arXiv 1206.40652012-06-18hep-thcond-mat.stat-mechgr-qc

Lorentz不变非对易空间中的路径积分

Path integral formalism in a Lorentz invariant noncommutative space

Mario J. Neves, Everton M. C. Abreu

备注：: 21 pages. Pre-print style. Comments welcome

摘要：

我们引入了一个新的路径积分形式主义的非对易（NC）量子力学在最近开发的Doplicher-Fredenhold-Roberts-Amorim（DFRA）NC框架，可以被认为是早期宇宙的NC时空的替代框架。算子形式主义被重新审视，我们应用它的属性，以获得一个NC过渡振幅表示。讨论了两个DFRA系统：NC自由粒子系统和NC谐振子系统。在此NC空间中的一些温度的概念也被认为是。NC DFRA量子场论的推广是直接的，我们将其应用到一个有质量的标量场。我们构造了生成泛函和有效作用量，给出了单粒子不可约图。作为一个例子，我们设置了一个$n\;(n\geq 3)$自相互作用$ϕ^{n}$的基础，以获得微扰理论对该模型的传播子和顶点的修正。我们从一开始就想强调的主要概念是，这里展示的形式主义将不会像往常一样在系统中引入NC参数。它将从已经存在的NC空间中自然生成。在这个额外维的NC空间中，我们还提出了降维恢复交换性的思想。

英文摘要：

We introduced a new formulation for the path integral formalism for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC framework that can be considered an alternative framework for the NC spacetime of the early Universe. The operators formalism was revisited and we apply its properties to obtain a NC transition amplitude representation. Two DFRA's systems were discussed, the NC free particle and NC harmonic oscillator. Some temperature concepts in this NC space are also considered. The extension to NC DFRA quantum field theory is straightforward and we apply it to a massive scalar field. We construct the generating functional and the effective action to give rise one-particle-irreducible diagrams. As an example, we set the basis for a $n\;(n\geq 3)$ self-interaction $ϕ^{n}$ to obtain the correction of the perturbation theory to the propagator and vertex of this model. The main concept that we would like to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.

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