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Register a new userIs gauge-invariant complete decomposition of the nucleon spin possible ?
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Masashi Wakamatsu
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Is gauge-invariant complete decomposition of the nucleon spin possible? Although it is a difficult theoretical question which has not reached a complete consensus yet, a general agreement now is that there are at least two physically inequivalent gauge-invariant decompositions (I) and (II) of the nucleon. %The one is a nontrivial gauge-invariant %generalization of the Jaffe-Manohar decomposition. %The other is an extension of the Ji decomposition, which allows %a gauge-invariant decomposition of the total gluon angular %momentum into the intrinsic spin and orbital parts. In these two decompositions, the intrinsic spin parts of quarks and gluons are just common. What discriminate these two decompositions are the orbital angular momentum parts. The orbital angular momenta of quarks and gluons appearing in the decomposition (I) are the so-called mechanical orbital angular momenta, while those appearing in the decomposition (II) are the generalized (gauge-invariant) canonical ones. By this reason, these decompositions are also called the mechanical and canonical decompositions of the nucleon spin, respectively. A crucially important question is which decomposition is more favorable from the observational viewpoint. The main objective of this concise review is to try to answer this question with careful consideration of recent intensive researches on this problem.
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The question whether the total gluon angular momentum in the nucleon can be decomposed into its spin and orbital parts without conflict with the gauge-invariance principle has been an object of long-lasting debate. Despite a remarkable progress achieved through the recent intensive researches, the following two issues still remains to be clarified more transparently. The first issue is to resolve the apparent conflict between the proposed gauge-invariant decomposition of the total gluon angular momentum and the textbook statement that the total angular momentum of the photon cannot be gauge-invariantly decomposed into its spin and orbital parts. We show that this problem is also inseparably connected with the uniqueness or non-uniqueness problem of the nucleon spin decomposition. The second practically more important issue is that, among the two physically inequivalent decompositions of the nucleon spin, i.e. the canonical type decomposition and the mechanical type decomposition, which can we say is more physical or closer to direct observation ? In the present paper, we try to answer both these questions as clearly as possible.
In almost all the past analyses of the decomposition of the nucleon spin into its constituents, surface terms are simply assumed to vanish and not to affect the integrated sum rule of the nucleon spin. However, several authors claim that neglect of surface terms is not necessarily justified, especially owing to possible nontrivial topological configuration of the gluon field in the QCD vacuum. There also exist some arguments indicating that the nontrivial gluon topology would bring about a delta-function type singularity at zero Bjorken variable into the longitudinally polarized gluon distribution function, thereby invalidating a naive partonic sum rule for the total nucleon spin. In the present paper, we carefully examine the role of surface terms in the nucleon spin decomposition problem. We shall argue that surface terms do not prevent us from obtaining a physically meaningful decomposition of the nucleon spin. In particular, we demonstrate that nontrivial topology of the gluon field would not bring about a delta-function type singularity into the longitudinally polarized gluon distribution functions. We also make some critical comments on the recent analyses of the role of surface terms in the density level decomposition of the total nucleon angular momentum as well as that of the total photon angular momentum.
The gauge dependence in the anomalous dimension of the gauge-invariant-canonical-energy-momentum tensor for proton is studied by the background field method. The naive calculation shows the problem, the absence of the counter term in the gluonic sectors. The analysis shows that the result [Chen et al., Phys. Rev. Lett. 103, 062001 (2009)] is derived from the background field method after we introduced a trick to avoid the problem except for the gluon-to-gluon sector; it is gauge dependent. The possible reason of this gauge-dependent result comes from the nontrivial treatment of the condition $F^{mu u}_{pure}=0$ at a higher order. This result shows that one needs a further improvement in treating this condition with a covariant way at a higher order by the background field method. In particular, we have to focus on two checkpoints, the gauge independence and zero eigenvalue in the anomalous-dimension matrix, in order to test the validity of the gauge-invariant-canonical-energy-momentum tensor.
The traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important qo{Pauli Exclusion Principle} but by the adoption of the complex standard relativistic quantum field theory. In a recent paper (Ref. [1]) we presented a proof of the spin-statistics problem in the nonrelativistic approximation on the basis of the qo{Conformal Quantum Geometrodynamics}. In the present paper, by the same theory the proof of the Spin-Statistics Theorem is extended to the relativistic domain in the general scenario of curved spacetime. The relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. No relativistic quantum field operators are used and the particle exchange properties are drawn from the conservation of the intrinsic helicity of elementary particles. It is therefore this property, not considered in the Standard Quantum Mechanics, which determines the correct spin-statistics connection observed in Nature [1]. The present proof of the Spin-Statistics Theorem is simpler than the one presented in Ref. [1], because it is based on symmetry group considerations only, without having recourse to frames attached to the particles.
There are different operators of quark and gluon momenta, orbital angular momenta, and gluon spin in the nucleon structure study. The precise meaning of these operators are studied based on gauge invariance, Lorentz covariance and canonical quantization rule. The advantage and disadvantage of different definitions are analyzed. A gauge invariant canonical decomposition of the total momentum and angular momentum into quark and gluon parts is suggested based on the decomposition of the gauge potential into gauge invariant (covariant) physical part and gauge dependent pure gauge part. Challenges to this proposal are answered. keywords{Physical and pure gauge potentials; Gauge invariant canonical quark and gluon momenta, orbital angular momenta and spins; Homogeneous and non-homogeneous Lorentz transformations; Gauge invariant decomposition and gauge invariant extension; Classical and quantum measurements.
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