Geometric algebra

The term “geometric algebra” usually refers to a relatively recent resurgence of interest in Clifford algebras, with an emphasis on geometric interpretations and motivations, and a variety of newly defined operations on the algebra. Here we will provide a brief synopsis of some of these ideas, with an eye towards potential usage in physical models.

Δ It should be noted that definitions, notation, and terminology in geometric algebra vary quite significantly from author to author.

A general element of the Clifford algebra is called a multivector (AKA Clifford number), while a general element of \({\Lambda^{k}V}\) is called a k-vector (AKA homogeneous multivector) and is said to be of grade \({k}\) (note that this is its weight as an element of the algebra graded over the exterior product, but not over the Clifford product).

An element that can be written as the exterior product of \({k}\) vectors (or equivalently, as the Clifford product of \({k}\) orthogonal vectors) is called a k-blade. So for example, \({e_{1}\wedge\left(e_{2}+e_{3}\right)}\) is a 2-blade, while \({\left(e_{1}\wedge e_{2}\right)+\left(e_{3}\wedge e_{4}\right)}\) is a 2-vector and \({e_{1}+\left(e_{2}\wedge e_{3}\right)}\) is a multivector. Note that if \({V}\) has dimension 3 or less, every \({k}\)-vector is a \({k}\)-blade; in higher dimensions, they are in general sums of \({k}\)-blades. 2-vectors are sometimes referred to as bivectors, and 3-vectors as trivectors.

The \({k}\)-vector part of a general Clifford algebra element \({A}\) is denoted \({\left\langle A\right\rangle _{k}}\) (the scalar part \({\left\langle A\right\rangle _{0}}\) is often written \({\left\langle A\right\rangle }\)), and is a sum of \({k}\)-blades. Thus any multivector can be decomposed into a sum of \({k}\)-vectors: \({A=\sum\left\langle A\right\rangle _{k}}\). The unit \({n}\)-vector \({\Omega}\) associated with a choice of orientation on \({V}\) is sometimes called the pseudo-scalar.

Various operations can be defined on the entire Clifford algebra by defining them for \({k}\)-blades and then using linearity. Below, we consider an \({a}\)-blade \({A}\) and a \({b}\)-blade \({B}\).

  • Dot product (AKA inner product): \({A\bullet B\equiv\left\langle AB\right\rangle {}_{\left|a-b\right|}}\) (the lowest grade part of the Clifford product \({AB}\))
  • Exterior product (AKA outer product): \({A\wedge B\equiv\left\langle AB\right\rangle _{\left(a+b\right)}}\) (the highest grade part of \({AB}\))
  • Reversion: the reverse \({\widetilde{A}}\) of \({A=v_{1}\wedge v_{2}\wedge\dotsb\wedge v_{a}}\) reverses the order of its components \({v_{i}\Rightarrow\widetilde{A}=\left(-1\right)^{a\left(a-1\right)/2}A\Rightarrow*A=\widetilde{A}\Omega,\:\widetilde{AB}=\widetilde{B}\widetilde{A}}\)

The definition of exterior product here can easily be shown to be equivalent to the usual one given in the exterior algebra, and thus shares the same symbol. In contrast, \({A\bullet B}\) is similar but not identical to the inner product \({\left\langle A,B\right\rangle \equiv\textrm{det}\left(\left\langle v_{i},w_{j}\right\rangle \right)}\) defined on the exterior algebra: \({A\bullet B}\) does not vanish for two elements of different grade, and for two \({k}\)-blades one obtains the result \({A\bullet B=\left\langle \widetilde{A},B\right\rangle =\left(-1\right)^{k\left(k-1\right)/2}\left\langle A,B\right\rangle }\). \({\widetilde{A}}\) is sometimes denoted \({A^{\dagger}}\), since under any representation of the Clifford algebra generated by hermitian matrices as vectors, the reverse corresponds to the hermitian conjugate.

Various relations then follow from these definitions. Below, we consider an \({a}\)-blade \({A}\) and a \({b}\)-blade \({B}\), where \({V}\) is \({n}\)-dimensional and the pseudo inner product has signature \({\left(r,s\right)}\).

  • \({v\bullet A=\left(vA-\left(-1\right)^{a}Av\right)/2}\)
  • \({v\wedge A=\left(vA+\left(-1\right)^{a}Av\right)/2\Rightarrow vA=v\bullet A+v\wedge A}\)
  • In particular, \({v_{0}\bullet\left(v_{1}\wedge v_{2}\right)=\left(v_{0}\bullet v_{1}\right)v_{2}-\left(v_{0}\bullet v_{2}\right)v_{1}}\), and
  • \({v_{0}\bullet\left(v_{1}\wedge v_{2}\wedge v_{3}\right)=\left(v_{0}\bullet v_{1}\right)\left(v_{2}\wedge v_{3}\right)-\left(v_{0}\bullet v_{2}\right)\left(v_{1}\wedge v_{3}\right)+\left(v_{0}\bullet v_{3}\right)\left(v_{1}\wedge v_{2}\right)}\)
  • \({\Omega\bullet A=\Omega A=\left(-1\right)^{a\left(n-1\right)}A\Omega=\left(-1\right)^{a\left(n-1\right)}A\bullet\Omega}\)
  • \({\Omega^{2}\equiv\Omega\Omega=\left(-1\right)^{n\left(n-1\right)/2+s}\Rightarrow\widetilde{\Omega}\Omega=\left(-1\right)^{s}\Rightarrow*A=\left(\widetilde{\Omega}\Omega\right)\widetilde{A}\Omega}\)
  • In particular, \({\Omega^{2}=-1}\) for the signatures \({\left(2,0\right),\left(3,0\right),\left(3,1\right)}\) and \({\left(1,3\right)}\)
  • \({\left\langle AB\right\rangle _{0}=\left\langle BA\right\rangle _{0}\Rightarrow\left\langle AB\dotsm C\right\rangle _{0}=\left\langle B\dotsm CA\right\rangle _{0}}\)
  • If \({B}\) is a bivector, the commutator \({\left[A,B\right]=AB-BA}\) has the same grade as \({A\Rightarrow}\) the commutator of two bivectors is another bivector
  • If \({B}\) is a bivector, \({BA=B\bullet A+\left[B,A\right]/2+B\wedge A\Rightarrow}\) two bivectors \({\hat{e}_{i}\wedge\hat{e}_{j}}\) anti-commute if they share exactly one vector, commute otherwise
Δ It should be noted that some authors in geometric algebra define a “dual” that differs by a (grade- and signature-dependent) sign from the usual Hodge dual used in most texts. For example, \({A\Omega}\) is sometimes defined as the “dual” of \({A}\), in which case the “swapping of the inner and exterior products” property can be generalized to the form \({A\bullet\left(B\Omega\right)=\left(A\wedge B\right)\Omega}\), valid for all \({a+b\leq n}\).

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