[Paper Review] "Field-shell" of the self-interacting quantum electron

Peter Leifer, Taha Massalha|arXiv (Cornell University)|Aug 23, 2010

Quantum Mechanics and Applications18 references3 citations

TL;DR

This paper proposes a geometric model of the self-interacting quantum electron as a cyclic motion in complex projective space $CP(3)$, where the electron's field-shell arises from affine parallel transport of energy-momentum under local conservation laws. The model derives quasi-linear PDEs for the field-shell from SU(4) coset dynamics, linking curvature of $CP(3)$ to emergent electromagnetic fields and offering a dynamical origin for electron mass through spin-charge self-interaction.

ABSTRACT

Self-interacting dynamics of non-local Dirac's electron has been proposed. This dynamics was revealed by the projective representation of operators corresponding to spin/charge degrees of freedom. Energy-momentum field is described by the system of quasi-linear ``field-shell" PDE's following from the conservation law expressed by the affine parallel transport in $CP(3)$ \cite{Le1}. We discuss here solutions of these equations in the connection with the following problems: curvature of $CP(3)$ as a potential source of electromagnetic fields and the self-consistent problem of the electron mass.

Motivation & Objective

• To develop a non-perturbative, deterministic model of the electron based on intrinsic self-interaction rather than external potentials.
• To address the foundational issue of electron mass by deriving it dynamically from the geometry of $CP(3)$ and spin/charge degrees of freedom.
• To replace the standard quantum mechanical framework with a state-dependent dynamical space-time (DST) arising from unitary group actions on $CP(3)$.
• To formulate a self-consistent field-theoretic description of the electron using quasi-linear PDEs derived from conservation laws in projective Hilbert space.
• To explore the emergence of classical-like behavior (e.g., field-shell structure) from quantum dynamics via affine parallel transport in $CP(3)$.

Proposed method

• Uses projective representation of $SU(4)$ generators to model spin and charge degrees of freedom in $CP(3)$, the coset space $SU(4)/S[U(1)\times U(3)]$.
• Applies affine parallel transport in $CP(3)$ to express energy-momentum conservation, leading to a system of quasi-linear PDEs for the field-shell.
• Models perturbations of generalized coherent states (GCS) via coset transformations analogous to Foldy-Wouthuysen transformations, inducing state-dependent local dynamical space-time (DST).
• Derives equations of motion for quantum accelerations and angular velocities using a linearized evolution operator $\hat{L}$ acting on spinor components $\eta^0, \eta^1$.
• Introduces complex coordinate $\pi = e^{-i\phi}\tan(\theta/2)$ to parametrize geodesics in $CP(3)$, enabling description of local dynamics via $\delta\theta/\delta\tau$ and $\delta\phi/\delta\tau$.
• Relates observable frequencies $F_1, F_2$ to physical quantities like $\Re(-\beta/\alpha)$ and $\Im(-\alpha + \beta/\alpha)$, linking dynamics to energy and phase evolution.

Experimental results

Research questions

• RQ1 How can the electron’s field-shell structure emerge from a geometric conservation law in $CP(3)$ without external potentials?
• RQ2 What is the dynamical origin of the electron’s mass in a self-interacting quantum model based on $CP(3)$ curvature?
• RQ3 Can the electromagnetic field be generated intrinsically from the geometry of the state space $CP(3)$ via spin/charge self-interaction?
• RQ4 How do the quasi-linear PDEs for the field-shell relate to the attractor dynamics of the characteristic equations in the state space?
• RQ5 What role does the affine parallel transport in $CP(3)$ play in defining the electron’s energy-momentum distribution and its self-consistent evolution?

Key findings

• The field-shell of the self-interacting electron is described by a system of quasi-linear PDEs derived from energy-momentum conservation via affine parallel transport in $CP(3)$.
• The curvature of $CP(3)$ acts as a potential source for electromagnetic fields, emerging from the non-trivial geometry of the state space.
• The electron’s mass is proposed to arise dynamically from the self-interaction of spin and charge degrees of freedom, encoded in the attractor structure of the characteristic equations.
• The model predicts that the frequencies $F_1 = \delta\theta/\delta\tau$ and $F_2 = \delta\phi/\delta\tau$ are proportional to $c/\hbar$ times real and imaginary parts of $-\beta/\alpha$ or $-\alpha + \beta/\alpha$, linking them to physical observables.
• The system of 6 real non-homogeneous equations (82)–(87) provides a closed-form expression for quantum accelerations and angular velocities as functions of spinor components, GCS coordinates, and dynamical frequencies.
• Numerical analysis of attractors in the characteristic equations is proposed as a path to solving the self-consistent problem of electron mass and field-shell structure.