Polyakov’s spin factor and new algorithms for efficient perturbative computations in QCD

Polyakov’s spin factor enters as a weight in the path-integral description of particlelike modes propagating in Euclidean space-times, accounting for particle spin. The Fock-Feynman-Schwinger path integral applied to QCD accomodates Polyakov’s spin factor in a natural manner while, at the same time, it identifies Wilson line (loop) operators as sole agents of interaction dynamics among matter and gauge field quanta. A direct application of such a separation between spin content and dynamics is the emergence of master expressions for the perturbative series involving either open or closed fermionic lines which provide new, comprehensive approaches to perturbative QCD. 1. Introductory remarks. The Fock[1]-Feynman[2]-Schwinger[3] path integral, also known as worldline formalism [4-7], constitutes a tool by which one is able to study a quantum field theoretical system in terms of the propagation of its particle quanta. Polyakov’s path integral [8], on the other hand, amounts to a direct formulation regarding the (Euclidean) space-time propagation of particle-like entities. Its novel ingredient is the so-called spin factor which enters the path integral as a weight accounting for the spin of a particle that traverses a given closed contour. Polyakov’s scheme is distinquished by its geometrical mode of description and serves, via its extension to two-dimensional worldsheets, as a prototype for discussing the propagation of quantum strings. Inspired by Polyakov’s work we have been able to reformulate [10] the Fock-FeynmanSchwinger path integral for gauge field systems with spin-1/2 matter fields so that the spin factor, associated with the propagation of the latter, explicitly makes its entrance. At the same time, the dynamics operating on the matter particles is carried by Wilson line (loop) operators. Let us display relevant expressions corresponding to the ‘effective’ action functionalW[A] and the full fermionic propagator G(x, y|A) in the presence of background gauge fields. They read, respectively, as follows: W[A] = − ∫ ∞ 0 dT T e 2 ∫ x(0)=x(T ) Dx(τ)e 1 4 ∫ T 0 dτẋ2(τ)trPexp { i 4 ∫ T 0 dτ σ · ω }