PHYS 5326, Spring 2003

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PHYS 5326 Lecture 13
Wednesday, Mar. 5, 2003 Dr. Jae Yu
Local Gauge Invariance and Introduction of
Massless Vector Gauge Field
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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.

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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.

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Non-relativistic Equation of Motion for Spin 0
Particle

• Energy-momentum relation in classical mechanics
  give

Provides non-relativistic equation of motion for
field, y, Schrodinger Equation
represents the probability of finding the
pacticle of mass m at the position (x,y,z)
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Relativistic Equation of Motion for Spin 0
Particle

• Relativistic energy-momentum relation

With four vector notation of quantum
prescriptions
Relativistic equation of motion for field, y,
Klein-Gordon Equation
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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 factor relativistic energy-momentum
  relation

This works for the case with 0 three momentum
But not for the case with non-0 three momentum
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Dirac Equation Continued

• The coefficients like g01 and g1 g2 g3i do
  not work since they do not eliminate the cross
  terms.

It would work if these coefficients are matrices
that satisfy the conditions
Or using Minkowski metric, gmn
Using Pauli matrix as components in coefficient
matrices whose smallest size is 4x4
The energy-momentum relation can be factored

w/ a solution
Acting it on a wave function y, we obtain Dirac
equation
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Euler-Lagrange Equation

• For conservative force, it can be expressed as
  the gradient of a scalar potential, U, as

The Euler-Lagrange fundamental equation of motion

In 1D Cartesian Coordinate system
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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,
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Klein-Gordon Largangian for scalar (S0) Field

• For a single, scalar field variable f, the
  lagrangian is

From the Euler-Largange equation, we obtain
This equation is the Klein-Gordon equation
describing a free, scalar particle (spin 0) of
mass m.
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Dirac Largangian for Spinor (S1/2) Field

• For a spinor field y, the lagrangian

From the Euler-Largange equation for y, we
obtain
Dirac equation for a particle of spin ½ and mass
m.
Hows Euler Lagrangian equation looks like for y?
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Proca Largangian for Vector (S1) Field

• For a vector field Am, the lagrangian

From the Euler-Largange equation for Am, we
obtain
Proca equation for a particle of spin 1 and mass
m.
For m0, this equation is for an electromagnetic
field.
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Local Gauge Invariance - I
Dirac Lagrangian for free particle of spin ½ and
mass m
No, because it adds an extra term from derivative
of q.
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Local Gauge Invariance - II
The derivative becomes
So the Lagrangian becomes
Since the original L is
L is
Thus, this Lagrangian is not invariant under
local gauge transformation!!
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Local Gauge Invariance - III
Defining a local gauge phase, l(x), as
where q is the charge of the particle involved, L
becomes
Under the local gauge transformation
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Local Gauge Invariance - IV
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
Where Am is a new vector gauge field that
transforms under local gauge transformation as
follows
Addition of this vector field to L keeps L
invariant under local gauge transformation, but
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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.
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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.
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Homework

• Prove that the new Dirac Lagrangian with an
  addition of a vector field Am, 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.