A Journey of Seeking Pressure and Forces in the Nucleon

TL;DR

The work critically examines momentum current density in classical and quantum contexts and challenges the mechanical interpretation of the nucleon's MCD as simple pressure and surface stress. By dissecting kinetic and interaction contributions across gases, liquids, solids, and EM fields, it shows that only in certain short-range, boundary-defined systems can MCD be read as pressure; in the nucleon, long-range color forces and the QCD trace anomaly yield a more nuanced picture where the trace anomaly acts as a vacuum pressure and divergences reveal color-Lorentz forces. The nucleon analysis, using QCD EMT decompositions and gravitational form factors, identifies the trace anomaly as a dominant isotropic contributor to confinement, with gluon tensor parts providing a repulsive component. Overall, the paper argues that Polyakov’s mechanical interpretation does not hold in QCD, and emphasizes that force densities and confinement emerge from divergences of MCD rather than a simple pressure-stress interpretation, offering a refined view of internal nucleon dynamics and momentum flow.

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 $δ^{ij}$, 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.

Figures (8)

• Figure 1: Figure (a) shows the force on a molecule (red dot) located at different positions within the container, from other molecules in a sphere of radius equal to the interaction range $a$. Figure (b) shows that the force density is non-vanishing only near the container boundary. The force range $a$ is effectively taken as zero at macroscopic level.
• Figure 2: A fictitious plane inside a gas, through which the isotropic kinetic MCD represents a momentum flow transported by particles going from one side to the other. No physical force or mechanical effect is present in this process.
• Figure 3: The flow of momentum $p_x$ in the $x-y$ plane for two opposite electric charges located at $(\pm a,0,0)$, indicating the Coulomb force between them. Total (left) and interference (right) MCDs have the same mechanical effects although differed by the self-field MCDs of charges equivalent to a superpotential.
• Figure 4: Traces of 4D traceless kinetic (black), interaction (green), trace (red) and the total (blue) MCDs $T^{ii}$ (top) Cebulla:2007ei within pion fields of the Skyrme model.
• Figure 5: Traces of the momentum flows in H-atom: electron's kinetic MCD $T^{ii}_{\rm e-K}$ (black solid), interference MCD from the electron and proton fields $T^{ii}_{\rm int}$ (green solid), their self-MCDs $T^{ii}_{e+p}=T^{ij}_e+T^{ij}_p$ (brown dashed) and the total electric MCD $T^{ii}_{\rm tot-E}$ (blue solid).