[Paper Review] Polydimensional Supersymmetric Principles

William M. Pezzaglia|ArXiv.org|Sep 22, 1999

Algebraic and Geometric Analysis7 references3 citations

TL;DR

This paper proposes a polydimensional supersymmetric framework extending Clifford calculus, where geometric elements (vectors, bivectors) have independent coordinates. It introduces a classical action minimizing path length plus swept area, resolving the 50-year-old problem of deriving the correct Lagrangian for Papapetrou equations in curved spacetime with torsion.

ABSTRACT

Systems of equations are invariant under "polydimensional transformations" which reshuffle the geometry such that what is a line or a plane is dependent upon the frame of reference. This leads us to propose an extension of Clifford calculus in which each geometric element (vector, bivector) has its own coordinate. A new classical action principle is proposed in which particles take paths which minimize the distance traveled plus area swept out by the spin. This leads to a solution of the 50 year old conundrum of `what is the correct Lagrangian' in which to derive the Papapetrou equations of motion for spinning particles in curved space (including torsion). Based on talk given at: 5th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999.

Motivation & Objective

• To resolve the longstanding ambiguity in the correct Lagrangian formulation for spinning particles in curved spacetime.
• To extend Clifford calculus by assigning independent coordinates to geometric elements such as vectors and bivectors.
• To develop a classical action principle that unifies geodesic motion and spin dynamics via minimization of path length and area swept by spin.
• To provide a consistent derivation of the Papapetrou equations, including effects of torsion, from a variational principle.
• To establish a framework for supersymmetric principles in a polydimensional geometric setting where geometric roles depend on the reference frame.

Proposed method

• Introduces polydimensional transformations that reassign geometric roles (e.g., line vs. plane) based on the observer's frame.
• Extends Clifford algebra by assigning distinct coordinates to different-grade geometric elements (e.g., vectors and bivectors).
• Proposes a new classical action functional that combines proper time (path length) and the area swept by the spin vector during motion.
• Applies variational calculus to this action to derive equations of motion for spinning particles in curved spacetime.
• Incorporates torsion into the geometric framework by allowing non-symmetric connections in the spacetime manifold.
• Derives the Papapetrou equations as the Euler-Lagrange equations from the proposed action, validating its consistency with known results.

Experimental results

Research questions

• RQ1 What is the correct variational principle that yields the Papapetrou equations for spinning particles in curved spacetime?
• RQ2 How can geometric elements of different grades (e.g., vectors and bivectors) be consistently assigned independent coordinates in a unified framework?
• RQ3 In what way do polydimensional transformations alter the interpretation of geometric objects like lines and planes across reference frames?
• RQ4 Can a single action principle unify geodesic motion and spin precession in curved spacetime with torsion?
• RQ5 How does the inclusion of spin-induced area in the action lead to a natural derivation of the Papapetrou equations?

Key findings

• The proposed action principle successfully derives the Papapetrou equations for spinning particles in curved spacetime, including torsion effects.
• The framework resolves the long-standing ambiguity in the choice of Lagrangian for spinning particles by unifying path and spin-area minimization.
• Polydimensional transformations allow geometric objects to be frame-dependent, enabling a richer, observer-dependent geometric structure.
• The extension of Clifford calculus to include independent coordinates for different-grade elements enables a more flexible and physically consistent description of spin and motion.
• The variational derivation confirms that the Papapetrou equations emerge naturally from the new action, validating its physical consistency.
• The model provides a classical foundation for supersymmetric principles in a geometric setting where spin and trajectory are dynamically coupled through a unified action.