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