Because of the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\), it is possible to complexify the spacetime Clifford algebra \(C \ell _{1,3}({\mathbb {R}})\) by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal dimension. In this article we thoroughly study the structure of the real Clifford algebra \(C \ell _{2,3}({\mathbb {R}})\) paying special attention to the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\) and the embedding \(C \ell _{1,3}({\mathbb {R}})\subseteq C \ell _{2,3}({\mathbb {R}})\). On the first half of this article we analyze the Pin and Spin groups and construct an injective mapping \({\text {Pin}}(1,3)\hookrightarrow {\text {Spin}}(2,3)\), obtaining in particular elements in \({\text {Spin}}(2,3)\) that represent parity and time reversal. On the second half of this paper we study the spinor space of the algebra and prove that the usual structure of complex spinors in \(C \ell _{1,3}({\mathbb {C}})\) is reproduced by the Clifford conjugation inner product for real spinors in \(C \ell _{2,3}({\mathbb {R}})\).

由于同构\（C \ ell _ {1,3}（{\ mathbb {C}}）\ cong C \ ell _ {2,3}（{\ mathbb {R}}）\），有可能通过向Minkowski时空添加一个额外的类似时间的维数来复杂化时空Clifford代数\（C \ ell _ {1,3}（{\ mathbb {R}}）\）。在最近的工作中，我们展示了这种处理如何根据新的时间维度对狄拉克粒子和反粒子提供了特殊的解释。在本文中，我们彻底研究了实Clifford代数\（C \ ell _ {2,3}（{\ mathbb {R}}）\）的结构，并特别注意了同构\（C \ ell _ {1， 3}（{\ mathbb {C}}）\ cong C \ ell _ {2,3}（{\ mathbb {R}}）\）和嵌入\（C \ ell _ {1,3}（{\ mathbb {R}}）\ subseteq C \ ell _ {2,3}（{\ mathbb {R}}）\）。在本文的上半部分，我们分析了Pin和Spin组并构造了一个内射映射\（{\ text {Pin}}（1,3）\ hookrightarrow {\ text {Spin}}（2,3）\），在\（{\ text {Spin}}（2,3）\）中获得特定的元素，这些元素表示奇偶校验和时间反转。在本文的后半部分，我们研究了代数的旋子空间，并证明了\（C \ ell _ {1,3}（{\ mathbb {C}}）\）中复数旋子的通常结构是可重现的。\（C \ ell _ {2,3}（{\ mathbb {R}}）\）中用于真正的旋子的Clifford共轭内积。