Quaternion Electromagnetism and the Relation with 2-Spinor Formalism

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary `$i$' should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the 2-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation.


I. INTRODUCTION
There are quite a few papers claiming that the quaternion or the octonion can be used to describe the laws of classical electromagnetism in a simpler way [1][2][3][4][5][6]. However, they are mainly limited to describing Maxwell equations. Furthermore, the meaning of quaternion and the reasons why electromagnetic laws can be concisely described by them have not been well discussed up to now. Here we list more diverse quaternion representations of the relations in electromagnetism than previously known and we introduce a new simpler notation to express quaternions. The proposed notation makes the quaternion representation of electromagnetic relations look similar to the differential-form representation of them.
Moreover, the classical electromagnetic mass density and the complex Lagrangian can be newly defined and used to represent electromagnetic relations as quaternions.
It has been already well known that the quaternion can describe the Lorentz transformations of four vectors [7]. We here rederive the quaternion representation of the Lorentz boost and the rotation, by using isomorphism between the basis of quaternion and the set of sigma matrices. Hence, we find that not only four vector quantities but also electromagnetic fields can be transformed simply in the quaternion representation. Starting from the 4 × 4 matrix representation of quaternion, we define a new complex electromagnetic field tensor. By using it, a complex energy-momentum stress tensor of electromagnetic fields and a complex Lagrangian can be nicely expressed. Interestingly, the eigenvalues of the complex energy-momentum stress tensor are the classical electromagnetic mass density up to sign. To define complex tensors, we introduce a new spacetime index called 'tilde-spacetime index'. Imaginary number i is usually linked to time so that it can be regarded as imaginary time, but we are going to insist that it is more natural for i to be linked to space. In our representation, we also find that the complex conjugate of a quaternion is equal to the quaternion consisting of the physical quantities with parity inversion.
The 2-spinor formalism is known to be a spinor approach which is useful to deal with the general relativity [8,9]. In the formalism, all world-tensors can be changed to even-indexed spinors and there we will derive spinor descriptions of electromagnetism [10]. We here prove that the quaternion representations including Maxwell's equations are equivalent to the spinor representations of electromagnetism. We also explain how spinors in 2-spinor formalism are generally linked to the quaternion. Finally, we explore the meaning of quaternion and more extended algebras like octonion as n-fold rotation.

II. COMPLEX QUATERNION
Let us denote quaternions by characters with a lower dot such as q . . Quaternions are generally represented in the form where s, v 1 , v 2 , v 3 are real numbers and i, j, k are the units of quaternions which satisfy Eq. (1) consists of two parts, namely a 'scalar' part s and a 'quaternion vector' part v 1 i + v 2 j + v 3 k. If we denote the quaternion vector part by v, (1) is written as All quaternion vectors, denoted by an over-arrow symbol , can be interpreted as coordinate vectors in R 3 . We don't distinguish between vectors and quaternion vectors in this paper.
If q . 1 = a + A and q . 2 = b + B are two quaternions, the multiplication of the quaternions can be described as by applying (2), where A · B is the dot product and A × B is the cross product. The dot product and the cross product which are operations for 3-dimensional vectors are used in quaternion vectors.
The components of quaternions can be extended to complex numbers. We call such a quaternion 'complex quaternion'. The general form of complex quaternion is where a, b and components of c, d are real numbers, and i is a complex number √ −1 which differs from the quaternion unit i.
We denote the operation of complex conjugation by a bar¯, and the complex conjugate For a quaternion vector q = q 1 i + q 2 j + q 3 k, the exponential of q is defined by since q . 2 = −| q| 2 [11].

A. Electromagnetic Quantities
We use the unit system which satisfies 0 = µ 0 = c = 1 where 0 is vacuum permittivity, µ 0 is vacuum permeability and c is speed of light. The sign conventions for the Minkowski In the classical electromagnetism, the density of electromagnetic field momentum p and the density of electromagnetic field energy u are defined by where E is a electric field, B is a magnetic field [12,13]. In our unit system, the electromagnetic momentum p ≡ 0 E × B (in SI units) is the same as the Poynting vector S ≡ 1 µ 0 E × B (in SI units).
We define a complex Lagrangian L and an electromagnetic mass density m by The electromagnetic mass density m is defined from the energy-momentum relation m 2 = u 2 − | p| 2 where (u, p) is four-momentum of a particle of mass m. The meaning of m should be investigated more in detail; however, it is not discussed here. Comparing L and m, we can see that

B. Complex Quaternion Representations of Electromagnetic Relations
Let us define a few physical quantities in the form of complex quaternion, where γ is 1/ √ 1 − v 2 for the velocity v, V is the electric potential, A is the vector potential, E is the electric field, B is the magnetic field, ρ is the charge density, and J is the electric current density. J . is equal to ρu . and F . is just a quaternion vector. The scalar part of f . is the rate of work done by electric field on the charge and the vector part is the Lorentz force.
We define a quaternion differential operator by where t is the time and ∇ = ∂ x i + ∂ y j + ∂ z k is the vector differential operator in the three dimensional Cartesian coordinate system.
The relations in electromagnetism can be described in the complex quaternion form simply as follows: where l .
We can check all quaternion relations by expanding multiplications of quaternions using Eq. (4). Some expansions are proven in Appendix A. Relations 1), 3) and 4) have been already known [1][2][3][4], but the others have not been known so far. Each quaternion equation in (13) contains several relations, which are known in classical electromagnetism: Let us discuss in more detail each relation in Eqs. (13).
2) A . = A . + d . λ describes the gauge transformation of gauge fields.
3) d .Ā . = F . contains three relations. One is Lorentz gauge condition and the others are the relations between fields strength and gauge fields, as shown in Eq. (A1), It can be the wave equations of gauge fields with sources in the Lorentz gauge,  8) 1 2 F .F . = p . is the quaternion representation of electromagnetic energy and momentum. It can be easily verified, by expanding the left side, that 9) d .p . is expanded as Substituting (22) into (23), we get is the Maxwell stress tensor and Eq. (26) is not a well-known relation. The proof of the expansion is given in Appendix B.
By looking at Eqs. (24), (25) and (26), it is difficult to find a simple quaternion formula like d .p . = f . + l . . The exact formula of d .p . is obtained as The proof is given in Appendix C.
10) 1 2 F . F . = L is the relation between the complex Lagrangian and electromagnetic fields.
. This is, in fact, the Euclidean Lagrangian including topological term [14,15]. The real part 1 2 The variation of this part gives the first two Maxwell's equations (17). The complex part E · B is 1 4 F µν * F µν which is the topological term of gauge fields where * F µν is Hodge dual of F µν . Its variation gives the other two Maxwell's equations (18). 11) p .p . = LL = m 2 is a Lorentz invariant and a gauge invariant quantity.
We can get the quaternion representation of Lorentz transformation by using isomorphism given above and the spinor representation of the Lorentz group. Let us denote by S[Λ] the spinor representation of the Lorentz group which acts on Dirac spinor ψ(x). Then Dirac In the chiral representation of the Clifford algebra, the spinor representation of rotations where φ = φφ, η =v tanh −1 | v|, φ is the rotation angle,φ is the unit vector of rotation axis, v is the boost velocity, andv is the unit vector of boost velocity.
Since it is known [16] that the following relation also holds: for any four-vector V µ .
The components of (33) are . This represents the quaternion Lorentz transformation for Let us define Lorentz transformation factor ζ(φ, η) by where cosh η = γ, sinh η = γv. Since (cosh η 2 +iη sinh η 2 ) = γ +i γ v is the quaternion velocity u . ( v) of a boosted frame with a boost velocity v, Eq. (36) can be rewritten as where R . ( φ) ≡ (cos φ 2 +φ sin φ 2 ). The inverse and complex conjugate of ζ(φ, η) are defined as From Eqs. (34) and (35), the Lorentz transformations of a quaternion which has the form Therefore, the Lorentz transformations of a quaternion gauge field A . and a quaternion strength field F . are As an example, if we boost a frame with a speed v along x axis, then which is a very efficient representation in computing rotations and boosts, because it is visually intuitive to choose proper coordinate for computation.

A. Complex Space and Real Time
In this Section, we explain that it is more natural to attach imaginary number i to the spatial coordinates rather than to the time coordinate. The infinitesimal version of the Lorentz transformation in one dimension is This can be manipulated to where v = dx s /dt s is a boost velocity, dx s is an infinitesimal displacement of the moving frame and dt s is an infinitesimal time it takes for the frame to move along the displacement.
If we put imaginary number 'i' to the spatial coordinate as (45) and (46), the Lorentz transformation can be seen as a kind of rotation, for pure imaginary angles α, β and r = (dt 2 s + (i dx s ) 2 ). In contrast, if we put i to the time coordinate rather than to the spatial coordinate, then i dt = r cos α, dx = r sin α, which means that (45) and (46) cannot be regarded as a kind of rotation.
and the vector matrix by # notation as where ijk are the Levi-Civita symbols.
Then the electromagnetic tensor F µν can be represented as where E, B are vector matrix of E, B and superscript E t means the transpose of a matrix E.
The dual tensor can be represented as Now we define tensor indices with tilde such as 'μνρ..', called 'tilde-spacetime indices'. Oμ and Oμ for any Applying this rule to electromagnetic tensors, we get

B. The 4 × 4 Represtation of Complex Quaternions
The basis elements of quaternion, 1, i, j, k, can be represented as 4 × 4 matrices A quaternion such as q . = a + b 1 i + b 2 j + b 3 k can be represented in the tensor representation where T means the tensor representation. When a = 0, T (q . ) has a simple form For a quaternion field strength F . = Ei − B = F 1 i + F 2 j + F 3 k, the tensor form of F . is where F ≡ iE − B which is a vector matrix of the vector F = i E − B. This is eventually identical to Fμν + iGμν.

C. Complex Electromagnetic Tensor and Electromagnetic Laws
Let us define F and its conjugate F * as A few complex tensors can also be defined as follows, where J is the vector matrix of J, (ρE + J × B) is the vector matrix of ρ E + J × B , ∇ is the vector matrix of ∇, p is the vector matrix of p, and ( Then the following tensor relations hold: where I = (1, 1, 1, 1) is unit matrix. L and m are the complex Lagrangian and the electromagnetic mass density, Eq. (9). All relations can be easily verified by simple calculations.
Actually the components of D, J and T are equal to the components of ∂μ, Jμ and Tμν EM , where ∂ µ is the four-gradient, J µ is the electric current density and T µν EM is the electromagnetic stress-energy tensor defined as Those relations of complex tensors can be verified by using several known tensor relations in electromagnetism, instead of the direct calculation. For example, Eq. (64), which represents Maxwell's equations can be easily verified from ∂ µ F µν = 0 and ∂ µ G µν = J ν . Eq. (68) again shows that the eigenvalues of complex electromagnetic stress-energy tensor are the electromagnetic mass density up to sign.
By differentiating both sides of the relation (63), we get since ∂ a (A ab B bc ) = (∂ a A ab )B bc = (∂ a A ab )B bc + A ab (∂ a B bc ). Substituting (64) into (70) and comparing it to (66), we further get the following relations:

A. The Correspondence of 2-Spinor Representations and Quaternion Representations in Electromagnetism
Let us start with some basic contents of 2-spinor formalism. Mathematically, any null-like spacetime four-vector X µ can be described as a composition of two spinors, where σ µ are sigma matrices (σ 0 , σ 1 , σ 2 , σ 3 ), the components of ψ A are ψ 1 = ξ, ψ 2 = η for proper complex numbers ξ and η, and (ψ A ) † =ψ A . It can be rewritten as by using the relationσ µC C = ε C B ε CB σ µ BB and σ µ AA σ BB µ = 2δ B A δ B A . We now define a spinor X AA as which is equivalent to X µ . The factor, which connects a four-vector to a corresponding spinor, is called 'Infeld-van der Waerden symbol', e.g. such as 1 √ 2 σ a AA in (75). And it can be generally written as g a AA . We can extend this notation not only to a null-like four-vector but also to any tensors by multiplying more than one Infeld-van der Waerden symbols: Any tensor such as T abc.. with spacetime indices a, b, c.. can be written as a spinor T AA BB .. with spinor indices A, A , B, B .., by multiplying T abc.. with g a AA , g a BB .., such as T AA BB .. = T ab.. g a AA g a BB .. . This can be simply written as Any antisymmetric tensor H ab = H AA BB can be divided into two parts where φ AB = 1 2 H C ABC and ψ A B = 1 2 H C C A B (unprimed spinor indices and primed spinor indices can be rearranged back and forth). If H ab is real, then ψ A B =φ A B and Since an electromagnetic field tensor F ab (51) is antisymmetric, it can be written as with an appropriate field ϕ AB . There we find closely related electromagnetic relations [10]: where ∇ AA = ∂ a (in Minkowski spacetime) is the four-gradient, Φ AA = Φ a is the electromagnetic potential and J AA = J a is the charge-current vector. The former is the relation of electromagnetic potentials and strength fields, and the latter is equivalent to the two of Maxwell's equations. Now we will prove that Since J a corresponds to J . = ρ + i J, Eqs.
(80) are exactly corresponding to quaternion relations in (13) as follows: Our proof starts from manipulating F AA BB as Then, Since where i , j , k are the 3-dimensional vector indices which have the value 1, 2 or 3, and ij k is pqk δ i p δ j q for the Levi-Civita symbol ijk . Einstein summation convention is understood for 3-dimensional vector indices i, j and k. Similar to (88) and (89), Finally, for an electromagnetic tensor F AA BB , Eqs. (81) and (82) hold. Eqs. (89) and (91) also show the link between Eq. (63) and the spinor form of the electromagnetic energy-stress tensor T ab = 1 2 ϕ ABφA B .

B. General Relations of Quaternion and 2-Spinor Formalism and the equivalence between Quaternion Basis and Minkowski Tetrads
Generally speaking, all spinors with spinor indices in 2-spinor formalism are directly linked to quaternion. Since σμ = (σ 0 , iσ 1 , iσ 2 , iσ 3 ) is isomorphic to quaternion basis (1, −i, −j, −k), . For any spinors with two spinor indices in the form X AA , it can be rewritten as X AA = X a g a AA = Xãgã AA . It means that we can think of all spinors of the form X AA to be obtained by multiplying the four-vector with gã AA .
Any spinor ψ A can be represented with spin basis o A , ι A like where o A , ι A is normalized so that o A ι A = 1. It is well known that Minkowski tetrads (t a , x a , y a , z a ), which is a basis of four-vectors, can be constructed from spin basis Therefore, where a bold index, which represents a 'components', is distinguished from a normal index.
Any spacetime tensor can be divided into components and basis like V a = V a δ a a . The component matrix of Minkowski tetrads with respect to the spin basis is We can replace g a by the tilde-tetrads gã. The component matrix of tilde-tetrads gã with respect to the spin basis is which is isomorphic to 1 √ 2 (1, −i, −j, −k). From this isomorphism, we can set gã AA (= gã a ) = 1 √ 2 (1, −i, −j, −k), which is equivalent tô Then any four-vector with tilde-spacetime index can be written as

TENDED ALGEBRA
A. The role of sigma matrices and quaternion basis as an operator Let us multiply one of sigma matrices with a tilde-spacetime index by gã as an operator: This is the operation of changing the spin basis as Since σμ = (σ 0 , σ 1 i, σ 2 i, σ 3 i) is isomorphic to (1, −i, −j, −k), gã AA (σ1) A B can be written as This corresponds to changing the spin basis as and can be written as −gãk.
This corresponds to changing the spin basis as and can be written as −gãj.
Since the component of (σ2) A B is equal to ε A B and we can interpret that raising or lowering indices means changing spacetime basis.
In summary, the quaternion basis roles as a basis of spacetime itself as well as works as an

B. General Discussion on Extended Complex Algebra, and Appropriate Meaning
Quaternion algebra H is isomorphic to C × C with non-commutative multiplication rule, and the elements of H can be represented with the secondary complex number j [18]. The set of elements of the form q = a + bi + (c + di)j = z 1 + z 2 j where i 2 = j 2 = −1, ij = −ji is isomorphic to the set of quaternions q . = a + bi + cj + dk. In a similar way, we can construct a larger algebraic system of quaternions which is called 'Octonion' O by introducing tertiary complex number l, like o = q . 1 + q . 2 l. 'Sedenion' S, which is an even larger algebraic system than octonion, can also be derived by performing analogous procedure. This procedure is called Cayley-Dickson construction.
It is still questionable how octonions and sedenions can be used in physics. Since octonions have the similar structure of complex quaternions, they can be used to describe electromagnetism. Furthermore, it is known that a specific octonion is useful to describe SU(3) group, which is the symmetry group of strong interaction [19]. Sedenion is an algebra which have 16 basis elements. We suggest that its basis can be written in the form q µ ⊗ q ν , where q µ is a quaternion basis (1, i, j, k). We also speculate that this may be related to SU(4) group, which has 15 generators, or even to the theory of gravity. Since electromagnetic strength field tensor F ab = F AA BB = ϕ AB ε A B + ε ABφA B can be expressed in Algebraic system Basis Products of basis Used Vector rotation and Lorentz boost, 1l, il, jl, kl Rotation of gluon, color charges (SU (3)), electromagnetic laws with magnetic monopole e i * e j (= q µµ * q νν ) = s µµ νν q µµ q νν Gravity?, SU(4)? quaternion representation, Weyl tensor C abcd = Ψ ABCD ε A B ε C D +Ψ A B C D ε AB ε CD may be expressed by using sedenion. The representation of the basis and possible uses of each algebraic system are listed in Table I.
We can think of physical meaning of the algebras made through Cayley-Dickson construction. Multiplying complex numbers by a field implies a change in scale and phase of the field. In this point of view, the spatial rotation can be interpreted as a kind of two-fold rotation because quaternions can describe 3-dimensional spatial rotation and they consist of two independent imaginary units i and j. Moreover it might be that the space itself is constructed from a kind of two-fold rotation. Similarly, since the basis of octonion can be represented with three complex numbers (one quaternion and one complex number), the rotation between gluon color charges can be considered as a three-fold rotation. Likewise, if sedenion has useful relation with the gravity, the metric of spacetime can be deemed as a four-fold rotation.

IX. CONCLUSION
We have seen that quaternions can describe electromagnetism very concisely and beautifully. They can also represent Lorentz boost and spatial rotation in a simpler way. The complex conjugation of complex quaternion corresponds to parity inversion of the physical quantities belonging to the quaternion. We can also take a hint from the 4 × 4 matrix representation of quaternion and apply it to define the complex tensor, which in turn provides a new representation of electromagnetism. We have verified that the quaternion representation is directly linked to spinor representation in 2-spinor formalism, and then investigated meaning of quaternions; not only as a basis but also as an operator.
The use of quaternion could be extended not only for actual calculations, but also to obtain deep insights and new interpretations of physics. Any null-like vectors can be described by 2-spinors, and futhermore, Minkowski tetrads can also be constructed within the 2-spinor formalism. This formalism has the implication that the spacetime may come from 2-spinor fields. The beautiful conciseness of quaternion representation of electromagnetism and the link between quaternion and 2-spinor formalism may imply that spinors are the fundamental ingredients of all fields and the spacetime also consists of 2-spinor fields. Conversely, if those conjectures are true, then it is natural to explain why the algebras formed by the Cayley-Dickson procedure, like quaternion, is useful in the description of nature.