[Paper Review] QED using the nilpotent formalism

Peter Rowlands, John P. Cullerne|ArXiv.org|Sep 14, 2001

Algebraic and Geometric Analysis3 references3 citations

TL;DR

This paper proposes that the nilpotent formalism—based on a 32-component algebra unifying spacetime and mass-charge units—automatically resolves quantum field theory divergences in QED without requiring renormalization. By encoding second quantization and supersymmetry inherently in the algebra, the formalism ensures finite physical quantities through built-in cancellations and a natural UV cutoff at the Planck scale, effectively eliminating the need for perturbative renormalization procedures.

ABSTRACT

The nilpotent formalism for the Dirac equation, outlined in previous papers,is applied to QED. It is shown that what is usually described as 'renormalization' is effectively a statement of the fact that the nilpotent formulation is automatically second quantized and constrains the field into producing finite values for fundamental quantities.

Motivation & Objective

• To investigate whether the nilpotent formalism for the Dirac equation can be extended to quantum electrodynamics (QED) without requiring renormalization.
• To resolve the issue of infinities in QED by showing that the nilpotent algebra inherently produces finite values for physical observables.
• To demonstrate that second quantization and supersymmetry emerge naturally from the algebra, removing the need for ad hoc field quantization and extra particles.
• To explain the success of renormalization in QED not as a corrective procedure, but as a consequence of the nilpotent structure’s built-in finiteness.

Proposed method

• The formalism uses a 32-part algebra combining 4-vector units (i, j, k) with quaternion units (1, i, j, k), defining a 'full product' ab = a·b + i a×b with specific multiplication rules for vector units.
• The Dirac equation is derived by factorizing the relativistic energy-momentum relation E² − p² − m² = 0 and introducing the exponential phase factor e⁻ⁱ(Et−p·r), leading to a nilpotent wavefunction ψ = (±kE ± ii p + ij m)e⁻ⁱ(Et−p·r).
• The gamma matrices are mapped to nilpotent algebra elements: γ₀ = ik, γ₁ = ii, γ₂ = ij, γ₃ = ik, γ₅ = ij, embedding the Dirac equation within the algebra.
• Fermion and antifermion states are represented as sets of four nilpotent terms, with spin reversal achieved by flipping the sign of momentum; bosons are formed as scalar products of fermion and antifermion states.
• The fermion propagator is reformulated as iSF(p) = i/(kE + iiσ·p + ijm), which combines particle and antiparticle solutions into a single expression, removing infrared divergences.
• The theory introduces a natural UV cutoff at the Planck mass, with divergences suppressed because the algebraic structure demands finite integrals when the propagator index D < 0.

Experimental results

Research questions

• RQ1 Can the nilpotent formalism for the Dirac equation be extended to QED without invoking renormalization?
• RQ2 Why do divergences in QED appear to be removable through renormalization, and is this due to a deeper algebraic structure?
• RQ3 Does the nilpotent algebra inherently encode second quantization and supersymmetry, eliminating the need for additional field-theoretic constructions?
• RQ4 How does the nilpotent formalism resolve the infrared and ultraviolet divergences in fermion and gauge boson propagators?
• RQ5 What is the physical origin of the finite values of mass and charge in this framework, and how does it relate to the Planck scale?

Key findings

• The nilpotent formalism automatically incorporates second quantization, so that quantum field operators, creation/annihilation operators, and supersymmetry emerge naturally from the algebra without additional postulates.
• The fermion propagator in the nilpotent formalism, iSF(p) = i/(kE + iiσ·p + ijm), unifies particle and antiparticle solutions and eliminates infrared divergences by construction.
• The theory ensures finite values for fundamental quantities like mass and charge at all energy scales due to the algebraic structure’s built-in finiteness, removing the need for perturbative renormalization.
• Ultraviolet divergences are suppressed because the nilpotent algebra demands that integrals over momentum space converge when the propagator index D < 0, with a natural UV cutoff at the Planck mass.
• The electroweak and QCD propagators are reformulated in the nilpotent formalism with the same structure as QED, showing consistency across gauge theories and suggesting a unified foundation.
• Renormalization is reinterpreted not as a correction but as a consequence of the algebra’s inherent finiteness, with coupling constants treated as energy-dependent scaling parameters rather than renormalizable parameters.