A Journey of Seeking Pressure and Forces in the Nucleon

Abstract

Momentum current density (MCD) $T^{ij}$ is a general physics concept describing the momentum conservation through momentum flow generated from both the kinetic motion of particles and the interacting forces among them. It has been suggested by M. Polyakov et al. that the MCD in the nucleon, characterized by the form factor $C/D$ of the QCD energy-momentum tensor, can be interpreted as the pressure and shear forces between adjacent parts of the system because the nucleon interior approximates a continuous medium. While intuitively appealing, we find that the interpretation is hard to justify from a detailed examination of the physical mechanisms for the momentum flow in QCD. After reviewing through a broad range of classical and quantum systems, we find that while thermal and/or quantum average of isotropic motion contributes to kinetic MCD a pressure term proportional to $\delta^{ij}$, when there is an anisotropic motion, the pressure cannot simply be identified from the MCD tensor. Furthermore, kinetic pressure cannot be considered as the surface force between adjacent parts of a system. More importantly, at the scale of the nucleon dimension, the color forces among quarks and gluons is by no means short-ranged as in a continuous medium, and the resulting interaction MCD cannot be interpreted as normal or shear ``stress'' force, although an isotropic term from the QCD trace anomaly may be interpreted as a ``vacuum pressure.'' Following our previous study of force densities through divergences of kinetic MCDs, we affirm that the vacuum pressure term provides a confining potential on the quarks through color Lorentz forces.