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.

中文翻译：

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朗道问题和非经典性

探索扩展伽利略群的概念[公式：见正文]。辛流形中的场论表示已经与 Wigner 函数的方法相关联。表示是写在五个维度的德西特时空的光锥中。构造一个希尔伯特空间，赋予辛结构，用作[公式：见正文]的李代数的表示空间。这种表示产生了相空间中波函数的自旋0薛定谔（Klein-Gordon-like）方程，使得因变量具有位置和线性动量的内容。这是保形理论的一个特殊例子，波函数通过 Moyal 积与 Wigner 函数相关联。我们在相空间中以显式协变形式构造泡利-薛定谔（类狄拉克）方程。此外，我们分析了相空间中自旋 1/2 粒子的规范对称性，并展示了在这种情况下如何实现最小耦合。我们应用于外场中的电子问题，并恢复了非相对论的朗道能级。最后，我们研究了与系统非经典性相关的负性参数。