arXiv 1407.60222014-07-22hep-th

弯曲相空间中的非对易分析与相干态量子化

Noncommutative analysis in a curved phase-space and coherent states quantization

B. F. Rizzuti, E. M. C. Abreu, A. C. R. Mendes, M. A. Freitas, V. Nikoofard

备注：: 26 pages

摘要：

在这项工作中，我们精确地表明，2-球的曲率通过引入非对易（NC）参数在系统中引入量子特征，该参数通过运动方程自然出现。为了得到这个结果，我们使用了量子力学可以被理解为NC辛几何的事实，它将经典力学的标准描述推广为辛几何。在本文中，我们还分析了NC相空间中二维球上的自由粒子模型的动力学。此外，我们还证明了运动方程的解可以证明粒子在2球上的物理自由度运动与中心场所描述的运动之间的等效性。我们认为质点所感受到的有效力是由空间的曲率引起的。为了得到牛顿第二定律的NC修正，我们分析了经典观测量的NC Poisson代数。我们已经精确地证明了空间的曲率对平坦相空间中的自由粒子起着有效势的作用。此外，通过NC相干态量子化，我们得到了理论的格林函数。结果证实了在NC核中存在大动量的UV截止。我们还讨论了仿射联络和狄拉克括号之间的关系，因为它们分别描述了拉格朗日和哈密顿形式主义中约束表面上模型的正确演化。作为应用，我们讨论了狄拉克电子的所谓的\textit{Zitterbewegung}，由于它被假定为一种可观测效应，因此我们通过假定电子具有内部结构来追踪它的物理起源。

英文摘要：

In this work we have shown precisely that the curvature of a 2-sphere introduces quantum features in the system through the introduction of the noncommutative (NC) parameter that appeared naturally via equations of motion. To obtain this result we used the fact that quantum mechanics can be understood as a NC symplectic geometry, which generalized the standard description of classical mechanics as a symplectic geometry. In this work, we have also analyzed the dynamics of the model of a free particle over a 2-sphere in a NC phase-space. Besides, we have shown the solution of the equations of motion allows one to show the equivalence between the movement of the particle physical degrees of freedom upon a 2-sphere and the one described by a central field. We have considered the effective force felt by the particle as being caused by the curvature of the space. We have analyzed the NC Poisson algebra of classical observables in order to obtain the NC corrections to Newton's second law. We have demonstrated precisely that the curvature of the space acted as an effective potential for a free particle in a flat phase-space. Besides, through NC coherent states quantization we have obtained the Green function of the theory. The result have confirmed that we have an UV cutoff for large momenta in the NC kernel. We have also discussed the relation between affine connection and Dirac brackets, as they describe the proper evolution of the model over the surface of constraints in the Lagrangian and Hamiltonian formalisms, respectively. As an application, we have treated the so-called \textit{Zitterbewegung} of the Dirac electron. Since it is assumed to be an observable effect, then we have traced its physical origin by assuming that the electron has an internal structure.

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