Lecture 4: Superstrings, vibrations

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Lecture 4 Superstrings, vibrations vacuum
jitteriness.

• Quantum chromodynamics (QCD) S-matrix
• Feynman diagrams, dispersion relations
• Regge Trajectories (Regge theory)
• Dual Resonance model (birth of strings)
• String equations
• Classical string vibrations
• Vacuum jitters
• Gabriele Veneziano 1968 (audio KITP lecture)

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Energy regime of string theory
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The strong nuclear force

• In the 1950s baryons and mesons were found to
  have many excited states or resonances.
• In the 60s it was discovered that their
  scattering amplitudes were related to Regge
  trajectories
• J a(s) where J is angular momentum and s
  is square energy in the COM frame.
• HISTORY follows..

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Table of Hadrons
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Scattering experiments

• Elementary particle physicists use scattering
  experiments to probe the innards of particles
• The experiments involve collisions of simple
  particles at high energy to produce a shower of
  new particles
• The math to explain the processes is complex but
  can be broken down into a v large number of
  simpler components, for example

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Examples strong interactionScattering
experiments depicted as Feynman diagrams
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Interactions Feynman diagrams

• Interactions are represented by many diagrams,
  only the simplest were shown previously.
• Each part of the diagram represents a math
  formula
• A vertex represents an interaction (integral)
• A curly or straight line of a (of a certain type)
• Represents a propagation or motion of a
  particle. This has an associated formula.

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The S-Matrix

• John Wheeler 1937 invented the S-matrix approach,
  it relates input states to probable output
  quantum states without saying what happens
  inbetween.

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Feynmans simplification

• Feynman showed that the complex calcs needed to
  get the S-Matrix could be broken down into an
  infinite number of simple pictures whose
  importance (probability) get progressively
  smaller.
• Eg. Consider the sequence of numbers
• 1/1 1/4 1/9 1/16 1/25 1/36 1/49
• keep summing to infinity

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Example of diagrams QEDthere are problems for
QCD strong force.
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Strong force Hadron-Hadron interactions

• Unfortunately, hadron-hadron interactions
  involving the strong force, do not decrease in
  probability as the diagrams get more loops ( or
  become more complex). They are non-perturbative
  and Feynmans method breaks down.
• Bummer! What to do??
• Dispersion relations and Tulio Regge to the
  rescue.

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S-Matrix dispersion relations

• It turned out the the S-Matrix (relating input
  states to output states in scattering events)
  must be well behaved. In math terms Analytic.
• Even if the S-Matrix could not be calculated it
  was possible to make guesses by internal
  relationships called dispersion relations.
• Dispersion relations relate input to absorbed
  states. In terms of light, dispersion refers to
  how light is broken up by a prism into many
  colors. A visible light dispersion relation would
  be written for refractive index and relate real
  and imag parts, or input to absorbed light.
  (Knowledge of scattering implies knowledge of
  atomic structureBootstrap theory, Geoffrey Chew
  no particle is more fundamental than any other,
  each is related to the others.)

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Regge trajactories

• Tulio Regge was interested in The S-Matrix
• Resonances. Collisions of 2 high energy
  particles often produces long lived particles,
  but sometimes for a short time the energy of
  collision is simply localized in space until it
  decays away. It behaves like a new short lived
  particle or resonance. The whole system seems to
  ring with energy like a bell.

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Regge Trajectories Picture taken from
Superstrings and the search for the theory of
everything, by Peat.N stands for either proton
or neutron. Graph of angular momentum J vs.
energy squared. Several resonances (higher
harmonics) lie on a line.
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Linear equation and slopeSee Introduction to
string theory lectures by tHoofthttp//www.phys.
uu.ul/thooft/
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First guess at force between quarks
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Better model vortex or string
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Angular momentum J and Regge slope String
interpretation
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Dual Resonance Model

• Gabriele Veneziano in 1968 showed how to relate
  various resonances, collisions and scattering
  processes within a single model.
• Veneziano noted that two models of scattering
  events, s and t channels as they are called, give
  the same description of the physics. What looked
  like ringing (a resonance) in one channel would
  look like an interaction in the other.
• (Mandelstam variables s ,t ,u, refer to
  different momentum measurements)

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Dual description s and t channel
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Venezianos scattering amplitudenotes by
tHooft.
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Venezianos string amplitudesConsider the
residue of one pole in s.
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Dual model problems

• The Dual model allowed for ghosts, that is it
  allowed output states of a collision with
  negative probability, which is ridiculous!
• To get rid of the ghosts you need to have 26
  dimensions, 22 of which we dont know about.
• Yoichiro Nambu Bosonic string theory, model
  Hadrons and more gravity became included.
• Green and Schwarz get rid of infinities
  (anomalies in the theory)
• 1971 Peirre Ramond modifies Dirac eqn to get
  Fermionic strings.

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String vibrationsRemember strings vibrate in 1-9
dimensions. In 3d think of air vibrating inside
the resonating chamber of a trumpet.
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Vacuum JittersProbability of this process
occurring depends on the coupling strength
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Super Symmetry

• Each particle Fermion or Boson has a
  super-partner with ½ hbar less spin.
• Vacuum vibrations have energy close to Planck
  energy which is HUGE. How can strings account for
  light particles?
• Bosons have a positive vacuum energy contribution
• Fermions have a negative vacuum energy
  contribution
• Super-symmetry therefore solves the energy
  surplus.
• In order to have Fermionic strings, the
  modification to the Dirac gamma function
  (spinors) causes fermions to change into bosons
  and vice verse.

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The End
Audio from Veneziano after the break http//onlin
e.kitp.ucsb.edu/online/colloq/veneziano1

• Next Lecture 5
• Multiple Dimensions and Topology
• 1st March