PHYS 5326 – Lecture #9
Wednesday, Feb. 28, 2007
Dr. Jae Yu
1. Quantum Electro-dynamics (QED)
2. Local Gauge Invariance
3. Introduction of Massless Vector Gauge
Field
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
1
Announcements
• First term exam will be on Wednesday, Mar. 7
• It will cover up to what we finish today
• The due for all homework up to last week’s is
Monday, Mar. 19
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
2
Prologue
• How is a motion described?
– Motion of a particle or a group of particles can be expressed in
terms of the position of the particle at any given time in classical
mechanics.
• A state (or a motion) of particle is expressed in terms of
wave functions that represent probability of the particle
occupying certain position at any given time in Quantum
mechanics
– With the operators provide means for obtaining values for
observables, such as momentum, energy, etc
• A state or motion in relativistic quantum field theory is
expressed in space and time.
• Equation of motion in any framework starts with
Lagrangians.
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
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Non-relativistic Equation of Motion for Spin 0 Particle
Energy-momentum relation in classical mechanics give
2
p
V  E
2m

Quantum prescriptions; p  , E  i .
i
t
provides the non-relativistic equation of motion for field, y,
the Schrödinger Equation


  V   i
2m
t
2
2

2
Wednesday, Feb. 28, 2007
represents the probability of finding the
particle of mass m at the position (x,y,z)
PHYS 5326, Spring 2007
Jae Yu
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Relativistic Equation of Motion for Spin 0 Particle
Relativistic energy-momentum relationship

E  p c  m c  p p  m c  0
2
2 2
2 4
2 2
With four vector notation of quantum prescriptions;
p 
i
  where   

;

x

1 


 


,


,


,


1
2
3
 0

c

t

x

y

z


Relativistic equation of motion for field, y, the Klein-Gordon Equation

2nd order
in time
      m2c2  0
2
1  
 mc 
2
 2
   
 
2
c t


Wednesday, Feb. 28, 2007
2
2
PHYS 5326, Spring 2007
Jae Yu
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Relativistic Equation of Motion (Dirac Equation) for
Spin 1/2 Particle
To avoid 2nd order time derivative term, Dirac attempted to
factor relativistic energy-momentum relation

p p  m c  0
2 2
This works for the case with zero three momentum
p 
0 2
 m c   p  mc  p  mc   0
2 2
0
0
This results in two first order equations
p  mc  0
0
p  mc  0
0
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
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Dirac Equation Continued…
The previous prescription does not work for the case with non-0 three
momentum
p  p  m2c2   k pk  mc   p  mc 



 k  pk p  mc   k   k  pk  m 2c 2
The terms linear to momentum should disappear, so  k   k
To make it work, we must find coefficients
k

k 
p
p


 pk p
to satisfy:

 p   p   p   p 
    p      p      p      p 
        p p        p p        p p Other Cross Terms
0 2
1 2
0 2
0 2
0 1
1 0
2 2
1 2
3 2
1 2
0 2
0 1
2 2
2 2
3 2
2 0
0 3
0
2
3 2
3 0
0
3
The coefficients like 0=1 and 1= 2= 3=i do not work since they do
not eliminate the cross terms.
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
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Dirac Equation Continued…
It would work if these coefficients are matrices that satisfy the conditions
 
0 2
 1, 
      
1 2
2 2
3 2
  n   n    0 when   n
Or using   ,  n  
 1
Minkowski   n   n   
metric, gn 2g n
where g n
1 0 0 0 
0 1 0 0 


0 0 1 0 


0 0 0 1
Using gamma matrices with the standard Bjorken and Drell convention
1 0 
0
 

0

1


 1

0


 0

 0
0 0 0 
 
 
1 0 0 
0   1 0  
 

0   0 1 
Where si are Pauli spin matrices
Wednesday, Feb. 28, 2007
 0 s 
  i

 s 0 
i
i
0 1
 0 i 
1 0
s1  
, s 2  
, s 3  

1
0
i
0
0
1






PHYS 5326, Spring 2007
Jae Yu
8
Dirac Equation Continued…
Using Pauli matrix as components in coefficient matrices whose smallest
size is 4x4, the energy-momentum relation can now be factored
p  p  m2c2    k pk  mc    p  mc   0
w/ a solution

 p  mc  0
By applying quantum prescription of momentum p  i  
Acting the 1-D solution on a wave
k
i    mc
function, y, we obtain Dirac equation
y 1 
 
y
where Dirac spinor, y y   2 
y 3 
 
y 4 

Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
y
y 0
9
Euler-Lagrange Equation
For a conservative force, the force can be expressed as
the gradient of the corresponding scalar potential, U
F  U
dv
Therefore the Newton’s law can be written m  U .
dt
1
Starting from Lagrangian L  T  U  mv 2  U
2
The 1-D Euler-Lagrange fundamental equation of motion

d  L


dt  q
i

 L

 qi

Wednesday, Feb. 28, 2007
L
In 1D Cartesian
Coordinate system
PHYS 5326, Spring 2007
Jae Yu


dT
 mvx
dvx
 q1
L
U

q1
x
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Euler-Lagrange equation in QFT
Unlike particles, field occupies regions of space.
Therefore in field theory, the motion is expressed
in terms of space and time.
Euler-Larange equation for relativistic fields
is, therefore,
Note the four
vector form
Wednesday, Feb. 28, 2007
 L  L

 
    i   i


PHYS 5326, Spring 2007
Jae Yu
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Klein-Gordon Largangian for scalar (S=0) Field
For a single, scalar field , the Lagrangian is
2
1
1  mc  2

L             
2
2 
Since
L
  i and
   i 
L
 mc 
 
 i
i


2
From the Euler-Largange equation, we obtain
2
 mc 
   
  0



This equation is the Klein-Gordon equation describing a
free, scalar particle (spin 0) of mass m.
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
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Dirac Largangian for Spinor (S=1/2) Field
For a spinor field y, the Lagrangian
L  i  c y  y   mc yy

L
Since   y
  
2
 0 and L  i  c     y  mc 2y
y
From the Euler-Largange equation for `y, we obtain
 mc 
i  y  
y  0



Dirac equation for a particle of spin ½ and mass m.
How’s Euler Lagrangian equation looks like for y?
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
13
Proca Largangian for Vector (S=1) Field
Suppose we take the Lagrangian for a vector field A
2
1
1  mc  n
 n
n 
L
 A   A     An  n A  


 A An
16
8 

2
1
1
mc

 n

F n Fn 

 A An
16
8 

Where Fn is the field strength tensor in relativistic notation, E and B in
Maxwell’s equation form an anti-symmetic second-rank tensor
 0  Ex  E y  Ez 


E
0

B
B
x
z
y 

n
F 
E y Bz
0  Bx 


Ez  By Bx
0 

Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
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Proca Largangian for Vector (S=1) Field
Suppose we take the Lagrangian for a vector field A
2
1
1  mc  n
 n
n 
L
 A   A     An  n A  


 A An
16
8 

2
1
1
mc

 n

F n Fn 

 A An
16
8 

2
L
1  n n 
Since   A   4  A   A and L  1  mc  An
  n
A 4 



n
From the Euler-Largange equation for A, we obtain
2
2
 mc  n
 mc  n
n
  A   A   
 A   F  
 A 0





n
n

Proca equation for a particle of spin 1 and mass m.
For
m=0,
for5326,
anSpring
electromagnetic
field.
Wednesday,
Feb. this
28, 2007equation is
PHYS
2007
Jae Yu
15
Lagrangians
• 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
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
16
Homework
• Prove that Fmn can represent Maxwell’s
equations, pg. 225 of Griffith’s book.
• Derive Eq. 11.17 in Griffith’s book
• Due is Wednesday, Mar. 7
Wednesday, Feb. 28, 2007
PHYS 5326, Spring 2007
Jae Yu
17