not subject to the derivative .
If the phase  varies as a function of space-time coordinate, x  , is L still invariant under the local e i
   y
No, because it adds an extra term from derivative of  .
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
6

L '
The derivative becomes

 y  i
  e i
   y  e i
   
 y
So the Lagrangian becomes

 i
  y 
 i

 i
  y  
 y 



 e mc i

2

  y yy

 e i
   c

 y y 




 mc

2
 yy
Since the original L is
L  i
  y  
 y 
 mc
2
 yy
L’ is
L ' 
L
   y 




Thus, this Lagrangian is not invariant under local gauge transformation!!
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
7

Defining a local gauge phase, l (x), as l     q c         q c l   where q is the charge of the particle involved, L becomes
L '  L

 q y y


 l
Under the local gauge transformation: y  e
 iq l  
/ c y
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
8

L
Requiring the complete Lagrangian be invariant under l (x) local gauge transformation will require additional terms to the free Dirac Lagrangian to cancel the extra term
 i
  y  
 y 
 mc
2
 yy 
 q y y

A

Where A
 is a new vector gauge field that transforms under local gauge transformation as follows:
A


A

 
 l
Addition of this vector field to L keeps L invariant under local gauge transformation, but…
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
9

The new vector field couples with the spinor through the last term. In addition, the full Lagrangian must include a “free” term for the gauge field. Thus, Proca
Largangian needs to be added.
L
Invariant
 
16
1


F F


8
1


 m c
A
 


2
Not-invariant

A A

This Lagrangian is not invariant under the local gauge transformation, , because




A
  

 l

A

   l


A A


A
 l
A
  l
    
In what ways can we make this L invariant?
Jae Yu
10

The requirement of local gauge invariance forces the introduction of a massless vector field into the free Dirac
Lagrangian  QED Lagrangian – Interaction between Dirac fields (e +) and Maxwell fields (photons)
L
Free L for gauge field.



 i
 

16
1
 y


F F


 y





 q mc
2

 yy y y

A

A
 is an electromagnetic potential.
A


A

 
Vector field for gauge invariance
And is a gauge transformation of an electromagnetic potential.
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
11

The last two terms in the modified Dirac Lagrangian form the Maxwell Lagrangian
L
Maxwell




1

16


F F

 

1 c

J A





1
16


F F
 with the current density



 q y y
J
 cq
 

A

Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
12

Local gauge invariance is preserved if all the derivatives in the Lagrangian are replaced by the covariant derivative
D

 

 i q c
A

Minimal
Coupling
Rule
The gauge transformation preserves local invariance
D
 y   
 iq c
A

 e
 iq l c y
 e
 iq l c

 covariant derivative iq
D


A

 
 

 i
 l y q c



A


  e

 iq
 l c c
Since the gauge transformation, transforms the l
D
 y
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
13

The global gauge transformation is the same as multiplication of y by a unitary 1x1 matrix y 
U y 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 the “U(1) gauge invariance”.
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
14

L
Free Lagrangian for two Dirac fields y
1 masses m

 i
1 c and m
 
2 y 
1 is
 
 y
1
 m c
1 and y
2
2
1 with y y
1

 i
  y 
2
 
 y
2


 m c
2
2

 y y
2 2
Applying Euler-Lagrange equation to L, we obtain
Dirac equations for two fields i
  
 y
1
 m c
1 y
1

0 i
  
 y
2
 m c
2 y
2

0
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
15

By defining a two-component column vector y y y
2
 component Dirac spinors
The Lagrangian can be compactified as
L  i
  y  
 y  c
2 y y
With the mass matrix
Monday, Apr. 2, 2007
M
 

 m
1
0
0 m
2



PHYS 5326, Spring 2007
Jae Yu
16

If m
1
=m
2
, the Lagrangian looks the same as one particle free Dirac Lagrangian
L  i
  y  
 y  mc
2 yy
However, y now is a two component column vector.
Global gauge transformation of y y 
U y
Where U is any 2x2 unitary matrix

U U

1 y  y transformation.
U yy
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
17

U
 e
H is a hermitian matrix (H + =H).
The matrix H can be generalized by expressing in terms of four real numbers, a
1
, a
2
, a
3 and  as;
H
 
1
a where 1 is the 2x2 unit matrix and t is the Pauli matrices.
Thus, any unitary 2x2 matrix can be expressed as
U
 i

1 e e i
 a
U(1) gauge SU(2) gauge
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
18

The global SU(2) gauge transformation takes the form y  e i
 a y e 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, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
19

The local SU(2) gauge transformation by taking the parameter a dependent on the position x
 and defining y l   c a q y
Where q is a coupling constant analogous to electric charge
S
 e
 iq t l   c
L is not invariant under this transformation, since the derivative becomes 
 y
S
 y
  y
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
20

Local gauge invariance can be preserved by replacing the derivatives with covariant derivative
D
 i q
A
μ c where the vector gauge field follows the transformation rule
D
 y 
S

D
 y

With a bit more involved manipulation, the resulting L that is local gauge invariant is
L  i
  y 
D
 y
  i
 
Monday, Apr. 2, 2007
 y  
 y  mc mc
2 yy
2 yy
PHYS 5326, Spring 2007
Jae Yu

 q y y

21

A


Since the intermediate L introduced three new vector fields A  =(A
1
 , A
2
 , A
3
 ), and the L requires free L for each of these vector fields
L
A
 
1
16


F F
1

1

1
16


F F
2

2

1
16


F F
3

3
 
1
16

F
μν 
F
3 μν3
Set the Proca mass terms in L
1
 mc
 2
A
ν 
A
ν

0
8
 preserve local gauge invariance, making the vector bosons massless.
  the L local gauge invariant due to cross terms.
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
22

By redefining
F
μν   
A
ν   
A
μ 
2 q 
A
μ 
A
ν c
The complete Yang-Mills Lagrangian L becomes

L 

 i
  y  
 y  mc
2 yy

 
1
16

This L
F
μν 
F
μν

 q y y


A

•is invariant under SU(2) local gauge transformation.
•describes two equal mass Dirac fields interacting with three massless vector gauge fields.
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
23

The Dirac fields generates three currents
J

 y y

These act as the sources for the gauge fields whose lagrangian is
L gauge
 
1
16

F
μν 
F
μν

 q y y


A

The complication in SU(2) gauge symmetry stems from the fact that U(2) group is non-Abelian (noncommutative).
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
24

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, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
25

Consider a free Lagrangian for a scalar field, f :
L 
1 

 f
 
  f

 e
  
2
No apparent mass terms unless we expand the second term and compare this L with the Klein-Gordon L :
L 
1
2


 f
 
  f

1
 
2 
1
2
 
4 
1
6
 
6
L
KG

1
2
    1
2
 mc
 2 f
2
  where m

2

/ c

....
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
26

Consider a Lagrangian for a scalar field, f, in a potential:
L 
1 

 f
 
  f


1
2 2 
1
2 4
2 2 4
Mass term ( f 2 term) has the wrong sign unless mass is imaginary. How do we interpret this L ?
In Feynman calculus, the fields are the fluctuation
(perturbation) from the ground state (vacuum).
Expressing L =T-U, the potential energy U is
U
 
1
2
 f
2 
1
4 l f
4
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
27

The field that minimizes U is f    l
To shift the ground state to occur at 0, we introduce a new variable, h :
Replacing field, f , with the new field, h , the L becomes
L 
1
2
   
  h
2  lh
3 
1
4 l h
4 
1 
4
Mass term
3point vtx
4point
Comparing this modified L with L
KG vector field is m

2

/ c vtx
, the mass of the
/

2
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
28

The original lagrangian, L ,
L 
1
2


 f
 
  f


1
2
2 2 
1
4
2 4 is even and thus invariant under f  f.
L
However, the new L

1
2
   
  h
2  lh
3 
1
4 l h
4 
1
4

 l has an odd term that causes this symmetry to break

2 since any one of the ground states (vacuum) does not share the same symmetry as L .
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
29

Not symmetric about this axis
 
1
2

2 f
2 
1
4 l
2 f
4
Symmetric about this axis
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
30

While the collection of ground states does preserve the symmetry in L , the Feynman formalism allows to work with only one of the ground states.  Causes the symmetry to break.
This is called “spontaneous” symmetry breaking, because symmetry breaking is not externally caused.
The true symmetry of the system is hidden by an arbitrary choice of a particular ground state. This is a case of discrete symmetry w/ 2 ground states.
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
31

Spontaneous Breaking of a Continuous Symmetry
A lagrangian, L , for two fields, f
1
L 
1
2


 f
1
 and f
2 can be written

  f
1


1
2


 f
2
 
  f
2


1
2

1
2  f
2
2


1
4

1 is even and thus invariant under f
1, f
2
4  f
4
2

 f
1,
f
2
.
The potential energy term becomes
Monday, Apr. 2, 2007
U
 
1
2

1
2  f
2
2


1
4 w/ the minima on the circle:

1
4  f
2
4
 f
2  f
2,min
Jae Yu
2 
/
2
32

Spontaneous Breaking of a Continuous Symmetry
To apply Feynman calculus, we need to expand about a particular ground state (the
“vacuum”). Picking f
1,min
  l and f 
2,min
0
And introducing two new fields, h and x, which are fluctuations about the vacuum: h f
1
  l and x f
2
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
33

Spontaneous Breaking of a Continuous Symmetry
The new L becomes
L 


1
2
   
  h
2




 2

 l
4
2 
 
 
1
2
    h
4  x
4 
2 h x
2
 




1
4

/

2
3point vtx
L
KG of the field the mass m h
 h
2

 w/
/ c
L free of the massless field x
4point vtx
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
34

Spontaneous Breaking of Continuous
Global Symmetry
One of the fields is automatically massless.
Goldstone’s theorem says that breaking of continuous global symmetry is always accompanied by one or more massless scalar (spin=0) bosons, called
Goldstone Bosons.
This again poses a problem because the effort to introduce mass to weak gauge fields introduces a massless scalar boson which has not been observed.
This problem can be addressed if spontaneous SB is applied to the case of local gauge invariance.
Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu
35

• Prove that the new Dirac Lagrangian with an addition of a vector field A

, as shown on page 9, is invariant under local gauge transformation.
• Describe the reason why the local gauge invariance forces the vector field to be massless
• Due is Wednesday, Apr. 11
36 Monday, Apr. 2, 2007 PHYS 5326, Spring 2007
Jae Yu