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We discuss the uniqueness or non-uniqueness problem of the decomposition of the gluon field into the physical and pure-gauge components, which is the basis of the recently proposed two physically inequivalent gauge-invariant decompositions of the nucleon spin. It is crucialy important to recognize the fact that the standard gauge fixing procedure is essentially a process of projecting out the physical components of the massless gauge field. A complexity of the nonabelian gauge theory as compared with the abelian case is that a closed expression for the physical component can be given only with use of the non-local Wilson line, which is generally path-dependent. It is known that, by choosing an infinitely long straight-line path in space and time, the direction of which is characterized by a constant 4-vector $n^mu$, one can cover a class of gauge called the general axial gauge, containing three popular gauges, i.e. the temporal, the light-cone, and the spatial axial gauge. Within this general axial gauge, we have calculated the 1-loop evolution matrix for the quark and gluon longitudinal spins in the nucleon. We found that the final answer is exactly the same independently of the choices of $n^mu$, which amounts to proving the gauge-independence and path-independence simultaneously, although within a restricted class of gauges and paths. By drawing on all of these findings together with well-established knowledge from gauge theories, we argue against the rapidly spreading view in the community that there are infinitely many decompositions of the nucleon spin.
We argue against the rapidly spreading idea of gauge-invariant-extension (GIE) approach in the nucleon spin decomposition problem, which implies the existence of infinitely many gauge-invariant decomposition of the nucleon spin.
The recent controversy on the nucleon spin decomposition problem is critically overviewed. We argue that there exist two and only two physically inequivalent gauge-invariant decompositions of the longitudinal nucleon spin, contrary to the rapidly spr
Monopole-like objects have been identified in multiple lattice studies, and there is now a significant amount of literature on their importance in phenomenology. Some analytic indications of their role, however, are still missing. The t Hooft-Polyako
A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2le lvert Xrvert <lvert V(G)rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm
A general consensus now is that there are two physically inequivalent complete decompositions of the nucleon spin, i.e. the decomposition of the canonical type and that of mechanical type. The well-known Jaffe-Manohar decomposition is of the former t