in the local current conservation relation represented by the delta function. In other words any particle that comes into the site must leave the site due to current conservation. In this world-line formulation the partition function is a sum over explicitly positive terms even in the presence of µ . Details of the “worm algorithm” that we have developed to up date the world-line configuration [ k ] is described in appendix A. We focus on four observables in this work: The average particle number is then given by 〈 N 〉 = ρL $^{2}$. Note that κ = L$_{t}$/L $^{2}$〈 N $^{2}$〉 . The superfluid density is known to be ρ$_{s}$ = 1 /L$_{t}$ 〈 W $^{2}$〉 where W is the spatial winding number of particles [27]. We define 〈 N$_{s}$ 〉 = L $^{2}$ρ$_{s}$ as the number of particles that are in the superfluid phase in a finite system. The first three observables are “diagonal” observables and c an be measured on each world-line configuration and then averaged. The condensate susceptibi lity χ on the other hand is a “non- diagonal” observable, but it can be related to the size of eac h worm update as discussed in ap- pendix A. We discuss some tests of the algorithm in appendix B. In particular we have been able reproduce earlier results of the O (2) non-linear sigma model at µ = 0 . One of these results is the estimate of the critical coupling β$_{c}$ = 0 . 45421 [28]. For β > β$_{c}$ the O (2) symmetry is spon- taneously broken, while for β < β$_{c}$ the model is in the symmetric phase. Our tests also show agreement with exact calculations on a 2 × 2 lattice at non-zero µ . 4. The condensate susceptibility χ : 3. The superfluid density (or particle current susceptibili ty) ρ$_{s}$ : 2. The particle number susceptibility κ : 1. The average particle density ρ : ρ = 1 L $_{2}$〈 ∑ x ∈ timeslice k$_{x,t}$ 〉 (5) κ = 1 L $^{2}$L$_{t}$ 〈 ( ∑ x k$_{x,t}$ ) 2 〉 (6) ρ$_{s}$ = 1 2 L $^{2}$L$_{t}$ 〈 ∑ α =1 , $_{2}$( ∑ x k$_{x,α}$ ) 2 〉 (7) χ = ∑ y 〈 e $^{iθ$_{x}$}$e − $^{iθ$_{y}$}$〉 (8) 5