Solutions for the Landau Problem Using Symplectic Representations of the Galilei Group

Abstract

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

The Landau Problem and non-Classicality

“…In both relativistic or non-relativistic representations, the solutions of equations in phase space, ψ(q, p) are related to Wigner function by the star-product, that is., f W (q, p) = ψ(q, p) ⋆ ψ † (q, p). This provides a fully physical interpretations for the representation, with an interesting perspective: This formalism provides a way to address gauge theories and perturbative methods in phase space context, which is not a simple task in the usual Wigner formalism [27][28][29]. The Wigner function is the main object calculated in this article, for this we use the formalism of the phase space, whose projections on the axis of the momenta or coordinates, reproduce the results of the usual quantum mechanics.…”

On scalar electromagnetism in phase space

“…Such a powerful method has been recently used to study Schwinger pair production in strong electric fields [6,7]. Particularly such a formalism has been used to construct the Galilean symmetry in phase space and its consequences, as an example, the Schrödinger equation in phase space is obtained [8][9][10][11][12][13][14]. An extension for the relativistic symmetry was accomplished in reference [9] where the Klein-Gordon and Dirac equation in the phase space was studied.…”

Hénon–Heiles interaction for hydrogen atom in phase space

“…The algebra of operators defined in H turns out to be an associative (but not commutative) algebra in Γ, given by the star product. This introduces a non-commutative algebraic structure in phase space, a result that has been explored in different ways since the paper by Wigner [24,25,[41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. A natural symplectic representation of Lie groups is introduced in Γ by considering star-operators defined as a = a W * [25,41].…”