Spacetime algebra

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}