Remarks on angular momentum

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Remarks on angular momentum
Trieste, November 2006

• Piet Mulders

pjg.mulders_at_few.vu.nl
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Comments

• Parton model is not frame dependent (IMF)!
• Angular momentum is space integral (but space
  ambiguous!)
• In QM wave packets are allowed (Gallilean
  invariance, c? infinity).
• In relativistic QM (Lorentz/Poincare invariance)
  there is a problem.
• Can one not avoid problem with spin vector
  (parameterisation of density matrix) by using
  explicit spin basis, e.g. helicity states?
• These are projections of the fermion fields. Make
  sure you use a good basis.
• Expansion of nucleon state in terms of partons
  dangerous. Do it in front form ? Lightcone wave
  functions, etc.
• Transverse spin sumrule can be written down, but
  use operator expressions.

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(Angular) momentum operators in QCD
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Kinematic operators
Front form quantization
Instant form quantization
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Local forward and off-forward
Local operators (coordinate space densities)
Form factors
Static properties
Examples (axial) charge mass spin magnetic
moment angular momentum
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Nonlocal - forward
Nonlocal forward operators (correlators)
Specifically useful squares
Momentum space densities of f-ons
Sum rules ? form factors
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Nonlocal off-forward
Nonlocal off-forward operators (correlators AND
densities)
Sum rules ? form factors
GPDs
b
Forward limit ? correlators
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Caveat

• We study forward matrix elements, including
  transverse momentum dependence (TMD), i.e.
  f(p,pT) with enhanced nonlocal sensitivity!
• This is not a measurement of orbital angular
  momentum (OAM). Direct measurement of OAM
  requires off-forward matrix elements, i.e. GPDs.
  
• One may at best make statements like
• linear pT dependence ? nonzero OAM
• no linear pT dependence ? no OAM

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Aspects of high energy processes

• Ability to access matrix elements of specific
  operators (incoherence) in inclusive processes
  and
• This is not a measurement of orbital angular
  momentum (OAM). Direct measurement of OAM
  requires off-forward matrix elements, i.e. GPDs.
  
• One may at best make statements like
• linear pT dependence ? nonzero OAM
• no linear pT dependence ? no OAM

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Densities and (spacelike) formfactors
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Forward limits of (spacelike) form factors
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Caveat

• We study forward matrix elements, including
  transverse momentum dependence (TMD), i.e.
  f(p,pT) with enhanced nonlocal sensitivity!
• This is not a measurement of orbital angular
  momentum (OAM). Direct measurement of OAM
  requires off-forward matrix elements, i.e. GPDs.
  
• One may at best make statements like
• linear pT dependence ? nonzero OAM
• no linear pT dependence ? no OAM