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Remarks on Angular Momentum in Quantum Mechanics: Insights from Trieste, November 2006

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12 diapositives · 253 Views · Nov 15, 2014

In this paper, we discuss the implications of angular momentum in the context of Quantum Mechanics, emphasizing the role of the parton model and its frame dependence. We highlight the ambiguity in space integral forms of angular momentum and the challenges in relativistic quantum mechanics related to spin and density matrix parameterization. Key points include the necessity for a good basis in theoretical descriptions, the importance of using operator expressions for angular momentum in Quantum Chromodynamics (QCD), and the implications of transverse momentum dependence. This exploration addresses the subtleties of measuring orbital angular momentum (OAM) and the necessary conditions to evaluate it effectively.

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Remarks on Angular Momentum in Quantum Mechanics: Insights from Trieste, November 2006
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  1. Trieste, November 2006 Remarks on angular momentum Piet Mulders pjg.mulders@few.vu.nl

  2. 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’.

  3. (Angular) momentum operators in QCD

  4. Kinematic operators Instant form quantization Front form quantization

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

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

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

  8. 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. GPD’s. • One may at best make statements like: linear pT dependence  nonzero OAM no linear pT dependence  no OAM

  9. 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 pT dependence  nonzero OAM no linear pT dependence  no OAM

  10. Densities and (spacelike) formfactors

  11. Forward limits of (spacelike) form factors

  12. back 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. GPD’s. • One may at best make statements like: linear pT dependence  nonzero OAM no linear pT dependence  no OAM