(Scalar, Vector) Notation

We have already determined how the basis elements of the algebra multiply with each other. However, if each element can have up to 8 components, then the product of two of these elements could have up to 64 terms. This is not very friendly.

In this post we will present the algebraic multiplication law in terms of familiar vector operations such as dot and cross products.

We can represent an arbitrary element of the algebra as a linear combination of 4 pieces, using (scalar, vector) notation

A = (a + ib, C + iD)

Here the scalar portions are designated by lowercase italicized symbols, while the vector portions are uppercase non-italicized symbols. These 4 quantities represent the scalar, pseudo-scalar, vector, and pseudo-vector quantities.

We will first consider the multiplication of purely elements that are purely real.

A = (a, A)
B = (b, B)

The multiplication of two such elements results in 16 terms. These terms can be written using the vector dot and cross products as

AB = (ab + A ∙ B, aB + bA + iA × B)

The right hand side of this equation contains 3 distinct portions, a scalar, a vector, and a pseudo-vector term. Now let A and B represent general elements. This can be done in the following way

A = C + iD
B = E + iF
AB = CB - DF + iCF + iDB

Here C, D, E, and F are all purely real algebraic elements.

If an algebraic element is purely real, it is called a para-vector. Some examples of para-vectors are position, or momentum

x = (ct, X)
p = (E/c, P)

We have applied the appropriate factor of c in both cases, in order for the scalar portion to have the correct scale with respect to the vector portion - relativistically speaking. We see that we have associated known relativistic four vectors with algebraic para-vectors.

The multiplication of two para-vectors results in a bi-para-vector. Such a quantity usually corresponds with a multi-indexed tensor in standard relativistic treatments.