  The jellium is a model, introduced by Wigner (1934), for a gas of electrons
moving in a uniform neutralizing background of positive charge. Wigner
suggested that the repulsion between electrons might lead to a broken
translational symmetry. For classical one-dimensional systems this fact was
proven by Kunz (1974), while in the quantum setting, Brascamp and Lieb (1975)
proved translation symmetry breaking at low densities. Here, we prove
translation symmetry breaking for the quantum one-dimensional jellium at all
densities.
