Clifford Algebras

Hassani, Sadri

doi:10.1007/978-3-319-01195-0_27

Sadri Hassani²

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Abstract

The last chapter introduced the exterior product, which multiplied a p-vector and a q-vector to yield a (p+q)-vector. By directly summing the spaces of all such vectors, we obtained a vector space which was closed under multiplication. This led to a 2ⁿ-dimensional algebra, which we called the exterior algebra (see Theorem 26.3.6).

In the meantime we revisited inner product and considered non-Euclidean inner products, which are of physical significance. In this chapter, we shall combine the exterior product with the inner product to create a new type of algebra, the Clifford algebra, which happens to have important applications in physics.

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Notes

1.
We are using the natural units for which the Planck constant (over 2π) and the speed of light are set to 1: ħ=1=c.
2.
Here we define an inner product simply as a symmetric bilinear map as in Definition 2.4.2.
3.
Many other notations are also used to denote this algebra. Among them are \(\mathbf {C}_{\mu,\nu}(\mathbb{R})\), C(μ,ν), \(C\ell_{\mu,\nu}(\mathbb{R})\), \(C\ell_{p,q}(\mathbb{R})\), and C(p,q) where q=ν and p=μ. Occasionally, we’ll use one of these notations as well.
4.
It is worth noting that, by using or , we could obtain formulas for \(\textbf{C}_{n+4}^{0}(\mathbb{R})\) and \(\textbf{C}^{n+4}_{0}(\mathbb{R})\) analogous to (27.50) and (27.51). However, as entries 5, 6, and 7 of Table 27.1 can testify, they would not be as appealing as the formulas obtained above. This is primarily because .
5.
Here we are using the physicists’ convention of numbering the basis vectors from 0 to 3 with \(\hat {\mathbf {e}}_{0}=\hat {\mathbf {e}}_{4}\) and using Greek letters for indices.