A REPRESENTATION OF DE MOIVRE S FORMULA OVER PAULI-QUATERNIONS

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1 ISSN Ann. Acad. Rom. Sci. Ser. Math. Appl. Vol. 9, No. /017 A REPRESENTATION OF DE MOIVRE S FORMULA OVER PAULI-QUATERNIONS J. E. Kim Abstract We investigate algebraic and analytic properties of Pauli-quaternions. Also, we compose a polar form and De Moivre s formula over Pauliquaternions and research their characteristics by using the isomorphism of the Pauli matrices. MSC: 3A99, 3W50, 30G35, 11E88 keywords: Pauli matri, quaternions, De Moivre s formula, isomorphism 1 Introduction In mathematical physics, the Pauli matrices are Hermitian and unitary which are elements of a set of three comple matrices as follows: Definition 1. [5] [p.13] The Pauli matrices are real ( )-matrices which are linear combinations of the basis matrices ( ) ( ) ( ) ( ) i =, σ =, σ 1 0 =, σ i 0 3 =, 0 1 whose multiplication rules are σ 1 = σ = σ 3 = 1, σ 1σ = iσ 3 = σ σ 1, σ σ 3 = iσ 1 = σ 3 σ and σ 3 σ 1 = iσ = σ 1 σ 3. Accepted for publication in revised form on June 13-th, 017 jeunkim@pusan.ac.kr Department of Mathematics, Dongguk University, Gyeongju-si 38066, Republic of Korea 145

2 146 J. E. Kim These matrices have been introduced by the physicist Wolfgang Pauli. In quantum mechanics, the Pauli matrices are comple matrices which has the interaction of the spin with an eternal electromagnetic field. The real linear span of {I, iσ 1, iσ, iσ 3 } is isomorphic to the real algebra of quaternions H. The isomorphism from H to this set is given by the following map which is reversed signs for the Pauli matrices: 1 I, i iσ 1, j iσ, k iσ 3 (see [1]). Alternatively, the isomorphism can be acted by a map using the Pauli matrices in reversed order such that 1 I, i iσ 3, j iσ, k iσ 1. Specially, many studies have been derived in physics and mathematics. Altmann [] presented a consistent description of the geometric and quaternionic treatment of rotation operators by the fundamentals of symmetries and matrices. Arvo [3] approached in graphics programming by using techniques for representing polar forms of matrices and variables. Ata and Yayli [4] gave one-to-one correspondence between the elements of the unit split three-sphere with the comple hyperbolic special unitary matrices. Cho [6] studied Euler s formula and De Moivre s formula for comple numbers are generalized for quaternions. Farebrother et. al [7] and Jafari and Yayli [8] established that 48 distinct ordered sets of three 4 4 skew-symmetric serve as the basis of an algebra of quaternions and studied their properties over quaternions. Kim and Shon [9] gave and refined a polar coordinate epression over split quaternions related to hyperholomorphic functions. Based on these studies, we give attention to apply to De Moivre s formula over quaternions consisting of Pauli matrices. We investigate algebraic and analytic properties of Pauli-quaternions. Also, we compose a polar form and De Moivre s formula over Pauli-quaternions by using the isomorphism of the Pauli matrices. Preliminaries We introduce definitions and notations of the Pauli matri and quaternions represented by Pauli matrices. For detailed definitions and properties of Pauli matri, we refer to [10]. A Pauli-quaternion is defined as p = σ 1 + σ + 3 σ 3 and a set of Pauli-quaternions is denoted by H P. The conjugate of a Pauli-quaternion, denoted by p, is defined as p = σ 1 σ 3 σ 3. We give the product for any Pauli-quaternions

3 De Moivre s formula over Pauli-quaternions 147 p, q H P as follows: pq = ( 0 y y 1 + y + 3 y 3 )1 + {( 0 y y 0 ) + i( y 3 3 y )}σ 1 +{( 0 y + y 0 ) + i( 3 y 1 1 y 3 )}σ +{( 0 y y 0 ) + i( 1 y y 1 )}σ 3. Also, the Pauli- Moreover, we have pp = p p = ( )1. quaternion product may be written as y i 3 i y 1 i 3 0 i 1 y. 3 i i 1 0 y 3 Let the inner product over Pauli-quaternions I(p) = and then, the norm of Pauli-quaternions is N (p) = pp = For p = σ 1 + σ + 3 σ 3, if N (p) = 1, then p is called unit Pauli-quaternions. Also, spacelike and timelike Pauli-quaternions have multiplicative inverse, denoted by p 1, of p H P, where H P = H P \ E with E = { σ 1 + σ + 3 σ 3 0 = }, and their property pp 1 = p 1 p = 1. On the other hands, lightlike Pauliquaternions have no inverses. Referring [8] and [11], we give the following definition: Definition. p is spacelike if I(p) < 0, p is lightlike if I(p) = 0, p is timelike if I(p) > 0. Proposition 1. The set of spacelike Pauli-quaternions is not closed under multiplication for Pauli-quaternions. On the other hand, the set of timelike Pauli-quaternions forms a group under multiplication for Pauli-quaternions. Proof. For any two elements p and q of the set of spacelike Pauli-quaternions, the equations I(p) = < 0 and I(q) = y 0 y 1 y y 3 < 0 are satisfied. However, the product pq satisfies the following equation: I(pq) = ( 0 y y 1 + y + 3 y 3 ) + {( 0 y y 0 ) + i( y 3 3 y )} +{( 0 y + y 0 ) + i( 3 y 1 1 y 3 )} +{( 0 y y 0 ) + i( 1 y y 1 )} = ( 0 1 3)(y 0 y 1 y y 3).

4 148 J. E. Kim Since I(p) < 0 and I(q) < 0, we have I(pq) > 0. Thus, the set of spacelike Pauli-quaternions is not closed under multiplication. On the other hand, if p and q are elements of the set of timelike Pauli-quaternions, then I(p) > 0 and I(q) > 0 are satisfied. Also, since I(pq) > 0 is satisfied, the set of timelike Pauli-quaternions is a group under multiplication for Pauli-quaternions. Similarly, the vector part of any spacelike quaternion is spacelike, but vector part of any timelike quaternion can be spacelike, timelike and null. 3 De Moivre s Formula for Pauli-quaternions Now, we epress any Pauli-quaternions in a polar form similar to Pauliquaternions and quaternions. There are two types of polar forms of the Pauli-quaternions which are referred by [8] and [10] as follows: 1) If p is spacelike, that is, I(p) = < 0, then p = N (p)(sinh θ + v cosh θ), where with N (p) = cosh θ = N (p) and sinh θ = 0 N (p) , v = 1 σ 1 + σ + 3 σ 3 and v = 1. Proposition. Let p = sinh θ + v cosh θ be a unit spacelike quaternion. Then we have p n = sinh nθ + v cosh nθ if n is odd, and p n = cosh nθ + v sinh nθ if n is even for n Z, where Z is the set of integers. Proof. For n = 1, it is trivial. By the properties of hyperbolic functions, we have p = sinh θ + cosh θ + v sinh θ cosh θ = cosh θ + v sinh θ and p 3 = sinh 3θ + v cosh 3θ. Furthermore, for n = 1, by the properties of the multiplicative inverse element of H P, we have p 1 = sinh θ + v cosh θ and p = (p ) 1 = cosh θ v sinh θ. If the calculation process is repeated, we can obtain the result. ) If p is timelike, I(p) = > 0, then p = N (p)(cosh θ + w sinh θ),

5 De Moivre s formula over Pauli-quaternions 149 where with sinh θ = N (p) N (p) = and cosh θ = 0 N (p), 0 1 3, w = 1 σ 1 + σ + 3 σ 3 and w = 1. Proposition 3. Let p = cosh θ + w sinh θ be a unit spacelike quaternion. Then we have p n = cosh nθ + w sinh nθ (1) for n Z, where Z is the set of integers. Proof. Like the proof process of Proposition (3), by applying the induction, we can obtain the epression of Equation (1). We introduce the R-linear transformations representing left and right multiplication in H P by using the De Moivre s formula for a corresponding matri representation. Let p be a Pauli-quaternion, then ϕ Lp : H P H P and ϕ Rp : H P H P defined as follows: for χ H P, ϕ Lp (χ) = pχ and ϕ Rp (χ) = χp, () respectively. The Hamilton s operator ϕ Lp and ϕ Rp can be written by the matrices: a 0 a 1 a a 3 A ϕlp = a 1 a 0 ia 3 ia a ia 3 a 0 ia 1 a 3 ia ia 1 a 0 and A ϕrp = a 0 a 1 a a 3 a 1 a 0 ia 3 ia a ia 3 a 0 ia 1 a 3 ia ia 1 a 0, respectively. For unit Pauli-quaternions χ, the mapping ϕ p : H P H P is defined by ϕ p = ϕ Lp ϕ Rp = ϕ Rp ϕ Lp. If p and q are Pauli-quaternions and λ is a real number and ϕ Lp and ϕ Rp are operators as defined in equations A ϕlp and A ϕrp, respectively, then the following properties hold:

6 150 J. E. Kim Proposition 4. Let χ, χ 1 and χ be Pauli-quaternions and let α and β be real constants. Then, 1) χ 1 = χ if and only if ϕ Lp (χ 1 ) = ϕ Lp (χ ) (ϕ Rp (χ 1 ) = ϕ Rp (χ )), ) ϕ Lp (αχ 1 + βχ ) = αϕ Lp (χ 1 ) + βϕ Lp (χ ), 3) ϕ Rp (αχ 1 + βχ ) = αϕ Rp (χ 1 ) + βϕ Rp (χ ), 4) ϕ Lp (χ)ϕ Rp (χ) = ϕ Rp (χ)ϕ Lp (χ). Proof. From the definition of ϕ Lp and ϕ Lp (see ()), we can obtain the above results. Theorem 1. Let ψ L be a mapping defined as ψ L : (H P, +, ) (M (4,R),, ); ψ L ( σ 1 + σ + 3 σ 3 ) is an isomorphism i 3 i i 3 0 i 1 3 i i 1 0 Proof. By the definition of the equality of a matri, we have that ψ L () = ψ L (y), where = σ 1 + σ + 3 σ 3 and y = y 0 + y 1 σ 1 + y σ + y 3 σ 3, implies = y, that is, ψ L is injective. For any elements of M (4,R), since there is an element of H P such that ψ L ( σ 1 + σ + 3 σ 3 ) i 3 i i 3 0 i 1 3 i i 1 0, (3) the mapping ψ L is subjective. Also, by using Equation (3), we calculate ψ L (y) and ψ L ()ψ L (y). From comparing with ψ L (y) and ψ L ()ψ L (y), we obtain that ψ L (y) = ψ L ()ψ L (y). Therefore, the mapping ψ L is an isomorphism. Theorem. Let ψ R be a mapping defined as ψ R : (H P, +, ) (M (4,R),, ); ψ R ( σ 1 + σ + 3 σ 3 ) is also an isomorphism i 3 i i 3 0 i 1 3 i i 1 0

7 De Moivre s formula over Pauli-quaternions 151 Proof. Using the similar process of the proof of Theorem 1, we can also obtain that ψ R is an isomorphism. Acknowledgement. This work was supported by the Dongguk University Research Fund of 017. References [1] G. Arfken. Mathematical Methods for Physicists. Academic Press, Orlando, [] S.L. Altmann. Rotations, Quaternions, and Double Groups. Clarendon Press, Oford, England, 3rd ed., [3] J. Arvo. Graphics Gems II. Academic Press, New York, [4] E. Ata, Y. Yayli. Split quaternions and semi-euclidean projective spaces. Chaos, Solitions and Fractals 41: , 009. [5] E.U. Condon, P.M. Morse. Quantum Mechanics. McGraw-Hill, New York, 199. [6] E. Cho. De-Moivre Formula for Quaternions. Appl. Math. Lett. 11(6):33 35, [7] R.W. Farebrother, J. GroB, S. Troschke. Matri Representaion of Quaternions. Lin. Alg. Appl. 36:51 55, 003. [8] M. Jafari, Y. Yayli. Matri Theory over the Split Quaternions. Int. Jour. Geom. 3():57 69, 014. [9] J.E. Kim, K.H. Shon. Polar coordinate epression of hyperholomorphic functions on split quaternions in Clifford analysis. Adv. Appl. Clifford Al. 5(4):915 94, 015. [10] P. Longe. The properties of the Pauli matrices A, B, C and the conjugation of charge. Physica 3(3): , [11] M. Ozdemir, A. Ergin. Rotation with Unit Timelike Quaternions in Minkowski 3-Space. J. Geom. Phys. 56:3 336, 006.