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 φ be a linear map from V into an associative algebra A (with unit 1), such that φ(x)² = B(x,x)1 for all x in V. Then φ can be extended to a unique algebra homomorphism  φ^~ from Cl(V,B) to A.

For the algebra A we choose Mat(2,C) - the algebra of 2x2 complex matrices. We select an orthonormal basis eⁱ (i=1,2,3) in V, and we select the following three Pauli matrices: 

To any vector x in V, written as x = x₁e¹+x₂e²+x₃e³, we associate the matrix φ(x) defined as

φ(x) = x₁σ₁+x₂σ₂+x₃σ₃,

It satisfies

φ(x)² = x⋅x I.

Thus, by the Universal Property, φ extends to a unique algebra homomorphism from Cl(V) to Mat(2,C). Since the products of the Pauli matrices span the whole Mat(2,C), and also the real dimension of the algebra Mat(2,C) is eight, the homonorphism is, in fact an isomorphism. In other words: we have a particular realization (or a model) of Cl(V)  as Mat(2,C). We will use the same symbol φ to denote this isomorphism. What we need now are the formulas that will enable us to see the original real Clifford algebra structure within Mat(2,C).

Realizing Clifford Algebra in Matrices

With the orthonormal basis eⁱ (i=1,2,3) in V we have a basis in Cl(V)

1, e¹, e², e³, e²e³,  e³e¹, e¹e², e¹e²e³

An arbitrary element u in Cl(V) can be written as

u = a + b₁ e¹ +b₂ e² + b₃ e³ +c₁ e²e³ + c₂ e³e¹ + c₃ e¹e² +d e¹e²e³.

Then

φ(u) = a I + b₁ σ₁ + b₂ σ₂ + b₃ σ₃ + c₁ σ₂σ₃ + c₂ σ₃σ₁ + c₃ σ₁σ₂ + d σ₁σ₂σ₃.

Explicitly, in matrix form:

a little bit of playing with the matrix algebra allows us to recover a,b,c,d from  X=φ(u) as follows

a = ½  Re(Tr(X)),

b_i  =  ½  Re(Tr(σ_iX)),

c_i  =  ½  Im(Tr(σ_iX)),

d = ½  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).

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.

P.S. 07-11-24 7:02 Reading "Paranormal Science TGD-physics and Life-after-Death the vision of Matti Pitkänen, PhD" by Mark McWilliams (692 pages). His site: Articles.htm. 

There: 

" ... One of  the  mathematical  experiences  was  that  the  number  '3'  is  somehow  the  basic  number  of  Mathematics  and  of  whole  existence.    This  is,  of  course,  the  Holy Trinity of religions and mystics." 

P.S. 07-11-24 10:09 A.S. send me while ago the paper "ПРЕДСВЕТ, ВРЕМЯ, МАТЕРИЯ" by Владимир В.Кассандров. There I am finding:

 "Возникающая картина Мира имеет глубокие связи с твисторной геометрией световых лучей Пенроуза и в качестве основных элементов содержит релятивистски-инвариантный "предсветовой" эфир и порождаемые потоком "Предсвета"  в  фокальных  точках  (каустиках)  частицеподобные  образования.    Временная  координата выделена  динамически,  поскольку  для  любого  решения  локально  существует  4-мерное  направление, вдоль которого первичное бикватернионное "эфирообразующее" поле постоянно. Поток Предсвета является также и Потоком Времени, концепция которого оказывается в данном аспекте близкой к концепции Козырева. Здесь, однако, время не "взаимодействует" с материей, а предшествует ей и порождает ее; "скорость  хода  Времени"  совпадает  с    единственной    фундаментальной  скоростью  –  скоростью  света ("Предсвета").  Важную  роль  в  теории  играет  комплексная  структура  алгебры  бикватернионов,  предопределяющая многозначный характер "эфирообразующего" поля и "тонкую" структуру первичного по-тока Времени-Предсвета как суперпозицию огромного числа локально независимых субпотоков. Комплексно-кватернионная  структура  предполагает  также  рассмотрение  полного  8-мерного  пространства-времени, динамика частиц-сингулярностей в котором оказывается неожиданно богатой.[...] Особое внимание уделено  светоподобной  структуре  (потоку  "Предсвета"),  естественно  возникающей  в  теории, тесно связанной с твисторной геометрией Р. Пенроуза и играющей роль релятивистски-инвариантного  эфира.  Все  материальные  образования  рассматриваются  при этом как фокальные точки или каустики предсветового потока. "  

What a "coincidence"!