World sheets of spinning particles

Abstract

The classical spinning particles are considered such that quantization of classical model leads to an irreducible massive representation of the Poincaré group. The class of gauge equivalent classical particle world lines is shown to form a [(d + 1)/2]-dimensional world sheet in d-dimensional Minkowski space, irrespectively to any specifics of the classical model. For massive spinning particles in d = 3, 4, the world sheets are shown to be circular cylinders. The radius of cylinder is fixed by representation. In higher dimensions, particle's world sheet turns out to be a toroidal cylinderProceeding from the fact that the world lines of irreducible classical spinning particles are cylindrical curves, while all the lines are gauge equivalent on the same world sheet, we suggest a method to deduce the classical equations of motion for particles and also to find their gauge symmetries. In d = 3 Minkowski space, the spinning particle path is defined by a single fourth-order differential equation having two zero-order gauge symmetries. The equation defines particle's path in Minkowski space, and it does not involve auxiliary variables. A special case is also considered of cylindric null curves, which are defined by a different system of equations. It is shown that the cylindric null curves also correspond to irreducible massive spinning particles. For the higher-derivative equation of motion of the irreducible massive spinning particle, we deduce the equivalent second-order formulation involving an auxiliary variable. The second-order formulation agrees with a previously known spinning particle model.

Variational principle for cylindrical curves and dynamics of spinning particles in $d=3$ Minkowski space

“…Once the classical trajectory of the particle lies on the cylinder (2), the linear momentum p and total angular momentum J of the particle are determined by the formula from the paper [15],…”

“…The geometry of irreducible classical paths has been studied in [15]. In this work, it has been shown that the gauge equivalence class of particle's paths necessarily forms a cylindrical surface in Minkowski space called the world sheet.…”

“…The cylinder position is determined by the particle momentum and total angular momentum, while the representation defines the cylinder radius. The geometrical equations describing the motion of the particle along the world sheet are obtained for space-time dimension d = 3, 4 in the papers [15], [16]. These equations involve higher derivatives, and they are non-Lagrangian.…”

Geometrical model of massive spinning particle in four-dimensional Minkowski space

“…In the paper [10], the concept of the world sheet is proposed to describe massive particles with spin. The idea of this concept is that the classical trajectories of irreducible particle lie on a certain cylindrical surface in Minkowski space.…”

“…These equations are non-Lagrangian and involve higher derivatives. The geometrical model of massive spinning particle traveling lightlike and isotropic paths in three-dimensional Minkowski space was developed in the original work [10]. In four-dimensional Minkowski space, the world sheet is considered in [11].…”