Basic Calculations on Clifford Algebras

Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2 with addition and multiplication being effectively computable relative to the computability of their own spaces. Here we emphasize the algorithmic notions of the Clifford algebras. We recall the reduction of Clifford algebras into isomorphic structures also suitable for symbolic manipulation. 1 Notation Let K = R,C be the field of real or complex numbers. K stands for the n-Cartesian power of K, and Kn×n for the space of square (n×n)-matrices with entries in K. Both K and Kn×n are vector spaces over K with natural structures. Kn×n is an algebra. With respect to matrix multiplication, the following collections of matrices are subgroups of general use: General Linear Group GL(n,K) = {A ∈ Kn×n| det(A) 6= 0}. Orthogonal Group O(n,K) = {A ∈ GL(n,K)| AA = 1}. Special Orthogonal Group SO(n,K) = {A ∈ O(n,K)| det(A) = 1}. Some other particular groups are the following: • O(n) = O(n,R) • SO(n) = SO(n,R) • U(n) = O(n,C) • SU(n) = SO(n,C) For any m,n ∈ N, by [[m,n]] we will denote the set of integers {m,m+ 1, . . . , n− 1, n}. 2 Division Algebras Let R be the field of real numbers. Let i = √ −1 be a square root of −1 and let C = R[i] be the field of complex numbers. 1 2.1 Quaternions Let i, j, k be three symbols with relations i2 = j2 = k2 = ijk = −1 ij = k jk = i ki = j ji = −k ik = −j kj = −i (1) The quaternion algebra is H = R[i, j, k]. It is a non-commutative associative division algebra, extending C, and it is a 4-dimensional real vector space. The conjugate map is H → H, h = x0 + x1i + x2j + x3k 7→ h = x0 − x1i − x2j − x3k and it is congruent with respect to addition and multiplication. The norm of a quaternion h ∈ H is |h| = √ hh and, whenever it is non-zero, its multiplicative inverse is h−1 = 1 |h|2 h. The unit sphere in H is SH = {h ∈ H| hh = 1}, (2) and it is a subgroup under multiplication. Let S2 = {x ∈ R3| ‖x‖ = 1} be the unit sphere in the 3-dimensional real space, and let φ : S2 → SH, (w1, w2, w3) 7→ w1i + w2j + w3k. Then each point h ∈ SH can be written in the form h = e φ(w) = cos a+ φ(w) sin a, with − π < a ≤ π , w ∈ S2. (3) Let R3 = {x0 + x1i+ x2j + x3k ∈ H| x0 = 0} be the copy of R3 consisting of quaternions with zero real part. Let A : SH × R3 → R3, (h, x) 7→ hxh. It is an action of the group SH over R3 and for each h = e φ(w) ∈ SH, the map x 7→ A(h, x) is a counterclockwise rotation of angle 2a of R3 along the axis w. In this way, it is said that SO(3,R) is covered twice by SH. There are several matrix representation of the quaternions. Let Φ0 : H → C , x0 + x1i+ x2j + x3k 7→ [ x0 + x1i x2 + x3i −x2 + x3i x0 − x1i ] (4) Then Φ0 is an embedding that preserves addition and multiplication. Similarly, let Φ1 : H → R , x0 + x1i+ x2j + x3k 7→ 