Clifford algebras

Given an \({n}\)-dimensional real vector space \({V}\) with a pseudo inner product, we can generate another associative algebra called the Clifford algebra (AKA geometric algebra). As we will see, this algebra subsumes both the exterior algebra and the Hodge star.

The Clifford algebra generated by \({V}\) is defined to be the tensor algebra modulo the identification \({vv\equiv\left\langle v,v\right\rangle }\), where juxtaposition denotes the vector multiplication operation on the algebra, called Clifford multiplication (AKA geometric multiplication). As a vector space, the Clifford algebra is isomorphic to the exterior algebra; in fact, the exterior product can be defined in terms of the Clifford product, leading immediately to simple relationships between these and the inner product:

  • \({\left\langle v,w\right\rangle =\left(vw+wv\right)/2}\)
  • \({v\wedge w\equiv\left(vw-wv\right)/2}\)
  • \({vw=\left\langle v,w\right\rangle +v\wedge w}\)

Each vector \({v}\) can be written in terms of an orthonormal basis \({\hat{e}_{\mu}}\) of \({V}\), and for any pair of orthogonal vectors we have \({\hat{e}_{1}\hat{e}_{2}=\hat{e}_{1}\wedge \hat{e}_{2}=-\hat{e}_{2}\hat{e}_{1}}\). The exterior product then naturally extends to any number of vectors by taking their completely anti-symmetrized sum under Clifford multiplication:

\(\displaystyle v_{1}\wedge v_{2}\wedge\dotsb\wedge v_{k}\equiv\frac{1}{k!}\underset{\pi}{\sum}\textrm{sign}\left(\pi\right)v_{\pi\left(1\right)}v_{\pi\left(2\right)}\dotsm v_{\pi\left(k\right)} \)

This completes the definition of the exterior product in terms of Clifford multiplication, carrying over all its properties from the exterior algebra. In particular, the Clifford algebra can be given the same basis as the orthonormal basis in the exterior algebra

\(\displaystyle \overset{n}{\underset{k=0}{\bigcup}}\left\{ \hat{e}_{\mu_{1}}\wedge\dotsb\wedge\hat{e}_{\mu_{k}}\right\} _{1\leq\mu_{1}<\dotsb<\mu_{k}\leq n} \)

where we take \({\left\{ 1\right\} }\) for \({k=0}\).

Given a choice of orientation with corresponding unit \({n}\)-vector \({\Omega}\), the Hodge star of an element \({A\in\Lambda^{k}V}\) may be written in terms of the Clifford product as

\(\displaystyle *A=\left(-1\right)^{\frac{k\left(k-1\right)}{2}}A\Omega \)

where the pseudo inner product has signature \({\left(r,s\right)}\). It is helpful to see how the Hodge star works out in terms of the Clifford product in common signatures.

\({*v}\)\({v\Omega=-\Omega v}\)\({v\Omega=\Omega v}\)\({v\Omega=-\Omega v}\)\({v\Omega=-\Omega v}\)
\({*B}\)\({-B\Omega=-\Omega B}\)\({-B\Omega=-\Omega B}\)\({-B\Omega=-\Omega B}\)\({-B\Omega=-\Omega B}\)
\({*T}\)\({-T\Omega=-\Omega T}\)\({-T\Omega=\Omega T}\)\({-T\Omega=\Omega T}\)

Note: here we have \({v\in V,\: B\in\Lambda^{2}V}\), and \({T\in\Lambda^{3}V}\). In particular, the three dimensional cross product is seen to produce the pseudo-vector \({v\times w=*\left(v\wedge w\right)=-\Omega\left(v\wedge w\right)=-\left(v\wedge w\right)\Omega}\).

Considering \({ae_{1}(be_{1}+ce_{2})=ab+ac(e_{1}\wedge e_{2})}\), we can think of Clifford multiplication as an operation that “scalar multiplies parallel components and exterior multiplies orthogonal ones.” In particular, the product \({A\Omega}\) will “turn all basis vectors in \({A}\) into scalars,” yielding a form of orthogonal complement, as is the Hodge star.

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