Lecture 10: Mathematical Introduction to QFT, the beginnings of Strings'

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Lecture 10 Mathematical Introduction to QFT,
the beginnings of Strings.

• Final CLE lecture, April 12th 2004.
• Hold on to your hats folks this is
• Gonna be a wild ride!

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What I am going to Assume you Know!

• Basic Newtonian mechanics Fma type stuff
• Special relativity, matrices and vectors (tensors
  a bonus)
• Bachelors knowledge of physics (even if you did
  it a while ago)

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Contents

• Introduction to quantum field theory
• Lagrange methods, Relativistic Energy expressions
• Schrodinger and Klein Gordon Equations.
• Antiparticles and negative energies,
• things moving forward and backward in time
• Dirac equation and spinors (electrons are
  Fermions with spin ½)
• Modification of the Dirac eqn to include SUSY
• sparticles fermions bosons.

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Introduction to QFT

• Quantum Field Theory is usually taught after a
  basic familiarity with quantum mechanics,
  relativistic quantum mechanics and Lagrange
  techniques of classical mechanics is attained.
• Usually students would have been introduced to
  both special and general relativity before
  encountering QFT.
• Oh well, so much for the pre-amble lets get
  started!
• Remember this is the language of science, without
  it you cannot hope to fathom the depths of string
  theory.

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Since were a little pushed for time Ill teach
you JUST enough to get by.

• QFT in 1 hour this has gotta be a record.
• First well look at Lagrange methods which are
  VERY important.
• Complex analysis is also very important but well
  have to make do without... Our results will
  mostly be in the form of complex integrals -
  youll just have to be happy to get those and
  understand that they can be solved!

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Home, home on Lagrange!

• Consider a simple problem, a mass attached to a
  spring sliding on a frictionless surface.

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Intro to Lagrange Mechanics
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Recall what we had With the simple Newtonian
approach This is the same thing
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Relativistic Energy equations
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Non-Relativisitic Schrodinger Eqn.This is a
quantum equation which give a wavefunction
description of a particle or system.
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The Schrodinger Eqn can be written in time
independent form as
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OK, now we have the basic stuff out of the way,
lets do the Klein-Gordan Eqn. next! This is a
relativisitic version of the Schrodinger Equation.
If the brain hurts Thats OK, a normal reaction
to Relativistic Quantum Mechanics.
Im using the (- - -) metric
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How to get the Klein-Gordan Eqn. from the Action
Principle?
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What on Earth are negative energy states doing
here??

• Because the energy eqn. is squared, that means
  energy solutions can have either sign. This is a
  problem.
• Problem? Well you could in principle gain energy
  by a particle transition from a positive energy
  state to a negative one, extract energy from this
  process and do work. Since the energy is not
  bounded from below you could keep doing this and
  have an infinite power source!!
• Well whats bad about that? It just aint so.
  There is no such power source - so something in
  the physics is just wrong. We have a difficulty
  in interpreting f as a wave-function or its
  modulus square as a probability. The prob density
  is NOT positive definite negative probabilities
  cause a slight problem.

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Forward and backward in time
Propagation of a energy particle Forward in
time.
Propagation of a energy particle Backward in
time.
The introduction of special relativity into
quantum mechanics introduces the idea of
causality, that an event cannot precede its
cause. This implies that no signal can travel
faster than the speed of light in vacuum. This
has far reaching consequences. Two local
operators that represent physical observables
(things you can measure) must commute at
space-like separations (outside light cone). In
order to accomplish this we need negative energy
states, anti-particles. Negative energy states
going backward in time look like positive energy
states going forward in time.
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Backward propagation in times leads to particle
creation/destruction.
This figure represents 1 particle only moving
forward in time.
Particle is created at A. Then pair creation
occurs at Y. Particle and anti-particle
annihilate at X. Remaining Particle moves off to
B. Start and finish with one particle AB.
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Quantize f for the KG field.
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Need a particle interpretation for the field,
annihilation/creation operators a and a
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Introduction to the Dirac formulation

• Klein-Gordan (KG) eqn. is not a single particle
  theory. IT has both particles and anti-particles.
  The conserved current associated with the KG eqn
  can be re-interpreted as a charge current (not a
  probability current density). We allow the
  negative energy solutions to go backward in time,
  so that they look like positive energy particles
  moving forward in time. These are interpreted as
  anti-particles.
• When Dirac wrote his eqn. in 1928 anti-particles
  had not yet been discovered (that came 5 yrs
  later). Dirac had abandoned the KG eqn. and wrote
  a new one, he wanted first order differential
  eqn. for energy and the time derivative. His new
  eqn. had to be Lorentz invariant and
  relativistically correct in energy, so its
  solutions must also satisfy the KG eqn. He took
  basically a square root of the KG equation.

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Introduction continued

• The negative energy states were still admitted
  even though Diracs eqn. was linear in energy.
  This was a bitter disappointment at first.
  However, Dirac saw that if he forced all the
  negative energy states to be filled by a sea of
  ve. particles (holes in the sea were positive
  particles) we could at least avoid the energy
  from nothing fiasco. Dirac, Oppenheimer and Tamm
  calculated that when an electron met a hole, the
  electron quickly jumped in to fill the negative
  energy state vacancy, resulting in the
  annihilation of the electron and the hole. The
  interpretation of the hole as an antiparticle
  followed the 1933 discovery of the positron.
• We now seek an equation, as Dirac did, which is
  1st order in energy and the time derivative, so
  that we can get a non-negative probability
  density. We want first order in the spatial
  derivative also, so that space and time will be
  on equal footing. Thus making Lorentz invariance
  obvious, the eqn. will have same form in
  different frames of reference.

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Basic Dirac Equation (spin 1/2)
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Dirac equation continued
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Yet more on Dirac Equation!!
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Modify the Dirac equation to include internal
degrees of freedom (string harmonics)
Seminal paper reproduced from The paper
collection opposite. Important paper by the
French Physicist Dr. Pierre Ramond follows.
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The VERY End
You should experience brain expansion Possibly in
extra dimensions by now!!

• Thank you and good night!
• See my webpage if you still want more
• http//chaos.fullerton.edu/heidi