论文标题
通过额外的时间尺寸使时空代数复杂化：PIN，旋转和代数旋转器
Complexifying the spacetime algebra by means of an extra timelike dimension: Pin, Spin and algebraic spinors
论文作者
论文摘要
因为同构$ { -0.1em \ ell} _ {1,3}（\ bbb {r}）$，通过在Minkowski Spacetime中添加一个附加的时间级维度。在最近的一项工作中，我们展示了这种治疗方法如何根据新的时间维度提供对狄拉克颗粒和反粒子的特殊解释。在本文中，我们彻底研究了真正的Clifford代数$ {C \ Kern -0.1em \ ell} _ {2,3}（\ bbb {r}）$特别关注同态属性$ {c \ kern -0.1em \ ell} \ kern -0.1em \ ell} _ {2,3}（\ bbb {r}）$和嵌入$ {c \ kern -0.1em \ ell} _ {1,3} _ {1,3}（\ bbb {r}） \ ell} _ {2,3}（\ bbb {r}）$。在本文的上半年，我们分析了销钉和旋转组，并构建了一个注射映射$ \ peripatorAtorName {pin}（1,3）（1,3）\ hookrightArrow \ propatatorArrow \ propatatorArrow \ propatatorAname {spin}（2,3）$，以$ \ operatateName {spin}（spin}（2,3）$表示代表平等和时间及时reversal reversal and parity和time reversal and $ \ operatateName {spin}（2,3）$。在本文的下半年，我们研究了代数的纺纱空间，并证明了$ {c \ kern -0.1em \ ell} _ {1,3}（\ bbb {c}）$通常在$ {c \ kern -0.1em \ ell} _ {c \ kern -0.1em \ ell} _ \ ell} _ {2,3}（\ bbb {r}）$。
Because of the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$, it is possible to complexify the spacetime Clifford algebra ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})$ 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 ${C \kern -0.1em \ell}_{2,3}(\Bbb{R})$ paying special attention to the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$ and the embedding ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})\subseteq{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$. On the first half of this article we analyze the Pin and Spin groups and construct an injective mapping $\operatorname{Pin}(1,3)\hookrightarrow\operatorname{Spin}(2,3)$, obtaining in particular elements in $\operatorname{Spin}(2,3)$ 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 ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})$ is reproduced by the Clifford conjugation inner product for real spinors in ${C \kern -0.1em \ell}_{2,3}(\Bbb{R})$.
