arXiv 1004.04642010-04-03hep-thhep-ph

非对易几何作为统一所有基本相互作用包括引力的框架。部件I

Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I

Ali H. Chamseddine, Alain Connes

备注：: 56 pages
发表信息：: Fortsch.Phys.58:553-600,2010
DOI：: 10.1002/prop.201000069

摘要：

我们研究的假设，时空是一个产品的连续四维流形倍的有限空间。一个新的张量符号的发展，提出了各种结构的非交换几何。特别地，该符号用于确定标准模型的光谱数据。粒子谱及其所有的对称性，几乎是唯一的，根据不可约性和6模8维的有限空间的假设。从自然对称群SU（2）xSU（2）xSU（4）到U（1）xSU（2）xSU（3）的约化是假设两层时空相距有限距离但非动力学的结果。狄拉克算子的平方，以及所有的几何不变量，出现在热核膨胀的计算进行评估。我们重新推导了谱作用量中的首阶项。几何作用量统一了所有基本相互作用，包括高能引力。我们做了如下预言：（1）每个费米子家族的费米子数是16。(ii)对称群为U（1）xSU（2）xSU（3）。(iii)在正确的表象中有夸克和轻子。(iv)有一个二重态希格斯打破了U（1）的电弱对称性。(v)顶夸克质量为170-175 Gev。(v)有一种右手中微子具有跷跷板机制。此外，用截断函数得到的零级光谱作用量与实验值在百分之几以内是一致的。我们讨论了一些悬而未决的问题。由于希格斯场的预测质量对高阶修正相当敏感，我们为计算高阶修正打下了基础。我们推测在普朗克能量的非对易空间的性质和可能的作用的基本群体的问题的代。

英文摘要：

We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of dimension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat kernel expansion are evaluated. We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions: (i) The number of fermions per family is 16. (ii) The symmetry group is U(1)xSU(2)xSU(3). (iii) There are quarks and leptons in the correct representations. (iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of 170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent. We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.

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英文摘要：

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