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Concept
Spacetime algebra
Summary
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime.
It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with 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. It is also the natural parent algebra of spinors in special relativity. 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.
The spacetime algebra may be built up from an orthogonal basis of one time-like vector and three space-like vectors, , with the multiplication rule
where is the Minkowski metric with signature (+ − − −).
Thus, , , otherwise .
The basis vectors share these properties with the Dirac matrices, but no explicit matrix representation need be used in STA.
This generates a basis of one scalar , four vectors , six bivectors , four pseudovectors and one pseudoscalar , where .
The spacetime algebra also contains a non-trivial sub-algebra containing only the even grade elements, i.e. scalars, bivectors, and pseudoscalars. In the even sub-algebra, scalars and pseudoscalars both commute with all elements, and act like complex numbers. However, the pseudoscalar anticommutes with all odd-grade elements of the spacetime algebra, corresponding to the fact that under parity transformations, vectors and pseudovectors become negated.
Associated with the orthogonal basis is the reciprocal basis for
satisfying the relation
These reciprocal frame vectors differ only by a sign, with , and for .
Official source
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