The action of the O (2) non-linear sigma model on a lattice with a finite chemical pot ential that we study here is given by where x is the lattice site on a three dimensional cubic lattice, α = 1 , 2 represent the spatial directions and α = t represents the temporal direction. We will use L to represent the spatial size and L$_{t}$ the temporal size and assume periodic boundary conditions. The constant β plays the role of the coupling. The chemical potential µ is introduced in the standard way and couples to the conserved charge of the global O (2) symmetry [26]. When µ = 0 the action becomes complex and Monte Carlo algorithms to generate configurations [ θ ] that contribute to the partition function suffer from a sign problem. In particular the Wolff cluster algorithm [7] is no longer useful at non-zero chemical potential. Hence the phase diagram of the model in the ( β, µ ) plane remains unexplored. It is possible to avoid the sign problem if we rewrite the partition function in the world-line representation [9]. Using the identity where I$_{k}$ is the modified Bessel function of the first kind, on each bond ( x, α ) , and performing the angular integration over θ$_{x}$ the partition function can be rewritten as where the bond variables k$_{x,α}$ describe “world-lines” or “current” of particles moving fr om lattice site x to the site x + ˆ α and take integer values. A configuration of these bond variab les, denoted by [ k ] , is thus a world-line configuration. The global U (1) symmetry of the model is manifest 4 Z = ∑ [ k ] ∏ x { I$_{k}$$_{x,α}$ ( β ) e µδ$_{α,t}$k$_{x,α}$ } δ ( ∑ α ( k$_{x,α}$ − k$_{x}$$_{−}$$_{α,α}$ ) ) , (4) exp { cos θ } = ∞ ∑ k = −∞ I$_{k}$ ( β ) e $^{ikθ}$, (3) Z = ∫ [ dθ$_{x}$ ]e − $^{S}$, (2) S = − β ∑ x,α { cos( θ$_{x}$ − θ$_{x}$$_{+}$$_{α}$ − iµδ$_{α,t}$ ) } , (1) II. MODEL AND OBSERVABLES the effective quantum mechanical description of section IV. In section VII we discuss the phase diagram of the O (2) model which emerges from our work.