# 时空代数

## 结构

${\displaystyle \gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }=2\eta _{\mu \nu }}$

## 互易框架

${\displaystyle \gamma _{\mu }\cdot \gamma ^{\nu }={\delta _{\mu }}^{\nu }.}$

{\displaystyle {\begin{aligned}a\cdot \gamma ^{\nu }&=a^{\nu }\\a\cdot \gamma _{\nu }&=a_{\nu }.\end{aligned}}}

{\displaystyle {\begin{aligned}\gamma _{\mu }&=\eta _{\mu \nu }\gamma ^{\nu }\\\gamma ^{\mu }&=\eta ^{\mu \nu }\gamma _{\nu }.\end{aligned}}}

## 时空梯度

${\displaystyle a\cdot \nabla F(x)=\lim _{\tau \rightarrow 0}{\frac {F(x+a\tau )-F(x)}{\tau }}.}$

${\displaystyle \nabla =\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}=\gamma ^{\mu }\partial _{\mu }.}$

${\displaystyle \partial _{0}={\frac {1}{c}}{\frac {\partial }{\partial t}},\quad \partial _{k}={\frac {\partial }{\partial {x^{k}}}}}$

## 时空分离

 时空分离 – 例子： ${\displaystyle x\gamma _{0}=x^{0}+\mathbf {x} }$ ${\displaystyle p\gamma _{0}=E+\mathbf {p} }$[1] ${\displaystyle v\gamma _{0}=\gamma (1+\mathbf {v} )}$[1] 其中${\displaystyle \gamma }$是洛伦兹因子 ${\displaystyle \nabla \gamma _{0}=\partial _{t}-{\vec {\nabla }}}$[2]

• 所选时间轴的坍缩，产生二重向量张成的3D空间，相当于物理空间代数中标准3D基向量；
• 4D空间到所选时间轴上的投影，产生表示标量时间的1D标量空间。[3]

{\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\gamma _{k}\gamma _{0}\\\gamma _{0}x&=x^{0}-x^{k}\gamma _{k}\gamma _{0}\end{aligned}}}

{\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\sigma _{k}=ct+\mathbf {x} \\\gamma _{0}x&=x^{0}-x^{k}\sigma _{k}=ct-\mathbf {x} \end{aligned}}}

## 洛伦兹变换

${\displaystyle v'=e^{-\beta {\frac {\theta }{2}}}\ v\ e^{\beta {\frac {\theta }{2}}}}$,

${\displaystyle v'=\left(\cos \left({\frac {\theta }{2}}\right)-\beta \sin \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cos \left({\frac {\theta }{2}}\right)+\beta \sin \left({\frac {\theta }{2}}\right)\right)}$.

${\displaystyle v'=\left(\cosh \left({\frac {\theta }{2}}\right)-\beta \sinh \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cosh \left({\frac {\theta }{2}}\right)+\beta \sinh \left({\frac {\theta }{2}}\right)\right)}$.

## 经典电磁学

### 法拉第二重向量

STA中，电场磁场可统一为单一的二重向量场，叫做法拉第二重向量，等价于电磁张量[4]定义为

${\displaystyle F={\vec {E}}+Ic{\vec {B}},}$

${\displaystyle F=E^{i}\sigma _{i}+IcB^{i}\sigma _{i}=E^{1}\gamma _{1}\gamma _{0}+E^{2}\gamma _{2}\gamma _{0}+E^{3}\gamma _{3}\gamma _{0}-cB^{1}\gamma _{2}\gamma _{3}-cB^{2}\gamma _{3}\gamma _{1}-cB^{3}\gamma _{1}\gamma _{2}.}$

{\displaystyle {\begin{aligned}E={\frac {1}{2}}\left(F-\gamma _{0}F\gamma _{0}\right),\\IcB={\frac {1}{2}}\left(F+\gamma _{0}F\gamma _{0}\right).\end{aligned}}}

${\displaystyle F^{2}=E^{2}-c^{2}B^{2}+2Ic{\vec {E}}\cdot {\vec {B}}.}$

### 麦克斯韦方程

${\displaystyle {\vec {J}}=c\rho \gamma _{0}+J^{i}\gamma _{i},}$

${\displaystyle \nabla F=\mu _{0}cJ}$

{\displaystyle {\begin{aligned}\nabla \cdot \left[\nabla F\right]&=\nabla \cdot \left[\mu _{0}cJ\right]\\0&=\nabla \cdot J.\end{aligned}}}

### 洛伦兹力

${\displaystyle {\mathcal {F}}=qF\cdot v}$

### 势公式

${\displaystyle A={\frac {\phi }{c}}\gamma _{0}+A^{k}\gamma _{k}}$

${\displaystyle {\frac {1}{c}}F=\nabla \wedge A.}$

${\displaystyle A'=A+\nabla \Lambda }$

${\displaystyle \nabla \wedge \left(A+\nabla \Lambda \right)=\nabla \wedge A+\nabla \wedge \nabla \Lambda =\nabla \wedge A.}$

{\displaystyle {\begin{aligned}{\frac {1}{c}}\nabla F&=\nabla \left(\nabla \wedge A\right)\\&=\nabla \cdot \left(\nabla \wedge A\right)+\nabla \wedge \left(\nabla \wedge A\right)\\&=\nabla ^{2}A+\left(\nabla \wedge \nabla \right)A=\nabla ^{2}A+0\\&=\nabla ^{2}A\end{aligned}}}

${\displaystyle \nabla ^{2}A=\mu _{0}J}$

### 拉格朗日公式

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\epsilon _{0}F^{2}-J\cdot A}$

${\displaystyle \nabla {\frac {\partial {\mathcal {L}}}{\partial \left(\nabla A\right)}}-{\frac {\partial {\mathcal {L}}}{\partial A}}=0.}$

${\displaystyle \nabla \cdot A=0.}$

## 泡利方程

${\displaystyle i\hbar \,\partial _{t}\Psi =H_{S}\Psi -{\frac {e\hbar }{2mc}}\,{\hat {\sigma }}\cdot \mathbf {B} \Psi ,}$

${\displaystyle \partial _{t}\psi \,i\sigma _{3}\,\hbar =H_{S}\psi -{\frac {e\hbar }{2mc}}\,\mathbf {B} \psi \sigma _{3},}$

## 狄拉克方程

${\displaystyle {\hat {\gamma }}^{\mu }(\mathbf {j} \partial _{\mu }-e\mathbf {A} _{\mu })|\psi \rangle =m|\psi \rangle ,}$

STA中的狄拉克方程：

${\displaystyle \nabla \psi \,i\sigma _{3}-e\mathbf {A} \psi =m\psi \gamma _{0}}$

### 狄拉克旋量

${\displaystyle \psi =e^{{\frac {1}{2}}(\mu +\beta i+\phi )},}$

${\displaystyle \psi =R(\rho e^{i\beta })^{\frac {1}{2}},}$

${\displaystyle \psi =e^{i\Phi _{\lambda }/\hbar },}$

### 物理可观测量

${\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}$

### U(1)规范对称

${\displaystyle \psi \rightarrow \psi '=e^{iq\Lambda /\hbar }\psi =\psi e^{iq\Lambda /\hbar },}$

${\displaystyle \left(\nabla \psi +iq(\nabla \Lambda )\psi \right)e^{iq\Lambda /\hbar },}$

## 广义相对论

### 广义相对论的新表述

${\displaystyle {\frac {d}{d\tau }}R={\frac {1}{2}}(\Omega -\omega )R}$

${\displaystyle D_{\tau }=\partial _{\tau }+{\frac {1}{2}}\omega ,}$

## 参考文献

1. Lasenby, A.N.; Doran, C.J.L. Geometric algebra, Dirac wavefunctions and black holes. Bergmann, P.G.; De Sabbata, Venzo (编). Advances in the interplay between quantum and gravity physics. Springer. 2002: 256-283, See p. 257. ISBN 978-1-4020-0593-0.
2. ^
3. ^ Arthur, John W. Understanding Geometric Algebra for Electromagnetic Theory. IEEE Press Series on Electromagnetic Wave Theory. Wiley. 2011: 180. ISBN 978-0-470-94163-8.
4. Doran, Chris; Lasenby, Anthony, Geometric Algebra for Physicists, Cambridge University Press, 2003, ISBN 978-0-521-48022-2
5. ^ Joot, Peeter. A multivector Lagrangian for Maxwell's equation (PDF). [2023-06-04]. （原始内容存档 (PDF)于2023-10-30）.
6. See eqs. (75) and (81) in: Hestenes & Oersted Medal Lecture 2002
7. See eqs. (3.43) and (3.44) in: Doran, Chris; Lasenby, Anthony; Gull, Stephen; Somaroo, Shyamal; Challinor, Anthony. Hawkes, Peter W. , 编. Spacetime algebra and electron physics. Advances in Imaging and Electron Physics 95. Academic Press. 1996: 272–386, 292. ISBN 0-12-014737-8.
8. See eq. (3.1) and similarly eq. (4.1), and subsequent pages, in: Hestenes, D. On decoupling probability from kinematics in quantum mechanics. Fougère, P.F. (编). Maximum Entropy and Bayesian Methods. Springer. 2012: 161–183 [1990]. ISBN 978-94-009-0683-9. (PDF 互联网档案馆存檔，存档日期2022-10-29.)
9. ^ See also eq. (5.13) of Gull, S.; Lasenby, A.; Doran, C. Imaginary numbers are not real – the geometric algebra of spacetime (PDF). 1993 [2024-01-15]. （原始内容存档 (PDF)于2022-10-24）.
10. See eq. (205) in Hestenes, D. Spacetime physics with geometric algebra (PDF). American Journal of Physics. June 2003, 71 (6): 691–714 [2012-02-24]. Bibcode:2003AmJPh..71..691H. doi:10.1119/1.1571836. （原始内容 (PDF)存档于2023-01-04）.
11. ^ Hestenes, David. Oersted Medal Lecture 2002: Reforming the mathematical language of physics (PDF). American Journal of Physics. 2003, 71 (2): 104 [2012-02-25]. Bibcode:2003AmJPh..71..104H. . doi:10.1119/1.1522700. （原始内容 (PDF)存档于2023-01-04）.
12. ^ Brage Gording, Angnis Schmidt-May. The Unified Standard Model. Advances in Applied Clifford Algebras. 2020-08-18, 30 (4). S2CID 202565534. . doi:10.1007/s00006-020-01082-8.