Transcript Wednesday, Nov. 15, 2006

PHYS 3446 – Lecture #19
Wednesday, Nov. 15, 2006
Dr. Jae Yu
1. Symmetries
•
•
Wednesday, Nov. 15, 2006
Local gauge symmetry
Gauge fields
PHYS 3446, Fall 2006
Jae Yu
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Announcements
•
2nd term exam
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Next Wednesday, Nov. 22
Covers: Ch 4 – whatever we finish on Nov. 20
Workshop on Saturday, Dec. 2
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I heard from Liquid supply system construction
But still don’t have requests for
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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
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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
equations of motion from d Li  Li  0 are
dt r
m1r1  1V  r1  r2     V  r1  r2 
r1

m2 r2   2V  r1  r2    V  r1  r2 
r2
2.
3.
r
Prove that if   r  is2 a solution for the Schrödinger


2
H

r




V
r
    r   E  r  , then ei  r 
equation    2m


is also a solution for it.
Due for this is Monday, Nov. 27
Wednesday, Nov. 15, 2006
PHYS 3446, Fall 2006
Jae Yu
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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
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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
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Local Symmetries
• Let’s consider time-independent Schrödinger Eq.
 2 2

H  r       V  r    r   E  r 
 2m

• If   r  is a solution, ei  r  should also be a
solution for a constant 
– 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
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Local Symmetries
• Let’s consider a local phase transformation
  r   ei  r   r 
– 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


i r
i r
i r
 e    r    e   i   r    r     r   e    r 




Wednesday, Nov. 15, 2006
PHYS 3446, Fall 2006
Jae Yu
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Local Symmetries
• Thus, Schrödinger equation
 2 2

H  r    
  V  r    r   E  r 
 2m

• 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
    iA  r 
Wednesday, Nov. 15, 2006
PHYS 3446, Fall 2006
Jae Yu
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Local Symmetries
• Now requiring the vector potential A  r  to change
under transformation as
Additional
A  r   A  r     r 
Field
• Makes
i  r 
i  r 


e


iA
r

i


r
e

r



iA
r

r

  iA r   r 
                
• And the local symmetry of the modified Schrödinger
equation is preserved under the transformation
2
 2

H '  r    
  iA  r   V  r  '  r   E '  r 
 2m


Wednesday, Nov. 15, 2006

PHYS 3446, Fall 2006
Jae Yu
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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
• The potential A  r  can be interpreted as the EM
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
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U(1) Local Gauge Invariance
Dirac Lagrangian for free particle of spin ½ and mass m;
L  i  c       mc 2 
is invariant under a global phase transformation (global
i


e
 since   ei .
gauge transformation)
However, if the phase, , varies as a function of
space-time coordinate, x, is L still invariant under
i  x 


e

the local gauge transformation,
?
No, because it adds an extra term from derivative of .
Wednesday, Nov. 15, 2006
PHYS 3446, Fall 2006
Jae Yu
11
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


L  i  c       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
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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.
2
1
1  m Ac  

L
F F 

 A A
16
8 

This Lagrangian is not invariant under the local gauge
transformation, A  A    l , because
A A   A    l  A    l 
 A A   A  l  A   l      l    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.



2

L  i  c     mc  


Free L for
gauge field.


 1 
  q  A

F F 

16

A is an electromagnetic potential.
Vector field for
gauge invariance
And A  A    l is a gauge transformation of an
electromagnetic potential.
Wednesday, Nov. 15, 2006
PHYS 3446, Fall 2006
Jae Yu
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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
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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: Z0
– 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, Z0
– Charged current: W+ and WWednesday, Nov. 15, 2006
PHYS 3446, Fall 2006
Jae Yu
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