Algebra of Paravectors, Which Contributes to Its Intuitive Understanding

Algebra of paravectors JÓZEF RADOMANSKI´ ul. Bohaterów Warszawy 9, 48-300 Nysa, Poland e-mail: [email protected] Abstract Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually similar to vector counterparts, known from the Euclidean geometry. Paravectors meet the idea of parallelogram law, Pythagorean theorem and many other properties well-known to everyone. Keywords: Paravector, multivector, Geometric Algebra This article refers to professor William Baylis’ researches and shows a surprising similarity between properties of paravectors and vectors in the Euclidean space. It can be confusing at first to get used to a column notation of paravectors, but this form has proved to be the most adequate. The main advantage of this notation is a transparency of mathematical transformations. The author hopes that in future paravector formalism will replace the multivectors formalism in physics as simpler, more intuitive and imaginable. The results presented in this article justify it fully. 1 Basic definitions Definition 1.1. The term paravector means a pair consisting of a complex number (α) and a vector (β ) belonging to a three-dimensional complex space. α a + id Γ =: = (1) β b + i c The number will be called a scalar. Paravectors will be denoted with capital letters, eg: A,X ,Γ. Greek letters arXiv:1601.02965v2 [math.RA] 7 May 2016 will mean a complex size, and Roman letters - a real one. A scalar component of paravector Γ will be denoted with index „S”, and a vector component with „V”, i.e.: ΓS = α and ΓV = β . Definition 1.2. The reversed element of paravector (1) is the paravector a + id Γ− =: (2) b i c − − Definition 1.3. The conjugated element of paravector (1) is the paravector a id Γ∗ =: − (3) b i c − 1 Definition 1.4. of the summation: α1 α2 α1 + α2 + =: (4) β 1 β 2 β 1 +β 2 Conclusion 1.1. The neutral element under addition (null element) is the paravector 0 0 =: (5) 0 Definition 1.5. The opposite element of paravector (1) with respect to the addition is α Γ =: − (6) − β − Definition 1.6. of the multiplication: α1 α2 α1α2 +β 1β 2 =: (7) β β α β + α β + iβ β 1 2 2 1 1 2 1 × 2 where β β is a scalar (dot) product of vectors, and β β is a vector (cross) product. 1 2 1 × 2 Conclusion 1.2. The neutral element under multiplication is the paravector 1 1 =: (8) 0 α Note: There is no difference if we write number α or paravector . 0 Conclusion 1.3. The operation of multiplication is associative but not commutative, because the vector product is non-commutative. Definition 1.7. We call an outcome of multiplication of any paravector Γ by the element conjugate Γ∗ the vigor of paravector: vigΓ =: ΓΓ∗ (9) Conclusion 1.4. To each paravector we can assign a vigor which is a real paravector and its scalar component is a positive number. Proof. αα∗ αα∗ +ββββ ∗ (a + id )(a id )+(b + i c)(b i c) ΓΓ∗ = = = − − = β β αββ + α β + iβ β (a + id )(b i c)+(a id )(b + i c) + i (b + i c) (b i c) ∗ ∗ ∗ × ∗ − − × − a 2 + b 2 + c 2 + d 2 = 2(a b + d c + b c) × Definition 1.8. We call an outcome of multiplication of any paravector Γ by the reverse element Γ− the determinant of a paravector detΓ =: ΓΓ− =Γ−Γ (10) 2 Conclusion 1.5. Each paravector has a determinant which is a complex number. Proof. α α α2 β 2 (a + id )2 (b + i c)2 a 2 b 2 + c 2 d 2 + 2i (ad bc) ΓΓ− = = − = − = − − − β β αββ αββ iβ β i (b + i c) (b + i c) 0 − − − × − × Conclusion 1.6. The reversion and conjugation have the following properties: Reversion of paravector Conjugation of paravector 1 (Γ−)− =Γ (Γ∗)∗ =Γ 2 (Γ+Ψ)− =Γ− + Ψ− (Γ+Ψ)∗ =Γ∗ + Ψ∗ 3 (ΓΨ)− = Ψ−Γ− (ΓΨ)∗ = Ψ∗Γ∗ 4 ΓΓ C ΓΓ R R 3 − ∈ ∗ ∈ + × 5 (Γ−)∗ =(Γ∗)− Conclusion 1.7. A set of pravectors together with an operation of summation forms an Abelian group, and with multiplication forms a semigroup. Therefore, we can conclude that a set of paravectors together with operations of summation and multiplication gives a ring with multiplicative identity. Definition 1.9. We call the paravector Γ proper if detΓ R 0 (the determinant is a positive real number). ∈ + \{ } Definition 1.10. We call the paravector Γ singular if detΓ = 0. By definition of the determinant it follows: Conclusion 1.8. Each proper or singular paravector (1) must fulfill the following condition: ad = bc (11) It is advisable to remember the above equation, because it will be used with many proofs on proper paravectors. Conclusion 1.9. Let Γ1, Γ2 be paravectors, then the following statements are true: det(Γ Γ ) = detΓ detΓ • 1 2 1 · 2 det(Γ ) = detΓ • − det(Γ )=(detΓ) • ∗ ∗ Definition 1.11. For each non-singular paravector Γ, there exists an inverse element under multiplication: 1 Γ− Γ− =: (12) detΓ Conclusion 1.10. A set of non-singular paravectors together with multiplication is a non-commutative group. Conclusion 1.11. A set of proper paravectors together with multiplication is a non-commutative group. Definition 1.12. For each proper and/or singular paravector we define the module of paravector: Γ =: detΓ (13) | | p Conclusion 1.12. The module of paravector (proper or singular!) satisfies the following conditions: 3 1. s Γ = s Γ where s R | | | || | ∈ 2. Γ Γ = Γ Γ | 1|| 2| | 1 2| Definition 1.13. We call the paravector Λ orthogonal if detΛ = 1 , or equivalently: Γ Λ =: , where Γ is a proper paravector (14) pdetΓ Directly from the above definition it follows: 1 Conclusion 1.13. If Λ is an orthogonal paravector, then Λ− =Λ−. Definition 1.14. We call the paravector Γ special if Γ− =Γ∗, or equivalently: a , where a R, and c R 3 ∈ ∈ i c Definition 1.15. We call the paravector Γ unitar if ΓΓ∗ = 1 To summarize the current knowledge about paravectors, we can say that very little is missing so that a set of paravectors with summation and multiplication operations is a field: multiplication of paravectors is not commutative, and the role of the null element under multiplication is played by singular paravectors. Multiplying any paravector by singular one, we get a singular paravector. Note that although there are many null elements under multiplication, only one element is neutral with respect to summation. Conclusion 1.14. The set of special paravectors together with operations summation and multiplication is a division ring. In the diagram (fig. 1) a ring of paravectors with some of its substructures is presented: Figure 1: The ring of paravectors with some substructures - A set of proper paravectors together with multiplication is a noncommutative group. 4 - A set of orthogonal paravectors together with multiplication is a subgroup of proper paravectors group. The black point means zero, and the blue point is one. To avoid misunderstandings, we recall the following names which we will use: Γ - The paravector reverse of Γ • − Γ 1 - The paravector inverse of Γ • − 2 Integrated product of paravectors [ ] Γ Γ− +Γ Γ− [ ] Synge 7 defines the scalar product of complex quaternions as 1 2 2 1 /2, Hestenes 6 as a scalar part of the product of multivectors Γ1Γ2. Analyzing the expression Γ1Γ2−, we can see that it plays a similar but more universal role in the paravectors algebra as the dot product in vector space. Properties of its scalar component are the same as properties of the scalar product of vectors, and its vector part as a cross product of vectors. The trouble is that there can be two different products which have the same properties (the second one is Γ1−Γ2), so we define two integrated products: Definition 2.1. For any paravectors Γ1 and Γ2, the right integrated product of paravectors Γ1 and Γ2, denoted (Γ ,Γ , is defined by 1 2〉 (Γ ,Γ =: Γ Γ− 1 2〉 1 2 α α α α β β 1 2 1 2 − 1 2 hence (Γ1,Γ2 =Γ1Γ2− = = 〉 β β α β + α β iβ β 1− 2 − 1 2 2 1 − 1 × 2 Definition 2.2. For any paravectors Γ and Γ the left integrated product of paravectors Γ and Γ , denoted Γ ,Γ ), 1 2 1 2 〈 1 2 is defined by Γ ,Γ ) =: Γ−Γ 〈 1 2 1 2 α α α α β β 1 2 1 2 − 1 2 or Γ1,Γ2)=Γ1−Γ2 = = 〈 β β α β α β iβ β − 1 2 1 2 − 2 1 − 1 × 2 In both cases, the scalar part is the same, then Definition 2.3. The scalar component of an integrated product of paravectors Γ1 and Γ2, is called a scalar product of paravectors. The scalar product will be denoted Γ ,Γ =: (Γ ,Γ = Γ ,Γ ) . 〈 1 2〉 1 2〉S 〈 1 2 S Definition 2.4. The vector component of an integrated product we call a vector product of paravectors. (Γ ,Γ =: (Γ ,Γ = Γ ,Γ )∗ 1 2} 1 2〉V −〈 1 2 V It is necessary to distinguish the orientation of a vector product, the same as at integrated product case. Therefore, we denote the right vector product (Γ ,Γ , and the left vector product: Γ ,Γ ) 1 2} { 1 2 For further consideration it does not matter much if a product is right or left, so talking about an integrated product we will mean the right product.

Load more