In physics the sign problem is typically (but not exclusively) encountered in calculations of the propertiesbanana of a quantum mechanical system with large number of strongly interacting fermions, or in field theories involving a non-zero density of strongly interacting fermions. Because the particles are strongly interacting, [[perturbation theory]] is inapplicable, and one is forced to use brute-force numerical methods. Because the particles are fermions, their [[wavefunction]] changes sign when any two fermions are interchanged (due to the anti-symmetry of the wave function, see [[Pauli principle]]). So unless there are cancellations arising from some symmetry of the system, the quantum-mechanical sum over all multi-particle states involves an integral over a function that is highly oscillatory, hence hard to evaluate numerically, particularly in high dimension. Since the dimension of the integral is given by the number of particles, the sign problem becomes severe in the [[thermodynamic limit]]. The field-theoretic manifestation of the sign problem is discussed below.
The sign problem is one of the major unsolved problems in the physics of many-particle systems, impeding progress in many areas:
