The 2008 construction of Chen, Lü, Sun, Wang and Goldman splits the gluon potential into a pure-gauge part and a physical part, thereby making every term of the Jaffe–Manohar decomposition of the nucleon spin gauge invariant. The authors themselves deferred two decisive consistency questions: whether these gauge-invariant quark/gluon angular-momentum and momentum operators, together with the interaction-containing Lorentz-boost generators, satisfy the full Poincaré algebra in canonically quantized QCD; and whether the defining constraints fix the pure-gauge/physical split uniquely in the presence of Gribov-copy topology. Later literature established that the split is classically Lorentz covariant under generalized Lorentz transformations, and the Leader–Lorcé review argued that formally infinitely many such gauge-invariant extensions exist, so that the choice among them is a matter of convention — a position Wakamatsu disputes — but the explicit quantum boost-commutator computation and the global topological uniqueness analysis of the specific constraint system have not been published. The problem is to settle these two jointly, thereby determining whether the instant-form gauge-invariant canonical spin/orbital-angular-momentum operators are physically meaningful operators of QCD.

Background

Experiments on polarized deep-inelastic scattering showed in the late 1980s that the helicities of the quarks inside a proton account for only a small fraction of the proton's spin-one-half, and the attempt to partition the remaining spin among gluon spin and quark/gluon orbital motion exposed a subtle theoretical problem. Angular momentum operators in a gauge theory are built from the gauge potential, so any split into spin and orbital pieces is gauge dependent unless extra care is taken. The classic Jaffe–Manohar decomposition (Nucl. Phys. B 337, 509 (1990)) gives a complete split into quark spin, quark orbital angular momentum (OAM), gluon spin and gluon OAM, but only in a fixed gauge; the Ji decomposition (Phys. Rev. Lett. 78, 610 (1997)) is manifestly gauge invariant but leaves the gluon total angular momentum unsplit. Chen, Lü, Sun, Wang and Goldman (Phys. Rev. Lett. 100, 232002 (2008)) proposed to split the gauge potential itself, in QED simply

A⃗=A⃗pure+A⃗phys,∇⃗×A⃗pure=0,∇⃗⋅A⃗phys=0,\vec{A}=\vec{A}_{\mathrm{pure}}+\vec{A}_{\mathrm{phys}},\qquad \vec{\nabla}\times\vec{A}_{\mathrm{pure}}=0,\quad \vec{\nabla}\cdot\vec{A}_{\mathrm{phys}}=0,

where A⃗pure\vec{A}_{\mathrm{pure}} is a pure gauge (it has vanishing field strength and absorbs the inhomogeneous part of a gauge transformation) and A⃗phys\vec{A}_{\mathrm{phys}} is gauge invariant; in QCD the curl condition becomes D⃗pure×A⃗pure=0\vec{D}_{\mathrm{pure}}\times\vec{A}_{\mathrm{pure}}=0 with the pure-gauge covariant derivative D⃗pure=∇⃗−igA⃗pure\vec{D}_{\mathrm{pure}}=\vec{\nabla}-ig\vec{A}_{\mathrm{pure}}. Replacing A⃗\vec{A} by A⃗phys\vec{A}_{\mathrm{phys}} and ∇⃗\vec{\nabla} by D⃗pure\vec{D}_{\mathrm{pure}} in the canonical Jaffe–Manohar operators then yields operators for quark spin, quark OAM, gluon spin E⃗×A⃗phys\vec{E}\times\vec{A}_{\mathrm{phys}} and gluon OAM that are all separately gauge invariant, a construction now understood as a gauge-invariant extension of the Stueckelberg type (the same trick that makes a massive gauge boson gauge invariant by introducing an extra field). The gauge in which A⃗pure=0\vec{A}_{\mathrm{pure}}=0, a generalized Coulomb gauge, was called the physical gauge, and in it the new operators coincide with the familiar canonical ones. Two consistency issues were explicitly left open in the paper's discussion. First, in the instant form of dynamics the rotation and translation generators of an interacting relativistic field theory are interaction free, but the Hamiltonian and the Lorentz-boost generators K⃗\vec{K} contain interactions; the paper guaranteed only that the new operators transform properly under spatial translations and rotations, stating that their boost behavior requires carrying out canonical quantization (preferably in the physical gauge) and explicitly computing commutators with the interaction-involving boost generators as future work. Second, the authors noted that whether their constraints uniquely determine the pure-gauge/physical split is a tricky question, because in nonabelian gauge theories gauge-fixing conditions suffer from Gribov copies — topologically distinct gauge potentials satisfying the same transversality condition, first discussed by Gribov in 1978 — and argued only that supplementary conditions could be appended without spoiling gauge invariance of the operators.

Problem Statement

For the gauge-invariant canonical decomposition of Chen et al. — obtained from the Jaffe–Manohar operators by the substitution A⃗→A⃗phys\vec{A}\to\vec{A}_{\mathrm{phys}}, ∇⃗→D⃗pure\vec{\nabla}\to\vec{D}_{\mathrm{pure}} with the split A⃗=A⃗pure+A⃗phys\vec{A}=\vec{A}_{\mathrm{pure}}+\vec{A}_{\mathrm{phys}} fixed by D⃗pure×A⃗pure=0\vec{D}_{\mathrm{pure}}\times\vec{A}_{\mathrm{pure}}=0 and [A⃗phys,E⃗]=0[\vec{A}_{\mathrm{phys}},\vec{E}]=0 — settle, in canonically quantized QCD:

  1. (Poincaré algebra) Compute the equal-time commutators of the gauge-invariant quark and gluon momentum and angular-momentum operators with the full interaction-containing boost generators K⃗\vec{K} (and with each other), and establish whether the quark and gluon pieces satisfy the transformation law required by the Poincaré algebra — equivalently, whether gauge-invariant spin and OAM matrix elements in a moving nucleon are related to rest-frame ones in the way the Poincaré algebra dictates, with the renormalization-scale behavior of the individual composite operators treated consistently; and

  2. (global uniqueness) Determine, by a global (non-perturbative) analysis of the defining nonlinear constraints including the Gribov-copy topology, whether these constraints fix the pure-gauge/physical split uniquely in the nonabelian case: identify what supplementary or boundary conditions are required, whether topologically distinct solutions exist (and if so, whether the resulting operators or their matrix elements depend on the chosen representative), and whether the operator content of the decomposition is invariant under this choice.

These two facets jointly answer the underlying source question: whether the gauge-invariant canonical quark/gluon spin and orbital-angular-momentum operators are algebraically consistent, unique-enough operators of the theory, and hence physically meaningful decompositions of the nucleon spin, or whether — as critics claim — the construction fails the Poincaré algebra at the quantum level or is rendered ambiguous by the Stueckelberg-fixing topology. Refuting one facet settles that facet, but does not settle the other.

References

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