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Commutative Segre's Quaternions

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Commutative Segre's Quaternions Appendix A Commutative Segre’s Quaternions Up to now, we have formalized the application of hyperbolic numbers for studying the space-time symmetry stated by special relativity by means of the analogy with complex numbers. This formalization has provided a mathematical tool which allows us an automatic solution of problems in two-dimensional space-time, in the same way as we do in Euclidean plane geometry. As we have repeatedly recalled, this was possible thanks to the coincidence between the multiplicative group of hyperbolic numbers with the two-dimensional Lorentz group. A relevant characteristic of these group is to possess divisors of zero, a prop- erty which was considered, for a long time, very far from the common geometrical intuition [59] and limited the studies of multidimensional commutative systems with this property. Actually, as we have reported in Section 2.1.2, p. 6, the exten- sion of real and complex numbers can be done only by releasing • the commutative property of the product; • to be a division algebra, i.e., not to have divisors of zero. From a geometrical point of view, relevant results [42] have been obtained by means of non-commutative systems without divisors of zero. In particular by the Hamilton quaternions for representing vectors in the three-dimensional Euclidean space and, more generally, by Clifford algebra [45]. On the other hand we have shown in Chapters 7–10, the relevance of the introduction of functions of hyperbolic variables and we know that the functions of complex variables are very important for the determination of two-dimensional physical fields (Section 7.1), described by Cauchy–Riemann partial differential equations, as pointed out in ([54], p. 1252): “The extraordinary apt way in which the properties of functions of a complex variable fit our need for solutions of differential equations in two dimensions, does not apply to equations in more dimensions for which our task will be, in general, more difficult”. Actually we have • the Hamilton quaternions can be considered as an extension of real and complex numbers as regards the property of being a division algebra; • the commutative systems can be considered as an extension of real and com- plex numbers as regards the commutative property of products and the pos- sibility of introducing their functions, which could represent physical fields [71]. 170 Appendix A. Commutative Segre’s Quaternions Also if many problems must be tackled and solved, the goal, in studying the multi- dimensional commutative systems, is in the possibility of giving a positive answer to [54] for problems in more than two dimensions. Then, since before looking for applications, we need a well-formalized mathematical theory, in these appendices we begin this task from a four-dimensional commutative number system, which has been called [20] Segre’s quaternions for the reasons summarized in Section A.1.1. Actually there are two peculiarities of this system which stimulate its study: 1. four coordinates are usually considered for describing the physical world; 2. these numbers can be considered as composed by two systems of complex numbers, which have a well-known physical relevance. We shall see that, thanks to this last property, many characteristics can be ob- tained by an extension of complex analysis. A.1 Hypercomplex Systems with Four Units The four-units commutative hypercomplex systems are fourteen in number; we recall, from [13] and [24], their multiplication tables. 1. Six of them are non-decomposable. e0 e1 e2 e3 e0 e1 e2 e3 e0 e1 e2 e3 e1 e2 e3 0 e1 −e0 e3 −e2 e1 0 0 0 (a) , (b) , (c) e2 e3 0 0 e2 e3 0 0 e2 0 0 0 e3 0 0 0 e3 − e2 0 0 e3 0 0 0 (A.1.1) e0 e1 e2 e3 e0 e1 e2 e3 e0 e1 e2 e3 e1 e3 0 0 e1 e3 0 0 e1 e2 0 0 (a) , (b) , (c) e2 0 e3 0 e2 0 − e3 0 e2 0 0 0 e3 0 0 0 e3 0 0 0 e3 0 0 0 (A.1.2) 2. Two systems (not reported) are obtained by adding a unit to the three-units systems of Tables (7.6.1). 3. Six systems are obtained by the three two-dimensional systems. Actually calling α,α two quantities that can be 0, ±1, we summarize these systems in the following table (three systems are repeated): e0 e1 0 0 e1 α e0 0 0 (A.1.3) 0 0 e2 e3 0 0 e3 α e2 A.1. Hypercomplex Systems with Four Units 171 A.1.1 Historical Introduction of Segre’s Quaternions Out of the fourteen systems of four-units commutative hypercomplex numbers mentioned above, five of them can be considered as a generalization of a historical method due to Corrado Segre [67] who studied a geometrical representation of complex numbers. He proposed the following complexification: let there be given the complex number z = x +iy and let us substitute the real variables x, y for the complex variables x ⇒ z1 ≡ x1 + ky1 and y ⇒ z2 ≡ x2 + ky2, where k is a unit vector with the same properties of i, i.e., k2 = −1, but different from i since, in a geometrical representation, it represents the unit vector of another axis. Then we have z = x1 + ky1 +ix2 +iky2, and calling j =ik the unit vector of a new axis, we obtain a four-dimensional commutative hypercomplex number with the following multiplication table: 1 k i j k −1 j −i (A.1.4) i j −1 −k j −i −k 1 This multiplication table is similar to the one of Hamilton quaternions; then we can call these numbers Segre’s (commutative) quaternions. A.1.2 Generalized Segre’s Quaternions From an algebraic point of view this process of complexification can be general- ized to 2n dimensions ([60], Chap. 5) as proposed by Segre, as well as to other two-dimensional systems. In this case we can take for z and zm,(m =1, 2) the hy- perbolic numbers (h), parabolic numbers (p) and elliptic numbers (e), as reported in Table A.1. It has been proposed [20] to call these numbers generalized Segre’s quaternions. Some years after their geometrical introduction these numbers were reconsid- ered by F. Severi [68] in association with functions of two complex variables. After this association G. Scorza-Dragoni [66] and U. Morin [53] began to study the ex- istence and differential properties of quaternion functions and, in recent time, has been also reconsidered by Davenport [27] and Price [60]. From a physical point of view, there is an important difference between the functions of two complex vari- ables and functions of commutative quaternions. Actually, while for the former, two real functions of four real variables are defined, for the latter four real func- tions of four real variables are introduced. These functions, as we have shown in Section 7.2.3, can represent vector fields, which automatically have the same sym- metries (groups, geometries) and then the same invariants of the quaternions, since the structure constants act as “symmetry preserving operator” between vectors and the corresponding vector fields. These symmetries are the ones of the geome- tries “generated” by hypercomplex numbers (Chapter 3). We shall see that when commutative quaternions are represented in their geometry, they keep the same 172 Appendix A. Commutative Segre’s Quaternions z zm TABLE Ref. h h (A.1.3),α = α =+1 h e (A.1.3),α = α = −1 [66] e h (A.1.3),α = α = −1 [66] e e (A.1.3),α = α = −1 [60], [66] h p (A.1.3),α = α =0 p h (A.1.3),α = α =0 e p (A.1.1,b) [71], [26] p e (A.1.1,b) [71], [26] p p (A.1.1,c) Table A.1: Generalized Segre’s quaternions. The first and second column indicate the two-dimensional numbers (h, p, e) considered. In the third column we report the table in Section A.1, to which the number system corresponds, and in the fourth column the references in which they have been studied or applied. Note that the three systems from the second to the fourth rows represent the Segre’s commutative quaternions which has been recently reconsidered [27] and [60]. differential properties of complex numbers, in particular for conformal mappings. Therefore if a physical field can be associated with their functions, the problem we consider can be simplified, as it can if we use polar coordinates for problems with spherical symmetry or hyperbolic numbers for space-time symmetry [71]. A.2 Algebraic Properties Here we study three systems among the generalized Segre’s quaternions, reported in Table A.1: in particular the systems that are decomposable into the two- dimensional hypercomplex systems considered in Section 2.2 and Chapter 6. Here these systems are indicated by H2. We also indicate by Q ⊂ H4 the set of quater- nions defined as {q = t + ix+ jy+ kz; t, x, y, z ∈ R; i, j, k∈ / R}, (A.2.1) where the versors i, j, k satisfy the following multiplication rules. 1 i j k i α k αj (A.2.2) j k 1 i k αj i α In particular, according to the value of α, the Q systems shall be named after the two-dimensional component systems: A.2. Algebraic Properties 173 1. for α<0: Elliptic quaternions, in particular canonical (Segre’s) if α = −1; 2. for α =0:Parabolic quaternions; 3. for α>0: Hyperbolic quaternions, in particular canonical if α =1.
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