[Paper Review] The Stability of the Vacuum Polarization Surrounding a Charged Particle

F. J. Himpsel|arXiv (Cornell University)|Dec 27, 2015

Atomic and Molecular Physics5 references3 citations

TL;DR

This paper investigates the local force balance in vacuum polarization around a charged particle using quantum field theory, extending classical force balance results to the Dirac sea. It analytically demonstrates that force densities from polarization charges and quantum degeneracy are balanced for all filled j-shells, with renormalized forces remaining in equilibrium to leading order in the fine structure constant α.

ABSTRACT

The internal stability of the electron has been debated for a century at both the classical and the quantum level. Recently, a local force density balance was established for the 1s electron in the H atom, based on the energy-momentum tensor of the classical Dirac field. This methodology is now extended to quantum fields by considering the force densities acting on the vacuum polarization induced by a point charge. Such a model is applicable to any charged particle at large distances, since the only vestige of its internal structure is the electric Coulomb field together with the vacuum polarization induced by it. While the polarization charge density is attracted to the point charge, it is kept from collapsing by repulsive forces due to confinement and degeneracy. It is shown analytically that the corresponding force densities are balanced for every filled shell of mj states at a given angular momentum j. The force densities are then summed over all single-electron states in the Dirac sea and renormalized by subtracting singular terms. In leading order of alpha, the force densities remain balanced. This result establishes a local force balance for a prototypical manybody system.

Motivation & Objective

• To establish local force balance in the vacuum polarization induced by a point charge using quantum field theory.
• To extend classical force balance results from the 1s electron in hydrogen to the full Dirac sea of quantum fields.
• To investigate whether polarization charges remain stable against collapse due to quantum confinement and degeneracy effects.
• To demonstrate force balance across all filled mj subshells at a given angular momentum j.
• To validate the stability of the vacuum polarization structure through renormalized force density summation.

Proposed method

• Uses the energy-momentum tensor of the quantum Dirac field to compute force densities acting on vacuum polarization.
• Analyzes force balance for each filled subshell of mj states at a given j, ensuring local equilibrium.
• Sums force densities over all single-electron states in the Dirac sea to assess total force balance.
• Applies renormalization by subtracting singular terms to handle divergences in quantum field theory.
• Performs analytical calculations in leading order of the fine structure constant α to assess stability.
• Incorporates confinement and degeneracy effects as repulsive forces counteracting attraction of polarization charge.

Experimental results

Research questions

• RQ1 Is the vacuum polarization around a charged particle in local force equilibrium at the quantum level?
• RQ2 How do quantum degeneracy and confinement effects prevent collapse of the polarization cloud?
• RQ3 Does the force balance persist across all filled j-subshells in the Dirac sea?
• RQ4 Can the divergent force densities in the vacuum polarization be renormalized while preserving balance?
• RQ5 What is the leading-order behavior of the force balance in the fine structure constant α?

Key findings

• Force densities from the polarization charge and quantum degeneracy are balanced for every filled j-shell.
• The force balance is maintained after summing over all single-electron states in the Dirac sea.
• After renormalization by subtraction of singular terms, the total force density remains balanced to leading order in α.
• The stability of the vacuum polarization is analytically confirmed for a prototypical many-body quantum system.
• The results extend classical force balance results to quantum fields, validating long-standing theoretical expectations.
• The methodology provides a consistent framework for analyzing local stability in quantum vacuum structures.