Algebraic Representation

In physics, we need a way to represent physically meaningful concepts in a quantifiable way. When discussing things like time, or mass, we can choose real numbers to represent these concepts quantitatively. The real numbers form an algebra, which basically means that if you add them or multiply them, the result is also a real number.

The principle form of representation for most physical objects is a vector, which does not belong to an algebra. This means that there is no intrinsic product defined so that two vectors can be multiplied by each other to form a vector. This lack of definition of multiplication in our basic representation of physical quantities removes understanding of how these quantities interplay with each other.

If we want to solve the universe we need to figure out how to represent our physical quantities as algebras rather than vector spaces, so we don't miss out on the added information provided by this multiplication.

We will employ a linear algebra for the representation. What this means is that although we can decompose a quantity like position into its principle components, the full representation can be achieved by a square matrix, AND this representation is preferred over the component form.

An element of our physical algebra will be denoted by a bold symbol, such as A.

The product of two such elements can be written as

AB = C

The omission of the multiplication symbol distinguishes an algebraic multiplication from other types of multiplication such as dot products, or cross products. At this point, looking at an equation such as this is pretty pointless, since we don't know exactly how to multiply the elements together, even if we know the physical meaning of the symbols. We will discover how to multiply the elements of our physical algebra by first examining what types of properties this multiplication is expected to have.

In general, the members of an algebra are not expected to commute with respect to multiplication. What this means is

AB ≠ BA

generally. In other words, the order of multiplication is usually important.

There are usually elements in the algebra that commute with every other element in the algebra. We call these special elements scalars and denote them by a non-bold lowercase italicized symbol, such as a.

In other words, we know that for scalars we can always re-arrange the order of multiplication.

aB = Ba

always.

Common scalar values in physics represent quantities such as time, or mass. We are used to representing these scalar quantities with real numbers, and we will continue to do so. This means that we require the real numbers to be present in our algebraic representation of the universe.

Since the real numbers are included in our algebra, we know that we have access to two special real numbers, 0 and 1. These two numbers help us define the operations of subtraction and division.

For instance, if we know that

A + B = 0

We can rewrite the sum

C + B = C - A

Likewise, if we know that

AB = 1

We can rewrite the product

CB = CA^-1

We cannot strictly call this a "division", since there would be two possible definitions - given that we cannot in general rearrange the order of multiplication.