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 Poincar\'e 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 classical model. For massive spinning particles in $d=3,4$, the world sheets are shown to be cylinders. The radius of cylinder is fixed by representation. In higher dimensions, particle's world sheet turn out to be a toroidal cylinder $\mathbb{R}\times \mathbb{T}^D$, $D=[(d-1)/2]$. Proceeding 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.