PHYS 3446 – Lecture #19
Wednesday, Nov. 15, 2006
Dr. Jae Yu
1. Symmetries
• Local gauge symmetry
• Gauge fields
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
1

• 2 nd term exam
– Next Wednesday, Nov. 22
– Covers: Ch 4 – whatever we finish on Nov. 20
• Workshop on Saturday, Dec. 2
– I heard from Liquid supply system construction
– But still don’t have requests for
• Kerosene pump + Liquid Nitrogen for cooling
• Blue filter films
• Mid-tem grade discussion in the last 20 minutes
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
2

Homework Assignments
1. Construct the Lagrangian for an isolated, two particle system under the potential that depends only on the relative distance between the particles and show that the m r
 m r


1

1
 r
2
 

2

1
 r
2
 



 r
1

 r
2


1 d dt
1


 r

2 r
L
2
 r
 i 

L
 r i 
 



2
2 m
  is also a solution for it.


E
   e i
   
3. Due for this is Monday, Nov. 27
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
3

Why symmetry?
• When does a quantum number conserved?
– When there is an underlying symmetry in the system
– When the quantum number is not affected (or is conserved) under changes in the physical system
• Noether’s theorem: If there is a conserved quantity associated with a physical system, there exists an underlying invariance or symmetry principle responsible for this conservation.
• Symmetries provide critical restrictions in formulating theories
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
4

Local Symmetries
• All continuous symmetries can be classified as
– Global symmetry: Parameters of transformation are constant
• Transformation is the same throughout the entire space-time points
• All continuous transformations we discussed so far are global symmetries
– Local symmetry: Parameters of transformation depend on space-time coordinates
• The magnitude of transformation is different from point to point
• How do we preserve a symmetry in this situation?
– Real forces must be introduced!!
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
5

Local Symmetries
• Let’s consider time-independent Schrödinger Eq.

H
   
 




2
 
2 m
   
 solution for a constant 


 
E
  
• If is a solution, should also be a
– Any quantum mechanical wave functions can be defined up to a constant phase
– A transformation involving a constant phase is a symmetry of any quantum mechanical system
– Conserves probability density  Conservation of electrical charge is associated w/ this kind of global transformation.
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
6

Local Symmetries
• Let’s consider a local phase transformation
    e i
   
– How can we make this transformation local?
• Multiplying a phase parameter with an explicit dependence on the position vector
• This does not mean that we are transforming positions but just that the phase is dependent on the position
• Thus under local transformation, we obtain


 e i
  


  e i
  

 i

    
       

  e i
  
 
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
7

Local Symmetries
• Thus, Schrödinger equation
H
  



2
2 m
     


E
  
• is not invariant (or a symmetry) under local phase transformation
– What does this mean?
– The energy conservation is no longer valid.
• What can we do to conserve the energy?
– Consider an arbitrary modification of a gradient operator
   
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
8

Local Symmetries
  under transformation as
         
Additional
Field

 
• Makes
  
   

     i
  
 

 e i
  
  

  e i
   
    

• And the local symmetry of the modified Schrödinger
  equation is preserved under the transformation
H

'
   
 2
2 m

    
2
    


E

'
 
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
9

Local Symmetries
• The invariance under a local phase transformation requires the introduction of additional fields
– These fields are called gauge fields
– Leads to the introduction of a definite physical force vector potential
 
• The symmetry group associated with the single parameter phase transformation in the previous slides is called Abelian or commuting symmetry and is called U(1) gauge group  Electromagnetic force group
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
10

U(1) Local Gauge Invariance
Dirac Lagrangian for free particle of spin ½ and mass m ;
L  i
    

 
 
 is invariant under a global phase transformation (global gauge transformation) since .
However, if the phase,  , varies as a function of space-time coordinate, x  , is L still invariant under
No, because it adds an extra term from derivative of  .
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
11

L
U(1) Local Gauge Invariance
Requiring the complete Lagrangian be invariant under l (x) local gauge transformation will require additional terms to free Dirac Lagrangian to cancel the extra term

 i
    

 
 mc
2

 
 q
 

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…
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
12

U(1) Local Gauge Invariance
The new vector field couples with 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  
16

1


F F



8
1



 m c
A
 


2

A A

This Lagrangian is not invariant under the local gauge transformation, , because
A A


A
  


A

  l
A

   l


A A



A
 l
A
  l
    
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
13

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
    

 
 mc
2


Free L for gauge field.

 16
1


F F




 q
 

A

A
 is an electromagnetic potential.
A


A

 
 electromagnetic potential.
Vector field for gauge invariance
And is a gauge transformation of an
Wednesday, Nov. 15, 2006 14 PHYS 3446, Fall 2006
Jae Yu

Gauge Fields and Local Symmetries
• To maintain a local symmetry, additional fields must be introduced
– This is in general true even for more complicated symmetries
– A crucial information for modern physics theories
• A distinct fundamental forces in nature arises from local invariance of physical theories
• The associated gauge fields generate these forces
– These gauge fields are the mediators of the given force
• This is referred as gauge principle, and such theories are gauge theories
– Fundamental interactions are understood through this theoretical framework
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
15

Gauge Fields and Mediators
• To keep local gauge invariance, new particles had to be introduced in gauge theories
– U(1) gauge introduced a new field (particle) that mediates the electromagnetic force: Photon
– SU(2) gauge introduces three new fields that mediates weak force
• Charged current mediator: W + and W -
• Neutral current: Z 0
– SU(3) gauge introduces 8 mediators (gluons) for the strong force
• Unification of electromagnetic and weak force SU(2)xU(1) gauge introduces a total of four mediators
– Neutral current: Photon, Z 0
– Charged current: W + and W -
Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006
Jae Yu
16