QUANTUM HOLONOMY AND LINK INVARIANTS

It is shown that, in a non-Abelian quantum field theory without an anomaly and broken symmetry, the set of all matrix-valued quantum holonomies PSI-[gamma] = [P exp(i-closed-integral-gamma-A dx)] for closed contours gamma form a commutative semigroup, whereas [P exp(i integral-alpha-A dx)] = 0 for every open path alpha. The eigenvalues PHI-[gamma] of PSI-[gamma] are classified according to the irreducible representations of the gauge group. In an irreducible representation rho, Tr(PSI-[gamma]) = PHI-[gamma]Tr(1-rho) is a Wilson loop. The equation solves a puzzle in the relation between link invariants and Wilson loops in the Chern-Simons theory in three dimensions when the gauge group is SU(N/N), and provides useful insight in understanding nonperturbative quantum chromodynamics as a string theory.