The Minkowski Metric

As we have seen, we can begin to establish a multiplication rule for the algebra, as long as we know what the metric of the underlying vector space is.

For our purpose, we will consider the Minkowski Metric of special relativity as our starting point. We will use the (1, -1, -1, -1) convention of this metric. Using this metric requires that we introduce 4 basis elements. The fundamental identity applied to a 4x4 matrix gives us 16 expressions that will encode how we multiply these 4 basis elements.

When we say we want to use the Minkowski metric, we are actually implying that we want to discuss space-time, not just any old vector space which has a Minkowski norm.

At this point, we have a decision to make regarding how we will represent space and time in this algebra. Although space and time have the same footing in special relativity, we know from practical experience that there is a physical distinction between space and time. So far, in our algebra, the only thing that we have which might encode this distinction is the ability to seperate quantities out into scalar and vector portions.

If we decide to associate the scalar portion of an element with the time component, and the vector portion of the element with the spatial components, then we also make a statement about the physical interpretation of the conjugate operation. We are saying that the algebraic conjugate does nothing to the time component since it is a pure scalar, and it flips the sign of any space component which are pure vectors.

If we make the decision to encode the space/time boundary with the scalar/vector boundary, then we can immediately determine the identity of e₀. The basis vector associated with time must be the basis associated with scalars.

e₀ = 1

Making this identification automatically solves 7 of the 16 expressions introduced by the fundamental identity. We now only need to figure out the last 9, which correspond with how the spatial basis elements multiply with each other.

e_ie_j = -e_je_i ; i ≠ j

e_ie_i = -1

Since the algebraic conjugate only applies an overall change of sign to the purely spatial basis elements, we can recast these equations into

e_ie_j = - e_je_i ; i ≠ j

(e_i)² = 1

The first equation states that if we swap the order of any 2 spatial basis elements, we need to change the sign of the product. The second equation states that any basis element multiplied by itself is equal to 1.

As of yet we have not determined how to represent the product of 2 spatial basis elements. Before we can do this, we need to take a look at the product of all 3 distinct spatial basis elements

e₁e₂e₃

We don't really know what this quantity is yet, but we do know that if we change the order of the basis elements, it will change the overall sign of the quantity. We also know that applying an algebraic conjugate to any individual factor will also change the sign of the product.

The first thing we are going to do is apply the algebraic conjugate to this triple product.

e₁e₂e₃ = e₃ e₂ e₁

Since each of these elements is a pure spatial quantity, the action of the conjugate accumlates to a single minus sign on the product. The application of the conjugate has applied an overall odd permutation to the order of the elements, which results in a second minus sign. Thus we have the identity

e₁e₂e₃ = e₁e₂e₃

In otherwords, the algebraic conjugate does nothing to the triple product. From our definition, then, the triple product is a scalar. We may assume that the triple product must therefore be 1 - the only scalar with unit magnitude, however consider what happens if we square the triple product

(e₁e₂e₃)(e₁e₂e₃) = -e₁e₂e₃e₃e₂e₁ = -e₁e₂e₂e₁ = -e₁e₁ = -1

Thus, the triple product is a scalar that results in -1 if it is squared. The only quantity that can do this is the imaginary unit, i. Thus we can make the identification

e₁e₂e₃ = i

Consider multiplying the triple product on the right by e₃

e₁e₂e₃e₃ = e₁e₂ = ie₃

We can perform a similar trick in order to determine the products of the other spatial basis vectors. The end result is the following list of multiplication rules

(e_μ)² = 1; all μ
e₀e_μ = e_μ; all μ
e₁e₂ = -e₂e₁ = ie₃
e₂e₃ = -e₃e₂ = ie₁
e₃e₁ = -e₁e₃ = ie₂

The good news is that these rules encapsulate the entire multiplication rule for the algebra. Since we are using the Minkowski metric, we have basically built special relativity right into the algebra itself. The bad news is that our scalars are no longer real numbers, rather they are complex numbers. Aside from not knowing up front what physical significance the imaginary unit has, we have also doubled the size of our algebra from 4 dimensions to 8. We still only have 4 basis vectors, but our scalars are now a 2 dimensional sub-algebra. No need to fear - the complexities of complex scalars will be addressed shortly.