Clifford Biquaternions

1 of 5 Wednesday, October 30, 2024 As the story goes, there was once a wise narrator, keen to recount the hidden mysteries of the cosmos, who sat under the shade of a sprawling, ancient fig tree, its roots winding down like silken threads into the very bedrock of knowledge itself. Around this narrator, an eager gathering of listeners leaned forward, caught between the veil of night and the secrets of the universe. Around this narrator, an eager gathering of listeners leaned forward, caught between the veil of night and the secrets of the universe. "And so we continue," began the narrator, voice calm yet filled with a spark of hidden wonder, "the chronicles of Spin, a journey which carries forth from the 'Clifford (or Geometric) Algebra of Space' where we last left our exploration. But before we delve too deep, let us first revisit our foundation, like a traveler retracing their steps to recall where they first glimpsed the hidden path." With a gesture, the narrator conjured up an image of a grand three-dimensional 30/10/2024, 10:52 2 of 5 real vector space, its expanse glimmering in the minds of the listeners. "This space we call V," the narrator intoned, "endowed with an Euclidean scalar product, denoted by (u⋅v), and, lurking yet unused, an anti-symmetric bilinear form, ϵ. Together, these form the bedrock upon which we built our grand construct, the Clifford Algebra, Cl(V). We fashioned Cl(V) upon the Grassmann algebra, also known as the exterior algebra, Λ(V), whose structure unfolds like petals in the night garden." "Consider," the narrator continued, "Λ(V) as a grand composite, divided as Λ0(V) ⊕ Λ1(V) ⊕ Λ2(V) ⊕ Λ3(V), each part filled with elements bearing degrees, like stars classified by their light. And indeed, the elements of Λi(V) are deemed homogeneous, each of a specific degree, yet each harmonizing within the multiplication of the Grassmann product, ∧. In this space, when one multiplies elements of degree i and j, their product retains a purity, remaining of degree i+j. Though Λi(V) holds naught but the zero vector when i>3, this remains an understanding we keep quietly, as one might mind an absent companion." As the listeners nodded, their imaginations vivid, the narrator spoke of vectors u and v in V, drawing them in midair, where they intersected like blades of shimmering light. "For these vectors, we defined the Clifford product, uv, by a deformation of their Grassmann product, u∧v, such that:" uv = u∧v+(u⋅v)1. "It is as if Clifford algebra contains a hidden chamber," said the narrator, eyes twinkling, "a 'memory cell' within Λ0(V), safeguarding the scalar product, while in Λ2(V) the very shape and span of u and v reside, protected by u∧v." And the narrator reminded them, uv + vu= 2(u⋅v)1, uv − vu = 2u∧v. "From this Clifford product," the narrator continued, "we are able to retrieve both the Grassmann and scalar products of these two wandering vectors. And when u and v stand orthogonal, their Clifford product simplifies to u∧v, such 30/10/2024, 10:52 3 of 5 that uv = − vu, a true mark of their anti-commutative nature." "Now, imagine," and with these words, the narrator’s hand waved over the darkened air, revealing an orthonormal basis for V, denoted by ei for i=1,2,3. "For these basis vectors," the narrator explained, "we see that: ei ej + ej ei = 2 δij . Thus, each distinct basis vector finds itself in an anti-commutative relationship with others, while its square, alone, is equal to one. In our prior journey, we spoke of an inner product within Λ(V), noting that these basis elements: 1, e1, e2, e3, e12, e23, e31, e123 stand as an orthonormal assembly within the eight-dimensional space of Λ(V)." Then, with a flourish, the narrator whispered, "And yet the square of e123 bears the mark of −1, such that we introduced it as ι, a phantom akin to the imaginary unit i: for indeed, ι2 = −1, and this spectral ι commutes with everything across the Clifford landscape." The listeners grew more intent as the narrator described how this mysterious ι is not immutable but changes with the choice of orthonormal basis, as if it too were under the spell of transformations. "If another orthonormal basis, say e'i , should appear," the narrator revealed, "we might calculate ι′ as: ι = (1/3!) δ123ijk eiejek, ι' = (1/3!) δ123ijk e'ie'je'k. By relating the bases through an orthogonal matrix, such that e'i = ej Rji, we find: ι′ = det(R)ι. And thus, if R preserves parity—as in SO(3)—then ι'=ι; otherwise, it reverses 30/10/2024, 10:52 4 of 5 under inversion, showing ι to be a pseudo-scalar, a keeper of chirality, not a scalar." "Now," the narrator intoned, voice brimming with the thrill of revelation, "behold as we introduce the quaternions and their ethereal kin, the biquaternions, hidden here within our Clifford structure." There was a rustle of anticipation among the listeners as the narrator drew forth three simple bivectors from thin air: i = - e23, j = - e31, k = - e12. The narrator explained how these bivectors, like the quaternions themselves, obeyed the peculiar rules i2 = j2 = k2 = -1, while each anti-commuted with the others, a dance of contradictions. Together, 1 and i,j,k, they formed a quaternion basis, while 1 and ι comprised the complex numbers. And so, as the narrator wove the conclusion, the listeners felt the meaning settle upon them like a cloak in the cool night air: "This entire algebra, denoted by Cl(V), mirrors the (tensor) product of complex numbers and quaternions—a realm we call 'complex quaternions,' or 'biquaternions', a space eight-dimensional in the real world, or four in the complex realm." Finally, the narrator offered a task to the eager minds: "Imagine now, and prove, that: e1 = ιi, e2 = ιj, e3 = ιk. May this challenge unlock yet another chamber within the algebraic universe." And with that, the narrator leaned back, inviting contemplation, their tale now a thread woven into the endless tapestry of cosmic discovery. 30/10/2024, 10:52 5 of 5 30/10/2024, 10:52