In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl_1,3(R), or equivalently the geometric algebra G(M⁴) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics."^[1]: ix

Spacetime algebra is a vector space that allows not only vectors, but also bivectors (directed quantities describing rotations associated with rotations or particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted.^{[2]: 40, 43, 97, 113} It is also the natural parent algebra of spinors in special relativity.^[2]: 333 These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.^[1]: v

In comparison to related methods, STA and Dirac algebra are both Clifford Cl_1,3 algebras, but STA uses real number scalars while Dirac algebra uses complex number scalars. The STA spacetime split is similar to the algebra of physical space (APS, Pauli algebra) approach. APS represents spacetime as a paravector, a combined 3-dimensional vector space and a 1-dimensional scalar.^{[3]: 225–266}

Structure

For any pair of STA vectors, ${\textstyle a,b}$, there is a vector (geometric) product ${\textstyle ab}$, inner (dot) product ${\textstyle a\cdot b}$ and outer (exterior, wedge) product ${\textstyle a\wedge b}$. The vector product is a sum of an inner and outer product:^[1]: 6

${\displaystyle a\cdot b={\frac {ab+ba}{2}}=b\cdot a,\quad a\wedge b={\frac {ab-ba}{2}}=-b\wedge a,\quad ab=a\cdot b+a\wedge b}$

The inner product generates a real number (scalar), and the outer product generates a bivector. The vectors ${\textstyle a}$ and ${\textstyle b}$ are orthogonal if their inner product is zero; vectors ${\textstyle a}$ and ${\textstyle b}$ are parallel if their outer product is zero.^{[2]: 22–23}

The orthonormal basis vectors are a timelike vector ${\textstyle \gamma _{0}}$ and 3 spacelike vectors ${\textstyle \gamma _{1},\gamma _{2},\gamma _{3}}$. The Minkowski metric tensor's nonzero terms are the diagonal terms, ${\textstyle (\eta _{00},\eta _{11},\eta _{22},\eta _{33})=(1,-1,-1,-1)}$. For ${\textstyle \mu ,\nu =0,1,2,3}$:

${\displaystyle \gamma _{\mu }\cdot \gamma _{\nu }={\frac {\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }}{2}}=\eta _{\mu \nu },\quad \gamma _{0}\cdot \gamma _{0}=1,\ \gamma _{1}\cdot \gamma _{1}=\gamma _{2}\cdot \gamma _{2}=\gamma _{3}\cdot \gamma _{3}=-1,\quad {\text{ otherwise }}\ \gamma _{\mu }\gamma _{\nu }=-\gamma _{\nu }\gamma _{\mu }}$

The Dirac matrices share these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers;^[1]: x explicit matrix representation is unnecessary for STA.

Products of the basis vectors generate a tensor basis containing one scalar ${\displaystyle \{1\}}$, four vectors ${\displaystyle \{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}}$, six bivectors ${\displaystyle \{\gamma _{0}\gamma _{1},\,\gamma _{0}\gamma _{2},\,\gamma _{0}\gamma _{3},\,\gamma _{1}\gamma _{2},\,\gamma _{2}\gamma _{3},\,\gamma _{3}\gamma _{1}\}}$, four pseudovectors (trivectors) ${\displaystyle \{I\gamma _{0},I\gamma _{1},I\gamma _{2},I\gamma _{3}\}}$ and one pseudoscalar ${\displaystyle \{I\}}$ with ${\textstyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$.^[1]: 11 The pseudoscalar commutes with all even-grade STA elements, but anticommutes with all odd-grade STA elements.^[4]: 6

Subalgebra

STA's even-graded elements (scalars, bivectors, pseudoscalar) form a Clifford Cl_3,0(R) even subalgebra equivalent to the APS or Pauli algebra.^[1]: 12 The STA bivectors are equivalent to the APS vectors and pseudovectors. The STA subalgebra becomes more explicit by renaming the STA bivectors $({\mathrm{\gamma <!--\; \gamma \; -->}}_{}$Zdroj:https://en.wikipedia.org?pojem=Spacetime_algebra
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