The Landau problem and nonclassicality

Abstract

Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.

Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

“…The product of two operators, AB, both defined in H , is associated to a Weyl product or star product of their phase space correspondents, i.e. [61][62][63][64][65][66][67][68][69],…”

“…Such result can be seen as the action of a star operator, a W ðq, pÞ⋆, in the function b W ðq, pÞ. This fact is important for define the star operators a W ð q, pÞ⋆, which are used to build up representations of symmetry groups in a symplectic manifold [50][51][52]64]. This gives rise, for example, to the Klein-Gordon and Dirac equations written in phase space [44,61,62,64].…”

“…This fact is important for define the star operators a W ð q, pÞ⋆, which are used to build up representations of symmetry groups in a symplectic manifold [50][51][52]64]. This gives rise, for example, to the Klein-Gordon and Dirac equations written in phase space [44,61,62,64]. The connection with Wigner function is derived, providing a physical interpretation for the representation.…”

Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

“…The product of two operators, AB, both defined in H, is associated to a Weyl product or star-product of their phase space correspondents, i.e. [41][42][43][44][45][46][47][48][49],…”

“…Such result can be seen as the action of a star-operator, a W (q, p) , in the function b W (q, p). This fact is important for define the star operators a W (q, p) , which are used to build up representations of symmetry groups in a symplectic manifold [30][31][32]44]. This gives rise, for example, to the Klein-Gordon and Dirac equations written in phase space [24,41,42,44].…”