Scalars and Vectors

The algebraic conjugate operation affords us the ability to observe a distinction between different kinds of quantities. First, we know from the definition of the conjugate

AA = a

that when the algebraic conjugate acts on a scalar, it does nothing. For instance

a = AA = A A = AA = a

We are going to make this another defining property of a scalar - it is invariant to the algebraic conjugate.

Now there are several ways to define a scalar in math and physics, and so far I have made reference to none of these. Rather I have so far stated that a scalar is

• able to commute with respect to multiplication by any element of the algebra
• invariant with respect to application of the algebraic conjugate

We will see that these defining properties of a scalar lead to the other definitions that we know so well. We will also see that these 2 defining properties end up being mutually satisfied - at least for the end result of our physical algebra.

AA is a characteristic scalar associated with A. We can find another characteristic scalar. Rewrite A into terms that have symmetric and anti-symetric combinations with A.

A = 1/2(A + A) + 1/2(A - A)

If you expand the expression on the right hand side, you will see that it reduces to A. The expression on the right hand side has 2 terms. The first term is invariant to the algebraic conjugate, and thus we see that the first term is a scalar term. The application of the word "scalar" in this context becomes very similar to the original historical usage introduced by Hamilton.

If we remove the scalar part, what are we left with? The algebra represents multi-component objects, and the scalar part only represents single component objects. Thus we can assume that there are several components associated with the remaining portion, if we strip the scalar portion away.

If we apply the algebraic conjugate to the second, or anti-symmetric term, it changes sign. This is to be the defining property of a Vector. Again, the term vector has a host of meanings, and definitions. The usage as applied here is different than the standard relativistic version of the word, but it is similar to the original usage coined by Hamilton. We are going to be talking about vectors much more in future posts.

Since we know that can always decompose any element into a scalar and a vector part, we will use a special notation to represent this

<A>_S = 1/2(A + A)

<A>_V = 1/2(A - A)

And we will use the algebraic conjugate to detect a pure scalar or a pure vector

• If the conjugate does nothing to the algebraic element, it must be a pure scalar
• If the conjugate flips the sign of the element, it must be a pure vector

The very naive idea of a scalar and a vector is that a scalar is a single component quantity, and a vector is a multi-component quantity. As the algebra develops we will see that this is a pretty reasonable first approach to these quantities.

Most elements in the algebra have a linear combination of both a scalar and a vector part. The word we use to talk about a linear combination of a scalar and a vector is "para-vector". The meaning behind a para-vector is closer to the meaning of the relativistic vector.