A novel geometric model of a noncommutative plane has been constructed. We demonstrate that it can be construed as a toy model for describing and explaining the basic features of physics in a noncommutative spacetime from a field theory perspective. The noncommutativity is induced internally through constraints and does not require external interactions. We show that the noncommutative space-time is to be interpreted as having an {\it internal} angular momentum throughout. Subsequently, the elementary excitations - {\it i.e.} point particles - living on this plane are endowed with a {\it spin}. This is explicitly demonstrated for the zero-momentum Fourier mode. The study of these excitations reveals in a natural way various {\it stringy} signatures of a noncommutative quantum theory, such as dipolar nature of the basic excitations \cite{jab} and momentum dependent shifts in the interaction point \cite{big}. The observation \cite{sw} that noncommutative and ordinary field theories are alternative descriptions of the same underlying theory, is corroborated here by showing that they are gauge equivalent. Also, treating the present model as an explicit example, we show that, even classically, in the presence of additional constraints, (besides the usual ones due to reparameterization invariances), the equivalence between Nambu-Goto and Polyakov formulations is subtle.
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机译:构造了非交换平面的新型几何模型。我们证明它可以被解释为一个玩具模型,用于从场论的角度描述和解释非交换性时空中物理学的基本特征。非交换性是通过约束在内部引起的,不需要外部交互。我们表明,非交换时空将被解释为始终具有{\ it内部}角动量。随后,在该平面上的基本激发-{\ it即}点粒子-被赋予{\ it spin}。零动量傅立叶模式已明确证明了这一点。对这些激发的研究以自然的方式揭示了非交换量子理论的各种{\ it stringy}签名,例如基本激发\ cite {jab}的偶极性质和相互作用点\ cite {big}的动量相关位移。观察\ cite {sw}认为非可交换理论和普通领域理论是同一基础理论的替代描述,此处通过证明它们是标准等效的,来证实这一点。同样,将本模型作为一个明确的例子,我们表明,即使在经典情况下,在存在附加约束的情况下(由于重新参数化不变性而导致的常规约束除外),Nambu-Goto和Polyakov公式之间的等效性也很微妙。
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