幂零对称性作为大统一的机制
Nilpotent symmetries as a mechanism for Grand Unification
摘要：
在经典的Coleman-Mandula no-go定理中，内部和时空对称性的统一被禁止，内部群的李代数上存在正定不变标积的假设是必不可少的。如果我们允许标积是半正定的，这就为规范和时空对称的统一开辟了新的可能性。从李代数的结构定理可以看出，在统一对称的情况下，半正定不变标积的退化方向必须对应于具有幂零生成元的局部对称。在本文中，我们构造了一个可行的最小玩具模型利用这一机制：它承认统一的局部对称有一个紧凑的- U（1）-组件，洛伦兹- SL（2，C）-组件，和一个幂零组件粘合在一起。这种结构使得完全统一的对称群局部地忠实地作用于物质场扇区，而对应于幂零生成元的规范场可以从理论中转换出来，只留下带紧荷的规范场。结果表明，对于上述规范场消去机制，普通狄拉克方程已经有一个非常简单的原型例子：它与相应的可消去的规范场具有局域对称性，与伸缩群有关。概述的对称统一机制可以用来绕过科尔曼-曼杜拉和相关的不去定理的方式是从根本上不同于超对称。特别是，该机制避免了对底层时空流形的超坐标或额外维度的调用。
英文摘要：
In the classic Coleman--Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive semi-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact -- U(1) -- component, a Lorentz -- SL(2,C) -- component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman--Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold.
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