[Paper Review] Massless particles and the geometry of curves. Classical picture

Armen Nersessian|arXiv (Cornell University)|Nov 3, 1999

Advanced Mathematical Theories and Applications3 citations

TL;DR

This paper investigates the classical description of D-dimensional massless particles using worldline Lagrangians linear in curvature invariants. It shows that only the action 𝒮 = c∫k_N d~s yields irreducible Poincaré group representations, corresponding to curves with maximal gauge freedom and equal Nth curvatures, while other models fail to produce irreducible representations or consistent physical states.

ABSTRACT

We analyze the possibility of description of D-dimensional massless particles by the Lagrangians linear on world-line curvatures k_i, {\cal S}=\sum_{i=1}^Nc_i\int k_i d{ ilde s}. We show, that the nontrivial classical solutions of this model are given by space-like curves with zero 2N-th curvature for N\leq[(D-2)/2]. Massless spinning particles correspond to the curves with constant k_{N+a}/k_{N-a} ratio. It is shown that only the system with action {\cal S}=c\int k_N d{ ilde s} leads to irreducible representation of Poincaré group. This system has maximally possible number (N+1) of gauge degrees of freedom. Its classical solutions obey the conditions k_{N+a}=k_{N-a}, a=1,..., N-1, while first N curvatures k_i remain arbitrary. This solution is specified by coinciding N weights of the massless representation of little Lorentz group, while the remaining weights vanish.

Motivation & Objective

• To explore whether massless particles in D dimensions can be described by Lagrangians linear in worldline curvatures k_i.
• To determine which curvature-based actions yield physically consistent, irreducible representations of the Poincaré group.
• To identify the geometric conditions on curves (worldlines) that correspond to massless spinning particles and irreducible representations.
• To clarify the role of gauge degrees of freedom in curvature-based models of massless particles.
• To establish the uniqueness of the action 𝒮 = c∫k_N d~s in producing irreducible representations via geometric constraints on curvature ratios.

Proposed method

• Formalism based on Frenet-Serret equations to describe worldline geometry in D dimensions.
• Construction of a worldline action 𝒮 = ∑c_i∫k_i d~s, linear in curvature invariants k_i.
• Analysis of classical solutions under constraints derived from Poincaré invariance and gauge symmetry.
• Derivation of conditions for irreducibility of Poincaré group representations via curvature ratios and vanishing weights.
• Use of the little Lorentz group to classify massless representations and relate them to curvature symmetries.
• Identification of the unique action 𝒮 = c∫k_N d~s as the only one yielding (N+1) gauge degrees of freedom and consistent with irreducibility.

Experimental results

Research questions

• RQ1 Which curvature-based worldline actions yield irreducible representations of the Poincaré group for massless particles?
• RQ2 What geometric conditions on the worldline curve (in terms of curvatures k_i) are required for consistency with massless particle dynamics?
• RQ3 Why does the action 𝒮 = c∫k_N d~s uniquely lead to irreducible representations, while others do not?
• RQ4 How do the ratios of curvatures k_{N+a}/k_{N-a} relate to the spin content of massless particles in this framework?
• RQ5 What is the role of gauge degrees of freedom in curvature-based models, and why is (N+1) the maximal number in the unique consistent model?

Key findings

• The only action that leads to an irreducible representation of the Poincaré group is 𝒮 = c∫k_N d~s.
• Classical solutions of this action satisfy k_{N+a} = k_{N-a} for a = 1, ..., N-1, while the first N curvatures remain arbitrary.
• The system has maximally possible (N+1) gauge degrees of freedom, corresponding to the highest symmetry in the class of curvature-linear models.
• Massless spinning particles are realized as curves with constant ratio k_{N+a}/k_{N-a}, linking spin to curvature symmetry.
• The representation is irreducible because the N weights of the little Lorentz group coincide, and the remaining weights vanish.
• For N ≤ [(D-2)/2], nontrivial classical solutions exist only for curves with zero 2N-th curvature, restricting the geometric configuration.