The Spin Chronicles (Part 9): Matrix representation of Cl(V)

 This post is a continuation of the last post in the series: The Spin Chronicles (Part 8): Clifford Algebra Universal Property

Embedded in 3D Space

For reasons we don’t fully understand, we experience the world in three dimensions—length, breadth, and width. Our senses anchor us in this 3D framework, and we often feel its presence deeply, even beyond simple dimensions. Consider these lines from Rainer Maria Rilke’s "The Night Sky":

"Who has not longed under a brooding sky
for the sound of falling stars?
And to be swallowed up in the wide, wide world,
to feel within himself the cosmos stretching,
endlessly, silently…"

This poetic vision echoes a feeling of spatial vastness, as if the cosmos itself could fit within our minds.

Similarly, William Blake’s famous lines from "Auguries of Innocence" hint at the mystical unity of smallness and immensity:

"To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour…"

And to be swallowed up in the wide, wide world,

Space and Time: A Relationship

While we think of space as separate from time, time appears to emerge from space. In mystical or out-of-body experiences, time sometimes halts, while space remains, albeit in a modified form—often with a 360-degree perspective but still fundamentally three-dimensional. For now, we’ll set aside time, treating space as the foundational structure from which time will eventually arise, shaped by the expansion and contraction of spatial volumes.

Defining Space Mathematically

In space, there are no absolute directions, but there are "lengths" and "angles" between intersecting lines. To formalize this, we define space as a real, 3-dimensional vector space $V$ and equip it with a scalar product—a nondegenerate symmetric bilinear form. Using this structure, we construct its Grassmann and Clifford algebras, which give us powerful mathematical tools to further analyze spatial properties.

The Problem of "Length"

Assuming that we can measure  the "length" is perhaps too much. We can compare different lengths, but to give to a vector in V a definite number, its "length" is somewhat risky. The number depends on a subjectively selected "unit" of length. Moreover, in alien abduction experiences "small" can feel "large" - the scale changes. We will return to this issue in the future. For now let us assume that the unit of length is fixed. It can be, for instance, the cubit. Cultural units like the cubit—the length from a person’s middle finger to their elbow—were historically practical choices.

Biblical Reference to the Cubit
In Genesis 6:13-22 (NKJV), the cubit is used as a measurement for Noah’s Ark:

“Make yourself an ark of gopherwood; make rooms in the ark, and cover it inside and outside with pitch. The length of the ark shall be three hundred cubits, its width fifty cubits, and its height thirty cubits.”

Royal cubit

The cubit, likely originating from Sumerian civilization, offers a historical reference point for measuring length in a way grounded in human scale.

Constructing Clifford Algebra

Once we define length and the associated bilinear form, we can construct the Clifford algebra by deforming the exterior product, as discussed in previous posts. With this algebraic foundation, we can start performing calculations. Several Computer Algebra Systems allow symbolic manipulations with Clifford, or "geometric," algebra, though here we’ll limit ourselves to three dimensions and real numbers.

Interestingly, in three dimensions, there’s no need for specialized software for Clifford algebra—simple $2 \times 2$ complex matrices suffice! Using the Universal Property of Clifford algebra, we can represent it in the form of complex matrices.

Universal Property of Clifford Algebra
Let $\phi$ be a linear map from $V$ into an associative algebra $A$ (with unit $1$), such that $\varphi (x{)}^2=B(x,x)1$ for all $x$ in $V$. Then $\phi$ can be uniquely extended to a homomorphism from $Cl(V, B)$ to $A$.

For our algebra $A$, we choose Mat(2, C), the algebra of $2\times 2$ complex matrices. Selecting an orthonormal basis $e_i$($i=1,2,3$) in $V$ (I will write $e_i$, with lower indices in this post), we define three Pauli matrices:

To any vector $x$ in $V$, written as $x = x_1e_1 + x_2e_2 + x_3e_3$, we associate the matrix $\phi(x)$:

$$\phi(x) = x_1\sigma_1 + x_2\sigma_2 + x_3\sigma_3,$$

It satisfies

$$\phi(x)^2 = x \cdot x.$$

Thus, by the Universal Property, $\phi$ extends uniquely to a homomorphism from $Cl(V)$ to $Mat(2, C)$, which turns out to be an isomorphism because the Pauli matrices generate $Mat(2, C)$, and both algebras share the same real dimension of eight.

Realizing Clifford Algebra in Matrices

Using this isomorphism, we express elements of $Cl(V)$ within the algebra of $2\times 2$ complex matrices. For an arbitrary element $u$ in $Cl(V)$:

$$u = a + b_1 e_1 + b_2 e_2 + b_3 e_3 + c_1 e_2e_3 + c_2 e_3e_1 + c_3 e_1e_2 + d e_1e_2e_3,$$

we associate

$$\phi(u) = a I + b_1 \sigma_1 + b_2 \sigma_2 + b_3 \sigma_3 + c_1 \sigma_2\sigma_3 + c_2 \sigma_3\sigma_1 + c_3 \sigma_1\sigma_2 + d \sigma_1\sigma_2\sigma_3.$$

Explicitly, in matrix form:

With a little manipulation, we can recover the components $a$, $b$, $c$, and $d$ from X=$\phi(u)$:

$$a = \frac{1}{2} \text{Re(Tr(X))},$$ $$b_i = \frac{1}{2} \text{Re(Tr(}\sigma_i X\text{))},$$ $$c_i = \frac{1}{2} \text{Im(Tr(}\sigma_i X\text{))},$$ $$d = \frac{1}{2} \text{Im(Tr(X))}.$$

Challenge: Expressing Clifford Algebra Involutions

Exercise 1: Express ι = e₁e₂e₃ as a 2x2 matrix using the above representation.

Exercise 2: As a challenge, try to express the three involutions of $Cl(V)$ in terms of Mat(2, C). This will deepen your understanding of how Clifford algebra structure manifests in matrix form.

Stay tuned for further exploration in the following posts!

P.S. 06-11-24 17:18 You should not take the story of the Noah's Ark literally. See "Hidden in Plain Sight" by The Ethical Skeptic.