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Quaternion Fields the Six Dimensions of Space-Time Quaternion fields The six dimensions of space-time Rastko Vukovic Gimnazija Banja Luka December 3, 2014 Abstract It is explored some possibilities of mathematical symmetry between physical space and time. We define the space plus time with (three) Pauli matrices plus Quaternions. Indicated the possibility of extending the ideas of Lorentz rotation to general method of transformation coordinates, for arbitrary force field. The theoretical possibility of inversion of coordinates of space and time was observed. Quaternions are number systems that extend the set of complex numbers. They were discovered by Irish mathematician William Rowan Hamilton in 1843 who has applied it to the mechanics. Hamilton defined quaternion as the quotient of two directed lines, or two vectors in three-dimensional space. 1 Algebra Algebra in the narrow sense is the branch of mathematics that studies the equation. In a broader sense, algebra is the part of mathematics that uses numbers and other quantities represented by letters and symbols for studying formulas and equations. 1.1 Group Theory A set S together with a binary operation ◦, which satisfies the first three (a1-a3) of the next axioms, is called a group and denoted by (S; ◦). if it meets the fourth axiom (a4), it is called the commutative or Abelian group. For all σ 2 S in Abel group (S; ◦) is valid: a1. associativity: (σ1 ◦ σ2) ◦ σ3 = σ1 ◦ (σ2 ◦ σ3), a2. exist neutral e 2 G that is e ◦ σ = σ ◦ e = σ, a3. each σ 2 G has inverse σ−1 2 G such that σ ◦ σ−1 = σ−1 ◦ σ = e, a4. commutativity: σ1 ◦ σ2 = σ2 ◦ σ1. 1 For example, a set of integers (Z), rational (Q), or real (R) numbers is the group together with the common relation plus (+) of which the zero (0) is neutral, and the number −σ is the inverse element to the number σ. The set of rational or real numbers is the group together with the common multiplication (×) of which the one (1) is neutral, and the number σ−1 is the inverse to the number σ 6= 0. A division algebra (S; +; ×), or division ring, is the set S together with the two binary operators, addition and multiplication, respectively + and ×, such that (S; +) is Abelian group, (S; ×) is group and apply the law of left and right distributivity: a5. σ1 × (σ2 + σ3) = σ1 × σ2 + σ1 × σ3, and (σ1 + σ2) × σ3 = σ1 × σ3 + σ2 × σ3. In the future we will use dot for the multiplication, or we will omit it as usual. In the ring (S; +; ·), the group (S; ·) is anti-commutative, if for all σ 2 S is σ1 · σ2 = −σ2 · σ1. Frobenius and Peirce in 1878, and Mishchenko and Solovyov in 2000, proved that the only associative real division algebras are real numbers, complex numbers and quaternions. Lemma 1. There is only one neutral element of the group. Proof. By contradiction. Suppose that e1; e2 2 S are neutral elements and e1 6= e2. Than e1 ◦ e2 = e2 since e1 is neutral, and e1 ◦ e2 = e1 since e2 is neutral. Therefore e1 = e2 which is contradictions with the assumption. Lemma 2. If σ ◦ x = σ ◦ y, then x = y. Proof. There exists inverse σ−1 2 S for each σ. Therefore: x = e ◦ x = (σ−1 ◦ σ) ◦ x = σ−1 ◦ (σ ◦ x) = σ−1 ◦ (σ ◦ y) = e ◦ y = y: The reducing the equations, so that from 3x = 3y follows x = y. We can see it as left multiplication to the both sides by the inverse element 3−1. Lemma 3. Inverse of inverse is the starting element. Proof. It is σ−1 ◦ (σ−1)−1 = e = σ−1 ◦ σ. For lemma 2 follows (σ−1)−1 = σ. That's why the minus of minus is plus. −1 −1 −1 Lemma 4. It is always (σ1 ◦ σ2) = σ2 ◦ σ1 . Proof. It is −1 −1 σ1 ◦ (σ2 ◦ (σ1 ◦ σ2) ) = (σ1 ◦ σ2) ◦ (σ1 ◦ σ2) = e: −1 Left-multiplying the both sides by σ1 gives successively: −1 −1 −1 σ1 ◦ (σ1 ◦ (σ2 ◦ (σ1 ◦ σ2) )) = σ1 ; 2 −1 −1 −1 (σ1 ◦ σ1) ◦ (σ2 ◦ (σ1 ◦ σ2) ) = σ1 ; −1 −1 e ◦ (σ2 ◦ (σ1 ◦ σ2) ) = σ1 ; −1 −1 σ2 ◦ (σ1 ◦ σ2) = σ1 : −1 Left-multiplying the both sides by σ2 gives: −1 −1 −1 −1 σ2 ◦ (σ2 ◦ (σ1 ◦ σ2) ) = σ2 ◦ σ1 ; −1 −1 −1 −1 (σ2 ◦ σ2) ◦ (σ1 ◦ σ2) = σ2 ◦ σ1 ; −1 −1 −1 e ◦ (σ1 ◦ σ2) = σ2 ◦ σ1 ; −1 −1 −1 (σ1 ◦ σ2) = σ2 ◦ σ1 : And so on. That is the method of building the abstract algebra whose one fundament is the group theory. 1.2 Matrix The square matrix of the second order is a group in relation to the matrix multiplication: µ µ ν ν σ σ µ^ν^ = 11 12 11 12 = 11 12 =σ; ^ µ21 µ22 ν21 ν22 σ21 σ22 P2 where all matrix coefficients µab, νab and σab = γ=1 µaγνγb are complex numbers. The neutral element of that group is the unit matrix: 1 0 e^ = : (1) 0 1 From linear algebra we know that the matrices multiplied by a complex number (scalar) gives the matrix of the same order with each coefficient of the matrix multiplied by that number. The sum of two matrices (of the second order) is the matrix of the same order whose coefficients are the sum of the corresponding coefficients. Matrixes are commutative group in relation to the addition of the matrix. Lemma 5 (Pauli matrices). Solutions of the matrix equation σ^2 =e ^ are e^ plus the three Pauli matrices: 0 1 0 −i 1 0 σ^ = ; σ^ = ; σ^ = ; (2) x 1 0 y i 0 z 0 −1 p where i = −1 is imaginary unit. 3 These matrices are known from quantum mechanics, from the Pauli equation which takes into account the interaction of the spin of a particle with an external electromag- netic field. The proof of this lemma follows from direct multiplication Pauli matrices. These are the only Hermitian1 and unitary2 solutions. Lemma 6 (Quaternion matrices). Solutions of the matrix equation σ^2 = −e^ are the Quaternion matrices: i 0 0 1 0 i σ^ = ; σ^ = ; σ^ = ; (3) i 0 −i j −1 0 k i 0 p with i = −1. Quaternion matrices are used in quantum mechanics too, and in the treatment of multi body problems. The assertion of Lemma 6 is easy to check by direct matrix multiplication. For example, multiplying the Pauli matrices we obtain: 2 2 2 σ^x =σ ^y =σ ^z =e; ^ σ^xσ^yσ^z = ie;^ (4) iσ^x =σ ^yσ^z = −σ^zσ^y; iσ^y =σ ^zσ^x = −σ^xσ^z; iσ^z =σ ^xσ^y: Multiplying the Quaternion matrices: 2 2 2 σ^i =σ ^j =σ ^k = −e;^ σ^iσ^jσ^k = −e;^ (5) σ^i =σ ^jσ^k = −σ^kσ^j; σ^j =σ ^kσ^i = −σ^iσ^k; σ^k =σ ^iσ^j = −σ^jσ^i: Together with the unitary matrix, each of these groups makes the base of the four- dimensional vector space. Indeed, the linear systemeξ ^ 0 +σ ^1ξ1 +σ ^2ξ2 +σ ^3ξ3 = 0 has only the trivial solution (0; 0; 0; 0) in both cases, Pauli or quaternionsσ ^γ, γ = 1; 2; 3. In the both cases, the determinant of the said system is 2. Example 1. Represent an arbitrary matrix of the second order by: a. Pauli, b. Quaternion matrices. Result. a. a11 a12 a11 + a22 a12 + a21 i(a12 − a21) a11 − a22 = e^ + σ^x + σ^y + σ^z: a21 a22 2 2 2 2 b. a11 a12 a11 + a22 i(a11 − a22) a12 − a21 i(a12 + a21) = e^ − σ^i + σ^j − σ^k: a21 a22 2 2 2 2 1The Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. 2The complex square matrixu ^ is unitary ifu ^∗u^ =u ^u^∗ = e. The analogue of a unitary matrix in real is an orthogonal matrix. 4 Returning to the complexp numbers,p imaginary unit will be indicated by σi; σj; σk, or simply by i; j; k = −1. So −16 = 4i or 4j or 4k. Quaternion is then complex numbers of the form q = q0 + iqi + jqj + kqk, where q0; qi; qj; qk 2 R and properties similar to (5). 1.3 Versor Simply said a versor is a unit quaternion. The word is from Latin versus (turned) and was introduced by Hamilton in the context of his quaternion theory [2]. The versor of a Cartesian axis is also known as a normalized vector or standard basis vector. 3 In a 3-dimensional Euclidean space (R ) in a Cartesian coordinate system Oxyz the versors of the axes are 8 < x : i = (1; 0; 0); y : j = (0; 1; 0); (6) : z : k = (0; 0; 1); as shown in Figure 1. They define the direction of −! the axes and represent any vector OA or ~a = ~ax + ~ay + ~az as a = axi + ayj + azk: −! Versor of the vector OA has the form vA = cos αxi + cos αyj + cos αzk; where cos αn = an=a for n 2 fx; y; zg. The length q a = k~ak = a2 + a2 + a2 is also called magnitude Figure 1: Versors i, j, k. x y z or tensor of the vector ~a. The both projections of 0 the vector vA and A to the plane Oxy lie on the same straight line OA , at Figure 1. Generally, each versor has the form v = cos α + r sin α; r2 = −1 (7) and represents the rotation for the angle α (in radian) about the axis r. Now let us try to explain.
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