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On Application of Algebra of Quaternions II International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 341 ISSN 2250-3153 On application of algebra of quaternions II Emran Sasi Althabit*, Ismail.M.Mohamed ** * Higher Institute for water Affairs ** Academic of Engineering and Medical Sciences DOI: 10.29322/IJSRP.11.02.2021.p11041 http://dx.doi.org/10.29322/IJSRP.11.02.2021.p11041 Abstract- We introduced algebraic formulations of time as vector producing the modern description of spacetime, as a four- quaternions, Clifford Algebras , we deduce the Lorentz transform- vector ation from the Minkowski Space. 푋 = [푡, 푥⃗] (1.3) Defining a involution 푋∗ = [푡, −푥⃗] we then Index Terms- Clifford Algebras, K-algebra , multivector, Affine produce the invariant distance vector space , Minkowski Space. 푋푋∗ = [푡, 푥⃗][푡, −푥⃗] = 푡2 − 푥2 (1.4) Comparing these two descriptions we can see that they I. ALGEBRAIC FORMULATIONS OF TIME provide some significant differences. To begin with, Hamilton’ s n order to proceed with our approach, we now wish to description views time as an intrinsic part of the description of I algebraically model three- dimensional physical space. The three-dimensional space. On the other hand, with the Minkowski distinct approaches are commonly utilized to model various formulation, the immed-iate implication is that time is an physical systems such as three-vectors,four-vectors, matrices, additional Euclidean-type dimension. Although this assu-mption complex numbers, quaternions and complex spinors, depending is qualified by the fact that time contributes an opposite sign to the on the specific application. This fact is indeed quite surprising, in metric distance and so distinct from a regular four dimensional that, one of the specific goals of nineteenth century science was to Cartesian vector. Note that the octonions, being the generalization find the correct algebra for physical space [6]. The objective was of quaternions, have also been considered as an expanded arena initially led by Hamiltonn who produced the quaternion algebra for spacetime. through generalizing the two-dimensional complex numbers to It is an historical fact that the vector quaternions were found three dimensions. Indeed, due to the general success of complex difficult to work with and not suitable to describe Cartesian numbers in algebraically describing the properties of the plane vectors and were replaced by the Gibbs vector system in use today Hamilton reasoned that quaternions should therefore properly [6]. The reason for Hamilton’s failed attempt to algebraically describe the algebra of three-dimensional space. This then lead to describe three-dimensional space, is that by generalizing the the first attempt at a rigorous mathematical definition of a unified complex numbers of the plane to three dimensions he actually only produced the rotational algebra for three dimensions. space and time.Hamilton wrote the quaternion as 푞 = 푡 + 푥1푖 + 3 To properly describe three-dimensional space we further 푥2푗 + 푥3푘 (1.1) where 푥1 , 푥2 , 푥3 ∈ ℝ and the three basis vectors are subject to the well known quaternionic relations 푖2 = need to generalize the quaternions to include a true Cartesian 푗2 = 푘2 = −1 vector component [13]. This generalization of quaternions was in fact achieved by Clifford with the eight dimensional Clifford Hamilton’s quaternions form a 4-D associative normed division 3 algebra over the real numbers represented by ℍ. Hamilton defined algebra over three dimensions of 퐶ℓ(ℝ )[14] 푥⃗ = 푥1푖 + 푥2푗 + 푥3푘 as a vector quaternion to take the role of Cartesian vectors in order to describe spatial vectors, and then also 2. Clifford Algebras Action the group on quaternions(2-1) proposed, around 50 years before Minkowski, that if the scalar “푡” We define an action of the group of unit quaternions, 푆푈(2), on was identified with time then the quaternion can be used as a 3 3 representation for a unified four-dimensional spacetime. Hamilton ℝ . For this, we use the fact that ℝ can be identified with the pure quaternions in ℍ, namely, the quaternions of the form 푥⃗ = 푥1푖 + stating “Time is said to have only one dimension, and space to 3 have three dimensions. ... 푥2푗 + 푥3푘 , where (푥1, 푥2, 푥3) ∈ ℝ . Then, we define the action 3 −1 ∗ The mathematical quaternion partakes of both these of 푆푈(2) over ℝ by 푍 . 푋 = 푍푋푍 = 푍푋푍 (2.1) elements; it may be said to be ‘time plus space’, or ‘space plus where 푍 ∈ 푆푈(2) and 푋 is any pure quaternion. [9] time’: and in this sense it has,or at least involves a reference to, Now, it turns out that the map 휌푍 where −1 four dimensions”. Indeed squaring the quaternion we find 휌푧(푋) = 푍푋푍 is indeed a rotation, and that the map 휌: 푍 ↦ 휌푧is 푞2 = 푡2 − 푥2 + 2푡푥 (1.2) the scalar component 푡2 − 푥2 thus a surjective homomorphism, 휌 ∶ 푆푈(2) → 푆푂(3), whose kernel producing the invariant spacetime distance. is {−1,1}, where 1 denotes the multiplicative unit quaternion. Minkowski, after indeed considering the quaternions, but We can also define an action of the group 푆푈(2) × 푆푈(2) over 4 4 viewing them as too restrictive for describing spacetime, chose ℝ by identifying ℝ with the quaternions. In this case, rather to extend the Gibbs-Heaviside thre-vector system with the (푌, 푍 ). 푋 = 푌 푋 푍∗ (2.2) addition of a time coordinate to create a four-component This publication is licensed under Creative Commons Attribution CC BY. http://dx.doi.org/10.29322/IJSRP.11.02.2021.p110XX www.ijsrp.org International Journal of Scientific and Research Publications, Volume 11, Issue 2, February 2021 342 ISSN 2250-3153 where (푌, 푍 ) ∈ 푆푈(2) × 푆푈(2) and 푋 ∈ ℍ is any quaternion. identity element, 1퐼퐴so that 퐼퐴 ∙ 푎 = 푎 ∙ 퐼퐴 = 푎, for all 푎 ∈ 퐴. ∗ Then, the map 휌푌,푍∗is a rotation where 휌푌,푍∗(푋) = 푌 푋 푍 ,and the Given two 퐾-algebras 퐴 and 퐵, a 퐾-algebra homomorphism, ℎ ∶ map 휌: (푌, 푍 ) ↦ 휌푌,푍∗is a surjective homomorphis,휌 ∶ 푆푈(2) × 퐴 → 퐵, is a linear map that is also a ring homomorphism, with 푆푈(2) → 푆푂(4), whose kernel is {(1,1), (−1. −1)}. ℎ(퐼퐴) = 퐼퐵. For example, the ring, 푀푛푛(퐾) , of all 푛 × 푛 matrices over a field, Thus, we observe that for 푛 = 2, 3, 4, the rotations in 푆푂(푛) 퐾, is a 퐾-algebra. can be realized via the linear action of some group (the case n = 1 Tensor products( 2-3) is trivial, since 푆푂(1) = {1, −1}. It is also the case that the action Let 퐸, 퐹 be two vector spaces over the field K (퐾 = ℝ, ℂ). of each group can be somehow be described in terms of Consider the (infinite) direct sum multiplication in some larger algebra “containing” the original 푇(퐸, 퐹) =⊕(푒,푓)∈퐸×퐹 퐾 (2.4) vector space ℝ푛 (ℂ for 푛 = 2, ℍ for 푛 = 3, 4). However, these i.e., A tensor product of 퐸 and 퐹 is a pair, groups appear to have been discovered in an ad hoc fashion, and (퐸 ⊗ 퐹,⊗), where 퐸 ⊗ 퐹 is a 퐾-vector space and ⊗ : 퐸 × 퐹 → there does not appear to be any universal way to define the action 퐸 ⊗ 퐹 is a bilinear map. of these groups on ℝ푛 . It would certainly be nice if the action was Proposition.1. For any bilinear map 휙 ∶ 퐸 × 퐹 → 퐺 there exists always of the form 푍 . 푋 = 푍푋푍−1(= 푍푋푍∗) a unique linear A systematic way of constructing groups realizing rotations map 훷 ∶ 퐸 ⊗ 퐹 → 퐺 such that the diagram below is in terms of linear action, using a uniform notion of action, does commutative. [15] exist. Such groups are the spinor groups, to be described in the following paragraphs. We explained in Paragraph.1how the rotations in 푆푂(3) can be realized by the linear action of the group of unit quaternions, 푆푈(2), on ℝ3, and how the rotations in 푆푂(4) can be realized by 4 the linear action of the group 푆푈(2) × 푆푈(2) on ℝ .The main reasons why the rotations in SO(3) can be represented by unit quaternions are the following: The vector space 퐸 ⊗ 퐹 is defined up to isomorphism. The vectors 3 (1) For every nonzero vector 푢⃗⃗⃗⃗ ∈ ℝ , the reflection 푠푢⃗⃗⃗about the 푢⃗⃗ ⊗ 푣⃗, where 푢⃗⃗ ∈ 퐸 and 푣⃗ ∈ 퐹 , generate 퐸 ⊗ 퐹 . hyperplane Perpendi-cular to 푢⃗⃗ is represented by the map 푣 ↦ CLIFFORD’S DESCRIPTION OF SPACE (2-4) −푢푣푢−1 (2.3) here 푢⃗⃗ and 푣⃗ are viewed as pure quaternions in ℍ A Clifford geometric algebra 퐶ℓ(ℝ푛) defines an associative real 3 (i.e., if 푢⃗⃗ = (푢1, 푢2, 푢3) , then view algebra over 푛 dimensions and in three dimensions 퐶ℓ(ℝ )is 푢⃗⃗, as 푢1푖 + 푢2푗 + 푢3푘, and similarly for 푣⃗). eight-dimensional [14]. In this case we can adopt the three (2) The group SO(3) is generated by the reflection. quantities 푒1, 푒2, 푒3for basis vectors that are defined to The group of unit quaternions 푆푈(2) turns out to be anticommute in the same way as Hamilton’s quaternions, but isomorphic to the spinor group unlike the quaternions these quantities square to positive one, that 3 2 2 2 푆푝푖푛(3). Because Spin(3) acts directly on ℝ , is 푒1 = 푒2 = 푒3 = 1. Also, similar to quaternions, we the representation of rotations in 푆푂(3)by elements of 푆푝푖푛(3) can combine scalars and the various algebraic components into a may be viewed as more natural than the representation by unit single number, called in this case a multivector quaternions. The group S푈(2) × 푆푈(2) turns out to be 푋 = 푡 + 푥1푒1 + 푥2푒2 + 푥3푒3 + 푛1푒2푒3 + 푛2푒3푒1 + 푛3푒1푒2 isomorphic to the spinor group Spin(4), but this isomorphism is + 푏푒1푒2푒3 (2.5) less obvious.
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