The concept of noncommutative space originates from the Wigner formulation of quantum mechanics in phase space in 1932. In parallel, Heisenberg was the first to propose noncommutative commutation relations between the components of the position operator. Such a possibility was mathematically described by Snyder, studying representations of the (4+1)-De Sitter group. A synthesis of such works is the concept of noncommutative geometry, established with the Moyal product, that arises in the Wigner formalism. In addition, such noncommutativity is found in some limits of string theory, giving rise to the possibility of measurements of spatial noncommutativity in high-energy physics. In this work, we present a pedagogical review of physical theories in noncommutative spaces, from a historical perspective. We emphasize the theory of symmetry group representations in phase space, and point out two important, but not so well-known, aspects: (a) the notion of amplitude of probability and the representation of the Schrödinger equation in phase space (usually, the phase space representation of quantum mechanics is derived from the density matrix and the Liouville-von Neumann equation); and (b) a work of Dirac in 1930, where a formulation of quantum physics was introduced by the first time.
quantum mechanics; Wigner functions; noncommutative spaces
