Clifford或Grassmann坐标费米子二次量子化理论的新途径和自旋电荷族理论
New way of second quantized theory of fermions with either Clifford or Grassmann coordinates and spin-charge-family theory
摘要：
在克利福德空间中描述的具有内部自由度的费米子在任何维度上都具有半整数自旋。Clifford空间中有两种自旋。自旋-电荷族理论，假设d=13+1，使用一种自旋来描述d=3+1中夸克、轻子、反夸克和反轻子的自旋和电荷，而另一种自旋则用来描述族。本文提出了自旋电荷族理论提出的二次量子化的新方法。它表明，1-费米子态的创建和湮灭运营商，写为产品的幂零和投影的奇克利福德字符，满足反交换关系所需的费米子的第二量子化过程：1-费米子态在Clifford空间中已经二次量化，任何n-费米子二次量化矢量的创建算子是一个费米子创建算子的乘积，在空真空状态下操作。它表明，在格拉斯曼空间中也存在创造和湮灭运营商的一个奇怪的格拉斯曼字符，产生“费米子”，这也满足反交换关系的费米子，相应地代表第二个量化的1-“费米子”状态，在这种情况下，与整数自旋。格拉斯曼空间不提供家庭。我们讨论了这两种空间中场的新的二次量子化过程。对于格拉斯曼的情况下，我们提出的行动，基本状态，自由无质量的“费米子”和离散对称运营商的外尔方程的解决方案。简要回顾了自旋电荷族理论的研究成果，并指出了该理论尚待解决的问题。格拉斯曼和克利福德的情况下进行了比较，以更好地理解物理学中的基本费米子和玻色子场和宇宙学中的开放问题。
英文摘要：
Fermions with the internal degrees of freedom described in Clifford space carry in any dimension a half integer spin. There are two kinds of spins in Clifford space. The spin-charge-family theory,assuming even d=13+1, uses one kind of spins to describe in d=3+1 spins and charges of quarks and leptons and antiquarks and antileptons, while the other kind is used to describe families. The new way of second quantization, suggested by the spin-charge-family theory, is presented. It is shown that the creation and annihilation operators of 1-fermion states, written as products of nilpotents and projectors of an odd Clifford character, fulfill the anticommutation relations as required in the second quantization procedure for fermions: 1-fermion states are in Clifford space already second quantized, the creation operators for any n-fermion second quantized vectors are products of one fermion creation operators, operating on the empty vacuum state. It is demonstrated that also in Grassmann space there exist the creation and annihilation operators of an odd Grassmann character, generating "fermions", which fulfill as well the anticommutation relations for fermions, representing correspondingly the second quantized 1-"fermion" states, in this case with integer spins. Grassmann space offers no families. We discuss the new second quantization procedure of the fields in both spaces. For the Grassmann case we present the action, basic states, solutions of the Weyl equation for free massless "fermions" and discrete symmetry operators. A short overview of the achievements of the spin-charge-family theory is done, and open problems of this theory still waiting to be solved are presented. The Grassmann and the Clifford case are compared in order to better understand open questions in physics of elementary fermion and boson fields and in cosmology.
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