Remarks on angular momentum Piet Mulders Trieste, November 2006

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’.

(Angular) momentum operators in QCD

Kinematic operators Front form quantization Instant form quantization

Local – forward and off-forward Local operators (coordinate space densities): PP’  Static properties: Examples: (axial) charge mass spin magnetic moment angular momentum Form factors

Nonlocal - forward Nonlocal forward operators (correlators): Specifically useful: ‘squares’ Momentum space densities of  -ons: Sum rules  form factors Selectivity at high energies: q = p

Nonlocal – off-forward Nonlocal off-forward operators (correlators AND densities): Sum rules  form factors Forward limit  correlators GPD’s b Selectivity q = p

Caveat We study forward matrix elements, including transverse momentum dependence (TMD), i.e. f(p ||,p T ) 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. GPD’s. One may at best make statements like: linear p T dependence  nonzero OAM no linear p T dependence  no OAM

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. GPD’s. One may at best make statements like: linear p T dependence  nonzero OAM no linear p T dependence  no OAM

Densities and (spacelike) formfactors

Forward limits of (spacelike) form factors

Caveat We study forward matrix elements, including transverse momentum dependence (TMD), i.e. f(p ||,p T ) 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. GPD’s. One may at best make statements like: linear p T dependence  nonzero OAM no linear p T dependence  no OAM back