相对论性相空间与Newman-Penrose基
The relativistic phase space and Newman-Penrose basis
摘要：
我们定义了一个复相对论相空间，它是一个具有Minkowski度量的空间$\mathbb{C}^4$，在这个空间上有一个几何三重积，这个几何三重积类似于Cartan分类中IV型有界对称域的三重积，称为自旋域.在相对论性相空间中，我们证明了三重积的基本元素--三势元有两种。我们构造了这个空间的元素的谱分解。给出了相对论性相空间元素的相容性描述。我们证明了相对论性相空间有两个自然基，它们由相容的三电位子组成。第一个是四向量的自然基，第二个是Newman-Penrose基。第二种方法确定相空间上的Dirac双旋量。因此，相对论性相空间具有与量子力学态空间相似的特征。
英文摘要：
We define a complex relativistic phase space which is the space $\mathbb{C}^4$ equipped with the Minkowski metric and with a geometric tri-product on it. The geometric tri-product is similar to the triple product of the bounded symmetric domain of type IV in Cartan's classification, called the spin domain. We show that there are to types of tripotents-the basic elements of the tri-product in the relativistic phase space. We construct a spectral decomposition for elements of this space. A description of compatibility of element of the relativistic phase space is given. We show that the relativistic phase space has two natural bases consisting of compatible tripotents. The fist one is the natural basis for four-vectors and the second one is the Newman-Penrose basis. The second one determine Dirac bi-spinors on the phase space. Thus, the relativistic phase space has similar features to the quantum mechanical state space.
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