A comparative study of quaternionic rotational Dirac equation and its interpretation
aa r X i v : . [ phy s i c s . g e n - ph ] N ov A comparative study of quaternionic rotationalDirac equation and its interpretation
B. C. Chanyal ∗ and Sandhya † Department of Physics, G. B. Pant University of Agriculture and Technology,Pantnagar-263145, Uttarakhand, IndiaEmail: ∗ [email protected], [email protected] † [email protected] Abstract
In this study, we develop the generalized Dirac like four-momentum equation for rotat-ing spin-half particles in four-dimensional quaternionic algebra. The generalized quater-nionic Dirac equation consists the rotational energy and angular momentum of particle andanti-particle. Accordingly, we also discuss the four vector form of quaternionic relativisticmass, moment of inertia and rotational energy-momentum in Euclidean space-time. Thequaternionic four angular momentum (i.e. the rotational analogy of four linear momentum)predicts the dual energy (rest mass energy and pure rotational energy) and dual momen-tum (linear like momentum and pure rotational momentum). Further, the solutions ofquaternionic rotational Dirac energy-momentum are obtained by using one, two and four-component of quaternionic spinor. We also demonstrate the solutions of quaternionic planewave equation which gives the rotational frequency and wave propagation vector of Diracparticles and anti-particles in terms of quaternionic form.
Keywords: quaternion, four-vector, energy-momentum, rotational motion.
Mathematics Subject Classification:
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The results of classical theory have a large impact on the fully appropriate description of matterbut this theory does not explain the behavior of subatomic particles. The information aboutthe small-scale behavior of particles created the idea of quantum mechanics. Generally, thequantum physics deals with the description of the state of particles associated with the wavefunction. The angular momentum is an observable essential element in quantum mechanics,where the quantum particles can involve the orbital angular momentum and intrinsic angularmomentum. Further, Erwin Schrï¿œdinger developed a fundamental wave equation for non-relativistic microscopic particles but this equation does not applicable to particles moving with1elativistic velocity. To combine the special theory of relativity with quantum mechanics, therehas been developed the relativistic quantum mechanics. In the same way, Klein-Gordon andDirac independently investigated the relativistic wave equations by combining special relativitywith quantum mechanics. These relativistic wave equations describe the various phenomenonsthat occur in high energy physics and are invariant under Lorentz transformations. Dirac [1] discussed a relativistic quantum wave equation by using Hamiltonian operator to overcome thedifficulties arising in Klein-Gordon equation. As we know that the conservation of energy andangular momentum are one of the mandatory conservation laws to check the validity of Diracparticles. Keeping in mind the conservation laws of energy and angular momentum of a rotatingparticle, in this paper, we proposed a quaternionic Dirac equation that consists not only theenergy representation but also shows the angular momentum representation of spin-1/2 particles.The quaternion number [2] is basically an extension of complex numbers. Although, there arefour types of norm-division algebras, i.e. real, complex, quaternion and octonion algebra. Thedivision algebra is defined as an algebra in which all non-zero elements have their inverse undermultiplication [3]. Nowadays, the quaternionic algebra is a popular algebra to study the varioustheories in modern theoretical physics. The quaternionic algebra is associative and commutativeunder addition but not commutative under multiplication. Thus, this algebra form a group undermultiplication but not an Abelian group, is also called the division ring.Many researchers have attempted the formulation of usual Dirac equation for free particles interms of quaternionic algebra. Firstly, Rotelli [4] developed the Dirac equation in quaternionicfour fields. The another version of quaternionic Dirac equation has been studied by Rawat et.al [5] with the description of quaternionic spinors for positive and negative energy solutions. Assuch, the quaternionic form of Dirac equation with the connection of supersymmetric quantummechanics has also been discussed [6]. Besides, the quaternionic algebra has also been usedto describe the rotational motion of a rigid body where the quaternionic unit elements are di-rectly connected with matrices of rotational group SO (3) [7]. Further, the quaternion algebraexplained the special theory of relativity [8], Dirac Lagrangian [9], superluminal transformationsfor tachyons [10], wave equation in curved space-time [11], electromagnetic-field equations [12],gravi-electromagnetism [13], quantum mechanics [14, 15], particle in a relativistic box [16] anddual magneto-hydrodynamics of dyonic cold plasma [17]. On the other hand, many authors[18]-[37] have used hyper-complex algebras to study the several theories in different branchesof physics. Chanyal [38, 39] proposed a novel idea on the quaternionic covariant theory of rel-ativistic quantized electromagnetic fields of dyons. Carmeli [40, 41, 42] discussed the variousfundamental equations of quantum mechanics viz. Klein-Gordon equation, Schrï¿œdinger equa-tion, Weyl equation and the Dirac field equation of rotating particles on R × S topological space.Keeping in mind the properties of rotating spin 1/2 particles, we generalize the Carmeli’s fieldtheory in terms of quaternionic field. The benefit of generalized quaternionic field is that, wecan analysis the four-momentum representation of a particle in a single equation i.e. energy (asa scalar component) and momentum (as a vector component). In present study, starting withquaternionic algebra and its representation to SU (2) group (i.e. an isomorphic to orthogonalgroup SO (3) ), we construct the generalized Dirac like energy-momentum equation for rotating2articles in four-dimensional quaternionic space-time. We define the quaternionic moment ofinertia and rotational energy-momentum of a rotating particle by using four-relativistic mass,four-spaces and four-momentum. A novel approach to unified quaternionic Dirac like equationcontains the rotational energy (corresponding to the coefficient of quaternionic pure scalar unitelement) and the rotational momentum (corresponding to the coefficient of quaternionic purevector unit elements). Accordingly, the rotational energy and momentum solutions are obtainedby using one, two and four-component spinor forms of quaternionic wave function. We havecalculated the solutions of rotational energy and rotational momentum for particles with spin upand spin down states. To considering the wave nature of spin half particles (or anti-particles),we have studied the general form of quaternionic plane wave equation and developed the quater-nionic form of rotational frequency and wave propagation vector for particles and anti-particles.This theory also point out the conservation law of quaternionic four-momentum of particles. The quaternionic field ( H ) is expressed by a linear algebra consist four unit elements known asquaternionic basis ( e , e , e , e ) . In a quaternionic field, the unit element e is used to expressscalar field while the other unit elements e , e , e are used to express vector fields. Thus, thequaternionic field algebra can be expressed as H = e ( H S ) + e j ( H V j ) , ( ∀ j = 1 , , ≡ e h + e h + e h + e h , (2.1)where ( h , h j ) are the real numbers corresponding to quaternionic scalar-field ( H S ) and vector-field ( H V ) . If the real part of a quaternion is zero then the quaternion have only vector fieldcomponents called a pure quaternion as H → H P = ( e h + e h + e h ) , and if the imaginarypart of a quaternion is zero then the quaternion involve only scalar component called a realquaternion as H → H R = e h . The addition of two quaternions ( A , B ) ∈ H produce a newquaternionic field as A + B = ( e a + e a + e a + e a ) + ( e b + e b + e b + e b )= e ( a + b ) + e ( a + b ) + e ( a + b ) + e ( a + b ) . (2.2) ( a , a j ) & ( b , b j ) ∈ R The quaternions are associative as well as commutative under addition, i.e. A + B = B + A , A + ( B + C ) = ( A + B ) + C , ∀ ( A , B , C ) ∈ H . (2.3)3he quaternionic algebra contains additive identity element zero as e e e e . (2.4)Every quaternion has its additive inverse which can be expressed in the form of − H = e ( − h ) + e ( − h ) + e ( − h ) + e ( − h ) . (2.5)The quaternionic conjugate of given equation (2.1) can be expressed as, H = e ( H S ) − e j ( H V j ) ≡ e h − e h − e h − e h . (2.6)As such, the square of a quaternion is written as, H = H ◦ H = (cid:0) h + h + h + h (cid:1) , (2.7)where ′ ◦ ′ indicated the quaternion multiplication. Here, the quaternionic unit elements ( e , e , e , e ) are followed the relations e = 1 , e = e = e = − ,e e i = e i e = e i , e i e j = − δ ij e + ǫ ijk e k , ∀ ( i, j, k = 1 , , (2.8)where the symbol δ ij is the Kronecker delta symbol having value one for equal indies and zerofor unequal indies, while ǫ ijk is the Levi Civita three-index symbol taking value ǫ ijk = +1 forcyclic permutation, ǫ ijk = − for anti-cyclic permutation and ǫ ijk = 0 for any two repeatedindices. The multiplication of two quaternions can be expressed by using Table-1 as, A ◦ B = e ( a b − a b − a b − a b )+ e ( a b + a b + a b − a b )+ e ( a b + a b + a b − a b )+ e ( a b + a b + a b − a b ) , (2.9)which can be further simplify as A ◦ B = e (cid:16) a b − −→ a · −→ b (cid:17) + e j (cid:18) a −→ b + b −→ a + (cid:16) −→ a × −→ b (cid:17) j (cid:19) , ( ∀ j = 1 , , . (2.10)The quaternionic multiplication identity element can define as, e e e e . (2.11)Rather, the quaternionic field is associative but non-commutative under multiplication opera-4 e e e e e e e e e e − e − e e e − e − e e e e − e − Table 1: Quaternion multiplication tabletion, i.e. ( A ◦ B ) ◦ C = A ◦ ( B ◦ C ) , A ◦ B = B ◦ A . (2.12)Because the cross product of two vectors are always non-commutative i.e. −→ a × −→ b = −−→ b × −→ a .The two quaternions will be commutative only when the cross product of their vectors are zeroor the vectors −→ a and −→ b are parallel to each other. Moreover, the norm of a quaternion A becomes N = p A ◦ A = q a + a + a + a . (2.13)The equation (2.13) also known the modulus of a quaternion i.e. | A | . Every non-zero elementof a quaternion has an inverse A − = A | A | . (2.14)Besides, a quaternion can also be represented in the form of split quaternionic basis elements u , u ∗ , u , u ∗ as, A = u a + u ∗ a ∗ + u b + u ∗ b ∗ , ∀ ( a, a ∗ , b, b ∗ ) ∈ C ≡ u ( h − ih ) + u ∗ ( h + ih ) + u ( h − ih ) + u ∗ ( h + ih ) , (2.15)where u = 12 ( e + ie ) , u ∗ = 12 ( e − ie ) ,u = 12 ( e + ie ) , u ∗ = 12 ( e − ie ) , are written in the form of × matrix value realization as u = ! , u ∗ = ! , u = − i ! , u ∗ = − i ! . (2.16)5hus, the quaternionic basis represents the following × matrix realization as e ! , e − i − i ! , e −
11 0 ! , e − i i ! , (2.17)where the determinant of each quaternionic basis gives unity i.e., | e | = | e | = | e | = | e | = +1 .Interestingly, to considering quaternion basis ( e , e j ), the algebra of ( e , ie j ) is isomorphic tothe algebra of tau-matrices ( τ e , τ e j for j = 1 , , ) in SU (2) group representation. Here, τ e is × identity matrix corresponding to quaternion basis e while τ e j are corresponding to purequaternionic basis called Pauli tau-matrices. In given equation (2.15), the SU (2) representationis the set of all × complex matrices of determinant positive one and satisfy AA † = A † A = I × ,i.e. A = h − ih − ( h + ih ) h − ih h + ih ! a − ( ib )( ib ) ∗ a ∗ ! , (2.18)where | a | + | b | = 1 . For quaternionic rotational group, if A ∈ SU (2) maps onto R ( A ) ∈ SO (3) , then we may write R ( A ) j,k = 12 Tr ( τ e j A τ e k A − ) , ( ∀ j, k = 1 , , . (2.19)As such, the simplest representations of the quaternionic basis can also be expressed by themultiplication of ( − i ) with 2 × 2 Pauli tau-matrices as, e = ! τ e , e = − i ! iτ e ,e = − i − ii ! iτ e , e = − i − ! iτ e , (2.20)where e j ∼ = ( − iτ e j ) j = 1 , , are also defined a realization of infinitesimal rotations of threedimensions pure quaternionic space. Moreover, the quaternions can be use to study the rota-tional motion of a spin-1/2 particles because it is identical to rotational τ − matrices [43]. Wesummarize the properties of quaternionic basis and tau-matrices in Table-2. Let us start with the quaternionic relativistic mass of Dirac particles (e.g. electrons) that canbe expressed in terms of four-masses [44], M = e E c + e | p v | + e | p v | + e | p v |≃ e m + X j =1 e j m j , (3.1)6 roperties Quaternion basis Tau-matrices Square e = 1 , e j = − τ e ) = 1 , ( − iτ e j ) = − Determinant
Eigen values ± i ± i Trace
Multiplication e i e j = − δ ij e + ǫ ijk e k τ e i τ e j = δ ij τ e + iǫ ijk τ e k Commutation [ e i , e j ] = 2 e k ǫ ijk (cid:2) τ e i , τ e j (cid:3) = 2 iǫ ijk τ e k Anti-commutation { e i , e j } = − e δ ij (cid:8) τ e i , τ e j (cid:9) = 2 δ ij τ e Table 2: Properties of quaternionic basis and τ − matriceswhere m ∼ E /c indicates the rest mass and m , m , m are the moving masses of parti-cle having velocities v , v , v corresponding to quaternionic basis e , e , e , respectively. Itshould be notice that the quaternionic scalar field associated with the coefficient of e while thequaternionic vector field associated with the coefficient of e j . If vector part is zero then the realquaternion expresses only the rest mass ( m ), and if rest mass-energy of a particle is zero thenthe pure quaternion experiences the motion of particle. As such, the quaternionic space-timeposition ( R ) can also be expressed as R = e r + e r + e r + e r , (3.2)where r is the scalar component considering as time, while r , r , r are the three spacialcomponents in Euclidean space-time. Now, the moment of inertia in quaternionic form becomes, I = M (cid:0) R ◦ R (cid:1) = M (cid:0) r + r + r + r (cid:1) = M R . (3.3)Thus, we have I = e I + e I + e I + e I , (3.4)where I = m R is considered the moment of inertia along scalar axis e while I = m R , I = m R and I = m R are the moment of inertia along pure-quaternionic axes e , e , e , respectively. Keeping in mind the rotational analog of translation motion, we can define thequaternionic four-angular momentum as, L = R ◦ P , (3.5)7here P
7→ { e p , e p , e p , e p } is the quaternionic linear four-momentum. Thus, fromequation (3.5), we get L = e ( r m c − r m v − r m v − r m v )+ e ( r m v + r m c + r m v − r m v )+ e ( r m v + r m c + r m v − r m v )+ e ( r m v + r m c + r m v − r m v ) . (3.6)It can be reduces as, L = e L + e L + e L + e L = e ( L − L − L − L ) + e ( L + L + L − L )+ e ( L + L + L − L ) + e ( L + L + L − L ) , (3.7)where L is purely a scalar component can be indicated by quaternionic rotational energy ( E ) ,while the vector components ( L j , j = 1 , , ) indicated the quaternionic rotational angular-momentum. Now, the quaternionic rotational energy E ∼ L can be expressed by E = r p − ( −→ r · −→ p ) , ( coefficient of e ) . (3.8)Here, in quaternionic formalism the quaternionic rotational energy consists a couple of energy.We may consider the first scalar term ( r p ) represents the rest mass-energy while the sec-ond scalar term ( −→ r · −→ p ) represents the moving (projectional) energy. In the same way, thequaternionic angular momentum can then be written as, L j = ( r −→ p + p −→ r ) + ( −→ r × −→ p ) j , ( coefficient of e j ) . (3.9)In equation (3.9), the first term indicates the longitudinal component (irrotational momentum)while the second term indicates the transverse component (rotational momentum) of quater-nionic four-momentum. Therefore, we may describe the quaternionic form of rotational energy-momentum as following specific cases: Case-1:
Conditionally, for the pure quaternionic field the scalar coefficient is taken zero i.e., r = p = 0 . Then, we get the usual three dimensional form of rotational energy and angularmomentum in vector field, i.e. | E | ≃ ( −→ r · −→ p ) , (pure rotational energy) (3.10) −→ L ≃ ( −→ r × −→ p ) , (pure angular momentum) . (3.11) Case-2:
If pure quaternion part is zero i.e. −→ r = −→ p = 0 , then we get the pure scalar field as E ≃ r p , −→ L ≃ , (3.12)8hich shows that there will be only rest mass-energy having no rotating motion. Case-3:
If the quaternionic variables R and P are mutually interchange as R → P and P → R , then the rotational energy ( E ) remains unaffected but the angular momentum ( −→ L ) become changed, i.e., E ( R ⇔ P ) = r p − ( −→ r · −→ p ) , −→ L ( R ⇔ P ) = ( p −→ r + r −→ p ) − ( −→ r × −→ p ) . (3.13) Let us start with the Dirac equation for free particle as (cid:0) −→ α · −→ p − βmc (cid:1) ψ = 0 , (4.1)where −→ α and β are the Dirac matrices. Now, to check the energy and momentum relations ofan electron rotating in quaternionic space-time, we can extend the Dirac equation (4.1) in termof QRD equation form, i.e. (cid:0) A ◦ L − B λ I (cid:1) ◦ Ψ = 0 , (4.2)where A , L , B , I and Ψ are quaternionic variables considering for the rotational analogy of −→ α , −→ p , β , m and ψ , respectively. For rotating particles the speed of light c can be replaced by themaximum speed λ [42] as λ = c (cid:0) MI (cid:1) = cR for rotating particle, where R = p r + r + r + r .Using irreducible representation of SU (2) group, we may write the rotational Dirac matrices A with quaternionic structure as A = (cid:0) D ( A ) , D j ( A ) (cid:1) , ( ∀ j = 1 , , , (4.3)where the matrices D ( A ) and D j ( A ) are quaternionic D − matrices, can be define as D ( A ) = τ e τ e ! , D j ( A ) = τ e j τ e j ! , ( ∀ j = 1 , , , (4.4)and the B − matrix is B = − ! . (4.5)9ow, using quaternion multiplication the first term A ◦ L of QRD equation (4.2) can be expressedby A ◦ L = e (cid:2) D ( A ) E − λD ( A ) L − λD ( A ) L − λD ( A ) L (cid:3) + e (cid:2) λD ( A ) L + D ( A ) E + λD ( A ) L − λD ( A ) L (cid:3) + e (cid:2) λD ( A ) L + D ( A ) E + λD ( A ) L − λD ( A ) L (cid:3) + e (cid:2) λD ( A ) L + D ( A ) E + λD ( A ) L − λD ( A ) L (cid:3) , (4.6)which can further reduces as A ◦ L = e h D ( A ) E − λ (cid:16) −→ D ( A ) · −→ L (cid:17)i + e j (cid:20) λD ( A ) L j + D j ( A ) E + λ (cid:16) −→ D ( A ) × −→ L (cid:17) j (cid:21) , ∀ ( j = 1 , , . (4.7)Correspondingly, the second term of QRD equation (4.2) i.e. B λ I can be written as, B λ I = e B λ I + e B λ I + e B λ I + e B λ I . (4.8)Therefore, from equations (4.7) and (4.8) the generalized QRD equation can be expressed as [ e { D ( A ) E − λ (cid:16) −→ D ( A ) · −→ L (cid:17) − B λ I } + e j { λD ( A ) L j + D j ( A ) E + λ (cid:16) −→ D ( A ) × −→ L (cid:17) j − B λ I j } ] ◦ Ψ = 0 . (4.9)Interestingly, the QRD equation (4.9) consist both scalar and vector components that gives notonly the rotational energy but also gives the angular momentum of electrons.Real quaternionic field ( corresponding to e ) : Dirac rotational energy , Pure quaternionic field ( corresponding to e j ) : Dirac rotational momentum .To explain the dual nature of quaternionic unified rotational energy-momentum solutions, westart with quaternionic spinor ( Ψ ) with scalar and vector fields as [6], Ψ = e Ψ + e Ψ + e Ψ + e Ψ = (Ψ + e Ψ ) + e (Ψ − e Ψ )= Ψ a + e Ψ b , (4.10)where Ψ a = (Ψ + e Ψ ) and Ψ b = (Ψ − e Ψ ) . Further, for two and four components form,the quaternionic spinors can be written as Ψ = Ψ a Ψ b ! , (two component form), (4.11)10nd Ψ = Ψ Ψ Ψ Ψ , (four component form) . (4.12)Now, in next section we shall use one, two and four-component spinors to determine the solutionsof quaternionic rotational energy and angular momentum. In order to attempt the energy solutions of QRD equation, we equate the scalar components(coefficient of e ) in given equation (4.9) as h D ( A ) E − λ (cid:16) −→ D ( A ) · −→ L (cid:17) − B λ I i Ψ = 0 , (5.1)substituting the value of D ( A ) , −→ D ( A ) and B from equations (4.4) and (4.5), and obtain thefollowing matrix form E − λ I − iλ (cid:16) −→ e · −→ L (cid:17) − iλ (cid:16) −→ e · −→ L (cid:17) E + λ I Ψ a Ψ b ! = 0 , (5.2)which gives (cid:0) E − λ I (cid:1) Ψ a − iλ (cid:16) −→ e · −→ L (cid:17) Ψ b = 0 , (5.3) (cid:0) E + λ I (cid:1) Ψ b − iλ (cid:16) −→ e · −→ L (cid:17) Ψ a = 0 , (5.4)where −→ e → ( e , e , e ) . Equations (5.3) and (5.4) are represented the positive and negativeenergies of a rotating Dirac particle, respectively. The values of Ψ a and Ψ b are coupled withthe function of rotational energy and angular momentum ( E , −→ L ), i.e. Ψ ( E , −→ L ) = λE − λ I i (cid:16) −→ e · −→ L (cid:17) Ψ ( E , −→ L ) , (5.5) Ψ ( E , −→ L ) = λE − λ I i (cid:16) −→ e · −→ L (cid:17) Ψ ( E , −→ L ) , (5.6) Ψ ( E , −→ L ) = λE + λ I i (cid:16) −→ e · −→ L (cid:17) Ψ ( E , −→ L ) , (5.7) Ψ ( E , −→ L ) = λE + λ I i (cid:16) −→ e · −→ L (cid:17) Ψ ( E , −→ L ) , (5.8)11hich yields, Ψ Λ ( E , −→ L ) − i Ω + ( E ) h −→ e · −→ L i Ψ Λ +2 ( E , −→ L ) =0 , ( Λ = 0 ,
1) ( for positive rotational energy )Ψ Λ ( E , −→ L ) − i Ω − ( E ) h −→ e · −→ L i Ψ Λ − ( E , −→ L ) =0 , ( Λ = 2 ,
3) ( for negative rotational energy ) . (5.9)Equation (5.9) shows the complex behavior of quaternionic quantum wave function associatedwith the interaction between quaternion spin and orbital angular momentum (cid:16) −→ e · −→ L (cid:17) . Itshould be notice that the imaginary unit i placed with quaternionic basis to represent therotational matrices, so that, ie j ∼ = τ e j , j = 1 , , . Here, Ω ± ( E ) = λE ∓ λ I is a constant usedfor a quaternionic rotational energy (i.e. Ω + ( E ) corresponding to positive energy and Ω − ( E ) corresponding to negative energy). In this case, we consider Ψ = 1 , Ψ = 0 for spin up and Ψ = 0 , Ψ = 1 for spin down positiveenergy solutions. Then, we obtain Ψ → Ψ ↑ + ( E , −→ L ) = N E + e iλ (cid:16) −→ e · −→ L (cid:17) E + λ I , (5.10) Ψ → Ψ ↓ + ( E , −→ L ) = N E + e e iλ (cid:16) −→ e · −→ L (cid:17) E + λ I , (5.11)where the normalization constant N E + = E + λ I q ( E + λ I ) + λ L j . Similarly, for negative energysolutions with spin up and spin down states, we get Ψ → Ψ ↑− ( E , −→ L ) = N E − iλ (cid:16) −→ e · −→ L (cid:17) E − λ I + e , (5.12) Ψ → Ψ ↓− ( E , −→ L ) = N E − e iλ (cid:16) −→ e · −→ L (cid:17) E − λ I + e . (5.13)where N E − = E − λ I q ( E − λ I ) + λ L j . It should be notice that in one component formalism, allpositive and negative energy (spin up and spin down) spinors associated with quaternionic basiswith the fields corresponding to particle and antiparticle.12 .2 Two component solutions The two component solutions corresponding to positive and negative energy with spin up stateare expressed in quaternionic from as, Ψ → Ψ ↑ + ( E , −→ L ) = N E + iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I (5.14) Ψ → Ψ ↑− ( E , −→ L ) = N E − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I (5.15)Similarly, for positive and negative energy with spin down states, Ψ → Ψ ↓ + ( E , −→ L ) = N E + e − iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I ≃ − iN E + − iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I . (5.16) Ψ → Ψ ↓− ( E , −→ L ) = − N E − e − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I ≃ iN E − − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I (5.17)Notice that, the quaternionic basis element e shows the diagonal matrix which does not able tochange the state of orientation of spin-1/2 particles, while the basis e and e show off-diagonalmatrices whose can transform the state of orientation of spin-1/2 particles. The quaternionicunified of two component solution can now be written as, Ψ Unified ( E , −→ L ) ≃ N E + iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I − i − iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I + N E − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I + i − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I . (5.18)The unified spinor-function of two component solution shows a complex behavior of quaternionicfield where the each component predicts the energy solution for particles (or anti particles) withtheir spinor. Like one and two component solutions, we can extent it into four component solutions. In thiscase, we obtain the quaternionic four component solutions for positive energy with spin up anddown states as Ψ → Ψ ↑ + ( E , −→ L ) ≃ N E + iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I , Ψ → Ψ ↓ + ( E , −→ L ) ≃ N E + − iλ (cid:16) −→ e ·−→ L (cid:17) E + λ I , (5.19)13imilarly, the negative energy solutions for spin up and down states are Ψ → Ψ ↑− ( E , −→ L ) ≃ N E − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I , Ψ → Ψ ↓− ( E , −→ L ) ≃ N E − − iλ (cid:16) −→ e ·−→ L (cid:17) E − λ I . (5.20)Interestingly, the one, two and four component solutions are isomorphic to each other. Inquaternionic formalism, the scalar term (cid:16) −→ e · −→ L (cid:17) can be used as the rotational helicity of par-ticle, which represents the quaternionic spin-orbit interaction energy. In right hand rotationalhelicity the direction of spin is along to the direction of quaternionic angular momentum, whilefor left hand rotational helicity the direction of spin is opposite to the direction of quaternionicangular momentum. To discuss the momentum like solutions of QRD equation (4.9), we compare the quaternionicvector coefficient e j ( ∀ j = 1 , , as, (cid:20) λD ( A ) L j + D j ( A ) E + λ (cid:16) −→ D × −→ L (cid:17) j − B λ I j (cid:21) Ψ = 0 . (6.1)Now, putting the values of D ( A ) , D j ( A ) and B from equations (4.4) and (4.5) in givenequation (6.1), we get λ ( L j − λI j ) i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) λ ( L j + λI j ) Ψ a Ψ b ! = 0 , (6.2)which gives λ ( L j − λI j ) Ψ a + i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ b = 0 , (6.3) λ ( L j + λI j ) Ψ b + i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ a = 0 . (6.4)Equations (6.3) and (6.4) are identical to vector analogy of Dirac’s energy equations called gen-eralized quaternionic angular momentum equations which can describe respectively the angularmomentum of a electron and positron. Substituting the values of Ψ a and Ψ b , we obtain the14ollowing equations: Ψ ( E , −→ L ) = 1 iλ ( L j − λI j ) (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ ( E , −→ L ) , (6.5) Ψ ( E , −→ L ) = 1 iλ ( L j − λI j ) (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ ( E , −→ L ) , (6.6) Ψ ( E , −→ L ) = 1 iλ ( L j + λI j ) (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ ( E , −→ L ) , (6.7) Ψ ( E , −→ L ) = 1 iλ ( L j + λI j ) (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ ( E , −→ L ) , (6.8)In the simplified form the above equations reduce to, Ψ Λ ( E , −→ L ) + i Ω + ( −→ L ) (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ Λ +2 ( E , −→ L ) =0 , ( Λ = 0 , Λ ( E , −→ L ) + i Ω − ( −→ L ) (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) Ψ Λ − ( E , −→ L ) =0 , ( Λ = 2 , . (6.9)Here, Ω ± ( −→ L ) = λ ( L j ∓ λI j ) is a rotational variable, can be used as Ω + ( −→ L ) for particle and Ω − ( −→ L ) for anti-particle angular momentum. The term (cid:16) −→ e × −→ L (cid:17) shows a directional interactionbetween quaternion spin and orbital angular momentum. Now, using equations (6.5)-(6.8) wecan analysis one, two and four component solutions for QRD equation. We can start the angular momentum solutions with spin up state as Ψ = 1 and Ψ = 0 andspin down states as Ψ = 0 and Ψ = 1 of Dirac-particle, and obtain Ψ → Ψ ↑ + ( E , −→ L ) = N L + − e i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) λ ( L j + λI j ) , (6.10) Ψ → Ψ ↓ + ( E , −→ L ) = N L + e + e e i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) λ ( L j + λI j ) , (6.11)where N L + = λ ( L j + λI j ) s [ λ ( L j + λI j )] − (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) . Correspondingly, the angular momentum solu-tions for rotating Dirac anti-particle, we take Ψ = 1 and Ψ = 0 for spin up state and Ψ = 0 Ψ = 1 for spin down states, then Ψ → Ψ ↑− ( E , −→ L ) = N L − − i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) λ ( L j − λI j ) + e , (6.12) Ψ → Ψ ↓− ( E , −→ L ) = N L − − e i (cid:20) e j E + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) λ ( L j − λI j ) − e e , (6.13)where N L − = λ ( L j − λI j ) s [ λ ( L j − λI j )] − (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) . Like energy solutions, the quaternionic angularmomentum solutions of one component describe the rotational momentum field for Dirac particleand anti-particle.
For the study of quaternionic two component angular momentum solutions, we extant onecomponent angular momentum solutions into two component solution by using equation (4.11).Thus, the quaternionic two component solutions corresponding to spin up and spin down statesof particle and anti-particle are expressed by, Ψ → Ψ ↑ + ( E , −→ L ) = N L + − i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j + λI j ) , (6.14) Ψ ↑− ( E , −→ L ) = N L − − i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j − λI j ) , (6.15) Ψ ↓ + ( E , −→ L ) = − iN L + i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j + λI j ) , (6.16) Ψ ↓− ( E , −→ L ) = iN L − i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j − λI j ) . (6.17)The two component solutions of quaternionic angular momentum show that how spinors areassociated with the rotational motion of particle and anti-particle.16 .3 Four component solutions Further, we also may write the four component solutions of quaternionic angular momentumfor spin up and spin down states of Dirac-particle as, Ψ → Ψ ↑ + ( E , −→ L ) ≃ N L + − i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j + λI j ) , (6.18) Ψ ↓ + ( E , −→ L ) ≃ N L + i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j + λI j ) , (6.19)and for anti-particle, we obtain Ψ → Ψ ↑− ( E , −→ L ) ≃ N L − − i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j − λI j ) , (6.20) Ψ ↓− ( E , −→ L ) ≃ N L − i (cid:20) e j E + λ (cid:16) −→ e ×−→ L (cid:17) j (cid:21) λ ( L j − λI j ) . (6.21)Thus, in quaternionic formalism, one, two and four component angular momentum solutions areisomorphic to each other. The interesting part in quaternionic description for Dirac equation is,it shows not only rotational energy solution but also shows the rotational momentum solutionsfor considering the four-dimensional Euclidean space-time. We know that an electron rotate in a permissible orbit exhibits the wave-nature. In orderto calculate the angular frequency and wave propagation vector of electron and positron inquaternionic space-time, let us start with the quaternionic wave function Ψ consisting to SU (2) group elements as [42], Ψ = ∞ Σ J = (2 J + 1) J Σ M = − J T µJM D JS,M ∀ ( µ = 0 , , , , (7.1)17here J, M and S are denoted the total angular momentum, magnetic quantum number dueto total angular momentum and spin quantum number, respectively. Here (2 J + 1) definesthe discrete value of possible total angular momentum called the statistical weight, T µJM is aquaternionic variable associated with SU (2) group and D JS = ± ,M is the Dirac spinor. Thus, inquaternionic form, we have Ψ = ∞ Σ J = (2 J + 1) J Σ M = − J T JM D J ,M , ( corresponding to e ) (7.2) Ψ = ∞ Σ J = (2 J + 1) J Σ M = − J T JM D J − ,M , ( corresponding to e ) (7.3) Ψ = ∞ Σ J = (2 J + 1) J Σ M = − J T JM D J ,M , ( corresponding to e ) (7.4) Ψ = ∞ Σ J = (2 J + 1) J Σ M = − J T JM D J − ,M , ( corresponding to e ) . (7.5)Using above quaternionic rotational wave functions on equations (5.5) - (5.8), we obtain i ℏ ˙ T JM − λ I T JM − iλ (cid:16) −→ e · −→ L (cid:17) T JM = 0 , (7.6) i ℏ ˙ T JM − λ I T JM − iλ (cid:16) −→ e · −→ L (cid:17) T JM = 0 , (7.7) i ℏ ˙ T JM − λ I T JM − iλ (cid:16) −→ e · −→ L (cid:17) T JM = 0 , (7.8) i ℏ ˙ T JM − λ I T JM − iλ (cid:16) −→ e · −→ L (cid:17) T JM = 0 , (7.9)where we used energy operator E ∼ i ℏ ∂∂t . On the other hand, we may consider the generalplane wave solution of equation (7.1) in terms of quaternionic form as, T µJM = G µJM exp (cid:20) − i ℏ ( P ◦ R ) (cid:21) , (7.10)Here G µJM is a constant, P and R are usual quaternionic four-momentum and four-space, respec-tively. The scalar and vector components of quaternionic equation (7.10) are T JM = G JM exp (cid:20) − i ℏ ( E t − −→ p · −→ r ) (cid:21) , ( Coefficient of e ) (7.11) T aJM = G aJM exp (cid:20) − i ℏ (cid:18) ct −→ p + E c −→ r − ( −→ p × −→ r ) (cid:19)(cid:21) , ( Coefficient of e a ) (7.12)18here a = 1 , , . Now, substituting equations (7.11) and (7.12) and their time derivative inequations (7.6) - (7.9), we found (cid:0) − ℏ ω − λ I (cid:1) G JM − iλ (cid:16) −→ e · −→ L (cid:17) G JM = 0 , (7.13) (cid:0) − ℏ ω − λ I (cid:1) G JM − iλ (cid:16) −→ e · −→ L (cid:17) G JM = 0 , (7.14) (cid:0) − ℏ ω + λ I (cid:1) G JM − iλ (cid:16) −→ e · −→ L (cid:17) G JM = 0 , (7.15) (cid:0) − ℏ ω + λ I (cid:1) G J − iλ (cid:16) −→ e · −→ L (cid:17) G JM = 0 . (7.16)Equations (7.13) and (7.14) are manifested to negative energy solution while equations (7.15)and (7.16) are manifested to positive energy of Dirac particle. These dual-energy equations canbe reduced in × matrix form as, − ℏ ω − λ I − iλ (cid:16) −→ e · −→ L (cid:17) − ℏ ω − λ I − iλ (cid:16) −→ e · −→ L (cid:17) − iλ (cid:16) −→ e · −→ L (cid:17) − ℏ ω + λ I − iλ (cid:16) −→ e · −→ L (cid:17) − ℏ ω + λ I G JM G JM G JM G JM = 0 , (7.17)which gives ω : −→ ω ± = ± λ r λ I − ℏ (cid:16) −→ e · −→∇ Θ (cid:17) ℏ . (7.18)Here, we used angular momentum −→ L → − i ℏ −→∇ Θ , where −→∇ Θ shows the rotational analog ofnabla operator corresponding to Euler angles ( θ, φ, ψ ) [45]. The term ω + represented the angularfrequency of the Dirac like particle while the ω − represented the angular frequency of the itsanti-particle. Similarly, for the general solution of angular momentum equations, we substituteequations (7.11) and (7.12) in equations (6.5) - (6.8), and obtain λ ( L j − λI j ) G JM + i (cid:20) e j (cid:16) c −→ k (cid:17) + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) G JM =0 , (7.19) λ ( L j − λI j ) G JM + i (cid:20) e j (cid:16) c −→ k (cid:17) + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) G JM =0 , (7.20) λ ( L j + λI j ) G JM + i (cid:20) e j (cid:16) c −→ k (cid:17) + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) G JM =0 , (7.21) λ ( L j + λI j ) G J + i (cid:20) e j (cid:16) c −→ k (cid:17) + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) G JM =0 . (7.22)19here the wave propagation vector −→ k ∼ −→ p ℏ . These equations also can be written in × matrix form as, A − B A − BB A + B A + G JM G JM G JM G JM = 0 , (7.23)along with A ± = λ ( L j ± λI j ) , B = i (cid:20) e j (cid:16) c −→ k (cid:17) + λ (cid:16) −→ e × −→ L (cid:17) j (cid:21) . (7.24)Therefore, from equation (7.23) we obtain −→ k : −→ −→ k ± = ± ie j λ r(cid:16) − ℏ ∇ j − λ I j (cid:17) + i ℏ λ (cid:16) −→ e × −→∇ Θ (cid:17) j c , ( ∀ j = 1 , , . (7.25)Equation (7.25) represented an expression for wave vector (cid:16) −→ k (cid:17) corresponding to quaternionicangular momentum that propagates along quaternionic basis e j . Accordingly, −→ k + can representthe wave propagation for Dirac like particle while −→ k − can represent the wave propagationcorresponding to the anti-particle. Here, we should be notice that ( ω, −→ k ) shows the quaternionicfour-wave vector for Euclidean space. In the present work, the generalized Dirac equation for rotating particle has been demonstratedin term of quaternionic division algebra. Split quaternion is the another variety of quater-nion. The interesting part of split quaternion is that, it can use not only the wave-mechanism(Schrï¿œdinger theory) but also use for matrix-mechanism (Heisenberg theory) of quantum for-malism. The matrix realization of split quaternion shows the Pauli’s spin state of fermionsor anti-fermions. Therefore, to visualizing the rotational properties of Dirac-like particles splitquaternions or quaternions algebra can be used for Euclidean space-time. In quaternionic field,we have discussed four angular momentum consisted rotational energy and rotational momen-tum of an electron in four dimensional Euclidean space-time. The connection between rotationalmatrices (tau-matrices) along with quaternionic basis ( e , e , e , e ) has been described. Fur-ther, the quaternionic four-masses have been associated with the rest mass corresponding toquaternionic scalar basis and the moving mass corresponding to quaternionic vector basis givenby equation (3.1). We have also written the quaternionic relativistic four-space, moment ofinertia and angular momentum. The components of quaternionic resultant rotational energyand rotational momentum are established in compact and simple manner given by equations(3.8) and (3.9). We have investigated a new form of QRD equation (4.9) that unifies the rota-tional form of Dirac-energy and Dirac- angular momentum of a particle in a single framework.20he solutions of QRD equation has been represented in one, two and four components formwhich described the rotational motion of Dirac particle and anti-particle with spin up and downstates. Further, we also have discussed a general form of quaternionic wave function and itsplane wave solutions in terms of quaternionic field. 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