[Paper Review] A Geometrical Description of Spinor Fields

Roman Sverdlov|ArXiv.org|Feb 13, 2008

Algebraic and Geometric Analysis6 references3 citations

TL;DR

This paper presents a geometric, non-Grassmannian model of spinor fields using four orthonormal vector fields (A, B, C, D) and two real scalar fields (ϕ, χ), redefining spinor dynamics via Lagrangian coupling to eliminate fermionic 'weirdness' and enable causal set compatibility. The model interprets spin as rotational motion in a dynamically defined local frame, with fermionic behavior emerging from vector field interactions and redefined transformation rules.

ABSTRACT

The goal of this paper is to present the way to define fermionic fields and their Lagrangians in terms of three orthogonal vector fields of norm 1 together with two real valued scalar fields. This paper is based on a toy model where there are no Grassmann variables.

Motivation & Objective

• To develop a geometric, non-Grassmannian formulation of spinor fields that avoids the abstract nature of Grassmann variables.
• To provide a framework compatible with the causal set approach to quantum gravity, where spacetime structure emerges from fermionic field dynamics.
• To replace the abstract notion of spinors with tangible geometric objects—orthonormal vector fields and real scalars—offering a more intuitive physical interpretation.
• To demonstrate that spinor behavior, including the 360° rotation sign flip, can emerge from vector field dynamics when transformation rules are redefined via phase factors.
• To establish a mechanism by which local spacetime structure (manifold-like topology) can emerge from fermionic field configurations, supporting causal set dynamics.

Proposed method

• The fermionic field is redefined using four orthonormal vector fields (A, B, C, D) and two real scalar fields (ϕ, χ), with D determined by the other three via orthogonality and norm constraints.
• Lagrangian terms are constructed using directional derivatives of the vector fields (e.g., A^μ ∂_ν B_μ), encoding rotational and boost-like dynamics.
• Lagrange multipliers enforce orthonormality and unit norm constraints (A^2 = 1, B^2 = C^2 = D^2 = -1, and all cross terms zero).
• The transformation rules for vectors are redefined by introducing a phase factor (e^{iθ/2}) to reproduce the spin-1/2 behavior under 360° rotations.
• The model interprets spin as rotation in the B-C plane around the D-axis, with A defining the rest frame, and uses vector field coupling to simulate spin and orbital motion.
• Inner products and dynamics are defined purely through vector and scalar fields, eliminating the need for Grassmann variables or spinor algebra.

Experimental results

Research questions

• RQ1 Can spinor fields be fully described using only real vector and scalar fields, without relying on Grassmann variables or spinor representations?
• RQ2 How can the 360° rotation sign flip of spin-1/2 particles be reproduced using only vector fields and modified transformation rules?
• RQ3 Can a geometric model of spinors be constructed that naturally supports the causal set approach to quantum gravity by allowing spacetime topology to emerge from fermionic fields?
• RQ4 What is the physical interpretation of the vector fields A, B, C, D in terms of spin and motion, and how do they collectively encode spinor degrees of freedom?
• RQ5 How can the Lagrangian be structured so that vector field derivatives encode spin, orbital motion, and Lorentz invariance in a geometrically intuitive way?

Key findings

• The model successfully encodes all 8 real degrees of freedom of a Dirac spinor using four orthonormal vector fields (A, B, C, D) and two real scalars (ϕ, χ), with D uniquely determined by the others.
• The Lagrangian terms A^μ ∂_ν B_μ, A^μ ∂_ν C_μ, and B^μ ∂_ν A_μ encode spin, orbital motion, and boost dynamics, respectively, with the first term representing rotational motion in the B-C plane.
• The 360° rotation sign flip of spin-1/2 particles is reproduced not by the vector fields themselves, but by redefining the transformation law to include a phase factor e^{iθ/2}, effectively embedding SU(2)×U(1) symmetry into the vector dynamics.
• The model provides a geometric interpretation of spin as rotation in a plane defined by B^μ and C^μ, with the axis of rotation given by D^μ, offering a tangible visualization of spinor behavior.
• The vector fields A^μ, B^μ, C^μ, D^μ collectively define a local reference frame at each spacetime point, suggesting that spacetime structure can emerge from fermionic field configurations in causal set theory.
• The absence of Grassmann variables and the use of real fields make the model suitable for discrete spacetime formulations, particularly in the causal set approach, where manifold-like structure arises from field dynamics.