The sign problem arises when ρ ρ [ σ σ ] is non-positive. This typically occurs in theories of fermions when the fermion chemical potential μ μ is nonzero, i.e. when there is a nonzero background density of fermions. If μ μ ≠ ≠ 0 , there is no particle–antiparticle symmetry, and det ( M ( μ μ , σ σ ) ) , and hence the weight ρ ρ ( σ σ ) , is in general a complex number, so Monte Carlo importance sampling cannot be used to evaluate the integral.