Transcript Lagrangians and Local Gauge Invariance

PHYS 5326 – Lecture #13
Wednesday, Mar. 5, 2003
Dr. Jae Yu
Local Gauge Invariance and Introduction
of Massless Vector Gauge Field
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Announcements
• Remember the mid-term exam next Friday, Mar.
14, between 1-3pm in room 200
– Written exam
• Mostly on concepts to gauge the level of your understanding
on the subjects
• Some simple computations might be necessary
– Constitutes 20% of the total credit, if final exam will be
administered otherwise it will be 30% of the total
• Strongly urge you to go into the colloquium today.
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Prologue
• Motion of a particle is express 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.  Operators provide means for obtaining
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, Mar. 5, 2003
PHYS 5326, Spring 2003
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  ,
i

E  i .
t
Provides non-relativistic equation of motion for field, y,
Schrodinger Equation
2 2


   V  i
2m
t

2
represents the probability of finding the
pacticle of mass m at the position (x,y,z)
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Relativistic Equation of Motion for Spin 0 Particle
Relativistic energy-momentum relation

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 



1 


 

  ; where     ;   0 
, 1 
, 2 
, 3 
i
x
c t
x
y
z 

Relativistic equation of motion for field, y, Klein-Gordon Equation
  2      m2c 2   0
2nd order
in time
1  
 mc 
2
 2
   
 
2
c t
  
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2
2
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Relativistic Equation of Motion (Direct Equation) for
Spin 1/2 Particle
To avoid 2nd order time derivative term, Direct attempted
to

2 2
p
p

m
c 0

factor relativistic energy-momentum relation
This works for the case with 0 three momentum
p   m c  p
0 2


 mc p 0  mc  0
But not for the case with non-0 three momentum
2 2

0




p  p  m2c 2   k pk  mc   p  mc   k  pk p  mc  k   k pk  m2c 2
k
k



The terms linear to momentum should disappear, so

k 
k
p
p


 pk p
To make it work, we must find coefficients  to satisfy:

p   p   p   p 
    p      p      p      p 
      p p       p p       p p
0 2
0 2
0 1
1 2
0 2
2 2
1 2
1 0
3 2
1 2
2 2
0
0 1
Wednesday, Mar. 5, 2003
2
2 2
3 2
2 0
3 2
0 3
0
2
PHYS 5326, Spring 2003
Jae Yu
3 0
0
3
 Other Cross Terms
6
Dirac Equation Continued…
The coefficients like 0=1 and 1= 2= 3=i do not work since
they do not eliminate the cross terms.
It would work if these coefficients are matrices that satisfy the conditions
 
0 2
 
     
Or using
Minkowski
metric, g
 1,  1   2   3  1
2

2
2

      0 when   



,    2 g 
Using Pauli matrix as components in coefficient matrices whose smallest size is 4x4
The energy-momentum relation can be factored



p  p  m2c 2   k pk  mc   p  mc  0 w/ a solution   p  mc  0
By applying quantum prescription of momentum p  i 
Acting it on a wave function
y, we obtain Dirac equation
Wednesday, Mar. 5, 2003
i  y  mcy  0
k
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Euler-Lagrange Equation
For conservative force, it can be expressed as the gradient
of a scalar potential, U, as
F  U
Therefore the Newton’s law can be written m d v  U .
dt
1
Starting from Lagrangian L  T  U  mv 2  U
2
The Euler-Lagrange fundamental equation of motion

d  L
dt   q
i


 L

 q
i

Wednesday, Mar. 5, 2003
L
In 1D Cartesian
Coordinate system
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dT

 mvx

dv x
 q1
L
U

q1
x
8
Euler-Lagrange equation in QFT
Unlike particles, field occupies regions of space.
Therefore in field theory motion is expressed in
space and time.
Euler-Larange equation for relativistic fields
is, therefore,
Note the four
vector form
Wednesday, Mar. 5, 2003
 L  L

 
      
 i 
i

PHYS 5326, Spring 2003
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Klein-Gordon Largangian for scalar (S=0) Field
For a single, scalar field variable , 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, Mar. 5, 2003
PHYS 5326, Spring 2003
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

Since
2
L
L
 0 and
 i c    y  mc 2y
  y
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, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Proca Largangian for Vector (S=1) Field
For a vector field A, the lagrangian


2
1
1  mc  
 
 
L
 A   A   A   A  

 A A
16
8   
2
1
1  mc  
F  F 

 A A
16
8   
2
L
1    

 A   A and L  1  mc  A
Since
   A
4
A 4   





From the Euler-Largange equation for A, we obtain


2
2
 mc  
 mc  

  A   A  
 A   F  
 A 0
  
  




Proca equation for a particle of spin 1 and mass m.
For
m=0,
for5326,
anSpring
electromagnetic
field.
Wednesday,
Mar. this
5, 2003 equation is
PHYS
2003
Jae Yu
12
Local Gauge Invariance - I
Dirac Lagrangian for free particle of spin ½ and mass m;


L  ic y  y  mc yy

2
Is invariant under a global phase transformation (global
 i
i
y

e
y .
y

e
y
gauge transformation)
since
However, if the phase, , varies as a function of
space-time coordinate, x, is L still invariant under
i  x 
y

e
y
the local gauge transformation,
?
No, because it adds an extra term from derivative of .
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
13
Local Gauge Invariance - II
The derivative becomes
 y  i  e
i  x 
y e
i  x 
 y
So the Lagrangian becomes


L'  i c y   i   ei  x y  ei  x  y  mc 2 yy
 i c y    y  mc 2 yy  c y     y
Since the original L is L  ic y   y  mc2 yy
L’ is L'  L  c y     y
Thus, this Lagrangian is not invariant under local gauge transformation!!
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Local Gauge Invariance - III
Defining a local gauge phase, (x), as
c
 x   
 x 
q
where q is the charge of the particle involved, L becomes


L  L  qy  y   
'

Under the local gauge transformation:
y e
Wednesday, Mar. 5, 2003
 iq  x  / c
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y
15
Local Gauge Invariance - IV
Requiring the complete Lagrangian be invariant under
(x) local gauge transformation will require additional
terms to free Dirac Lagrangian to cancel the extra term


  

L  ic y   y  mc2 yy  qy  y A
Where A is a new vector gauge field that transforms
under local gauge transformation as follows:
A  A    
Addition of this vector field to L keeps L invariant
under local gauge transformation, but…
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Local Gauge Invariance - V
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     , because



A A  A     A    
 A A  A     A            
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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Local Gauge Invariance - VI
The requirement of local gauge invariance forces the
introduction of massless vector field into the free Dirac
Lagrangian.


 
L  i c y   y  mc yy


2

 1




F F   qy  y A
16

Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
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Homework
• Prove that the new Dirac Lagrangian with an
addition of a vector field A, as shown on page
12, is invariant under local gauge transformation.
• Describe the reason why the local gauge
invariance forces the vector field to be massless.
Wednesday, Mar. 5, 2003
PHYS 5326, Spring 2003
Jae Yu
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