(2 + 1)-Maxwell Equations in Split Quaternions

Abstract

The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two $SO(2,1)$ subgroups of the full group of symmetry of the norms of split quaternions, $SO(2,2)$. One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second $SO(2,1)$ subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

1. Introduction

It is important to express physical variables and their relationships by maximally universal mathematical structures. In previous papers, I suggested that normed division algebras can play this role [1,2,3,4,5,6,7,8], since they provide a natural framework to describe space-time transformations, together with spinors and vectors and their equations.

It is known that besides of the usual real numbers, according to the Hurwitz theorem, there are three other unique normed division algebras $A_n$: complex numbers, quaternions and octonions [9,10,11]. These four division algebras have dimensions $n$ = 1, 2, 4, and 8, respectively. The real numbers are ordered, commutative and associative, but for each next mentioned algebra one such property is lost.

An element X of any n-dimensional normed real algebra, $A_n$, can be written as the linear combination of the basis elements $e_n$ with the real coefficients $X_n$,

$$X=X_n{e}_n.$$

(1)

The unit basis element, $e_0=1$, is real and the other $n-1$ elements are hyper-complex. The square of the unit element is always positive, while the squares of some hyper-complex basis units can be negative as well,

$$e_0^2=1,e_{n-1}^2=\pm 1.$$

(2)

In normed algebras the conjugation operation is defined, which does not affect unit element but changes the sign of all hyper-complex basis units,

$$e_0^{\ast }=e_0,e_{n-1}^{\ast }=-e_{n-1}.$$

(3)

Then, using the multiplication and conjugation rules of $e_n$, the quadratic form (norm) in $A_n$ can be defined,

$$N=X{X}^{\ast },$$

(4)

which for any two elements, X and Y, satisfy the condition

$$N(XY)=N(X)N(Y).$$

(5)

In applications of normed algebras mainly the basis elements with the negative square $e_{n-1}^2=-1$ (similar to the ordinary complex unit i), which correspond to Euclidean norms,

$$N=\sum _n{X}_n^2,$$

(6)

are used. Introducing the vector-like basis elements (with the positive squares $e^2=1$) leads to split algebras having an equal number of “+” and “–” sign terms in the definition of their norms. In particular, signatures of the quadratic forms of split algebras have the structures: (1 + 1) for hyper-numbers, (2 + 2) for split quaternions and (4 + 4) for split octonions.

Normed algebras $A_n$ are closely related to Lie algebras in various guises. The norm of division and split algebras are preserved under the transformations of their elements $X_n$ by the Lie algebras of $SO(n)$ and $SO(n/2,n/2)$, respectively; $n=2,4,8$. Algebras $A_n$ are characterized by the remarkable property of having vector, spinor and conjugate spinor representations of the same (real) dimensions, a property best known as triality. The vector, spinor and conjugate spinor, as well as the gamma matrices that transform them, are all represented by division algebra elements and their multiplication. One can show that there is a division algebra of dimension n if the double cover of the rotation group $SO(n)$ has a spinor representation whose dimension equals n.

The current paper concentrates on four-dimensional ($n=4$) normed algebra of split quaternions. On physical application of quaternions, one can find thousands of publications [12]. The most commonly quaternions are used in the areas of computer graphics, navigation systems, quantum physics and kinematics (see [13,14,15,16,17,18] and references therein).

It is known that ordinary (Hamilton’s) and split quaternions can be used to describe three-dimensional Euclidean and Minkowski spaces, respectively. Here, vector and spinor representations are considered in (2 + 2) space of split quaternions, which generates kinematics of three-dimensional Minkowski space-times [8]. There are several physical models employed in (2 + 1) spaces, such as the theory of graphene [19], black holes [20], quantum gravity [21], Anti-de Sitter/conformal field thery (AdS/CFT) correspondence [22], gauge theory of gravity [23,24], etc.

This paper is organized as follows. Section 2 briefly reviews properties of split quaternions and their applications to describe geometry of (2 + 2) space. It is shown that maximal rotations towards the second time-like coordinate in (2 + 2) space of split quaternions can be used to introduce Plank’s constant and generate quantum uncertainty relation. In Section 3, the property of triality—rotations between quaternionic vectors and spinors—is discussed. In Section 4, the generalized Cauchy–Riemann analyticity condition for quaternionic functions is introduced. It is shown that the requirement of analyticity for triality invariant quaternionic construction leads to the system of equations that are equivalent to the Dirac–Maxwell system in three-dimensional Minkowski space-time. Finally, conclusions are presented in Section 5.

The paper is accompanied by several Appendices. In Appendix A, the $SL(2,R)$ and $SU(1,1)$ matrix representations of the basis elements of split quaternions are presented. Appendix B is devoted to classification of split quaternions by the values of their norms. In Appendix C, a reminder on descriptions of vectors and spinors of (2 + 2) space is given. In Appendix D, quaternionic representations of rotations and boosts are considered. In Appendix E, decomposition of split quaternions by idempotent and nilpotent elements is described. In Appendix F, the quaternionic spinors are considered. It is shown also the validity of the so-called 3-$\psi$ rule, the Fierz identity for quaternionic spinors.

2. Split Quaternions

Algebra of split quaternions over reals is an associative, non-commutative ring, the general element of which can be written as the linear combination of four basis elements, $e_0=1$ and $e_k$, $k=1,2,3$, with the real coefficients $q_0$ and $q_k$,

$$q=q_0+q^k{e}_k=q_0+q_1{e}_1+q_2{e}_2+q_3{e}_3.$$

(7)

The first basis element $e_0=1$ is real, while the other three, $e_k$, are anti-commuting hyper-complex units. Using algebra of $e_k$ it can be shown that one of the hyper-complex basis elements, for example the last one $e_3$, is possible to define as the product of other two,

$$e_3=e_1{e}_2=-e_2{e}_1.$$

(8)

Matrix representations of quaternionic basis elements is introduced in Appendix A.

In analogy with complex numbers, the quaternionic conjugation, which changes the sign of the hyper-complex basis elements, is defined:

$$e_0^{\ast }=e_0,e_1^{\ast }=-e_1,e_2^{\ast }=-e_2,e_3^{\ast }={(e_1{e}_2)}^{\ast }=e_2^{\ast }{e}_1^{\ast }=-e_1{e}_2=-e_3.$$

(9)

Then, the inverted (conjugated) quaternion can be constructed:

$$q^{\ast }=q_0-q_1{e}_1-q_2{e}_2-q_3{e}_3.$$

(10)

Note that the two main basis units of split quaternions have the feature of unit polar vectors,

$$e_1^2=e_2^2=1,$$

(11)

while the third hyper-complex basis element, $e_3=e_1{e}_2$, behaves like a complex unit,

$$e_3^2=(e_1{e}_2)(e_1{e}_2)=-e_1^2{e}_2^2=-1,e_3^{\ast }=-e_3.$$

(12)

Because of these distinct properties of basis elements, the norm of a split quaternion,

$$N=\sqrt{|q{q}^{\ast }|}=\sqrt{|q_0^2-q_1^2-q_2^2+q_3^2|},$$

(13)

has (2 + 2)-signature and in general $q{q}^{\ast }$ is not positively defined.

For geometrical applications, one can define the line element in the space of split quaternions in the form [8]:

$$s=\lambda +e_{1}x+e_{2}y+e_{3}t,$$

(14)

where the four real parameters that multiply basis units, 1 and $e_k$, denotes: some time-like quantity $\lambda$, the spatial coordinates x and y and the time coordinate t (the speed of light is set as $c=1$). Using the conjugation rules one can find that the norm of quaternion (14) (the interval),

$$s{s}^{\ast }=s^{\ast }s={\lambda }^2-x^2-y^2+t^2,$$

(15)

has (2 + 2)-signature and in general is not positively defined. As in the standard relativity, it is required:

$$s{s}^{\ast }\ge 0,$$

(16)

i.e., the consideration is restricted to time-like split quaternions, defined in Appendix B.

2.1. Vector-Type Rotations

Now we want to represent rotations of a split quaternion q by the products of other quaternions.

There are a lot of methods to represent three-dimensional rotations (the method of orthonormal matrices is considered in Appendix C), however, quaternions are known to be the most convenient ones [14,15,16]. The vector part (time-like or space-like) of unit split quaternions (see Appendix B) can be thought of as a vector about which rotation should be performed and scalar part specifies the amount of rotation that should be performed about the vector part. That is, only four numbers are enough to represent a rotation by quaternions and there is only one constraint—the unity of norm. This makes it possible to find solutions to some optimization problems involving rotations in three dimensions, which are hard to solve when using orthonormal matrices because of the six non-linear constraints to enforce orthonormality, and the additional constraint—the unity of determinant.

Rotations in the vector (2 + 1) space of split quaternions about the time-like axis, $e_3$, or about the space-like axes $e_1$ and $e_2$, can be expressed with two side products of the unit time-like quaternions with time-like vector part (A37), or unit time-like quaternions with space-like vector part (A35), respectively. Every unit half-angle split quaternion $\alpha (\theta /2)$ represents a vector-type rotation by two-side multiplication. The result of the product,

$$q^{\prime }=\alpha \left(\frac{\theta }{2}\right)q{\alpha }^{\ast }\left(\frac{\theta }{2}\right),$$

(17)

is the quaternion $q^{\prime }$, which norm and scalar part are the same as for q and the vector part $V_{q^{\prime }}$ is obtained by revolving $V_q$ through the angle $\theta$ conically about the vector $ϵ$ defined in Equation (A34).

The set of unit split quaternions ${\alpha }_t$, ${\alpha }_x$ and ${\alpha }_y$ form the group of rotations in (2 + 1)-dimensional space-time, which algebra is isomorphic to $SU(1,1)$. Under this isomorphism the quaternion multiplication operation corresponds to the composition operation of rotations.

However, why vector-like quaternions have the ‘double-cover’ property (17), why there are two different quaternions ($\alpha$ and ${\alpha }^{-1}={\alpha }^{\ast }$) that represent the same rotation instead of $\alpha (\theta )q$ in analogy with complex numbers? It turns out that multiplication by a single quaternion do represent double rotations of spinors in four-dimensional quaternionic space (see Appendix F) and not in a plane of the three-dimensional vector space $V_q$. In the case of complex numbers, one has just one dimension of rotation. In three dimensions of quaternionic vector parts, we talk regarding to rotating about an axis, but in reality it means rotating in a plane perpendicular to that axis. In four-dimensional space of quaternions there are enough dimensions that it’s possible to rotate in two independent planes at once. These planes have no axes in common, they intersect only at a single point, which is the center of rotation. So, both rotations can take place without disturbing each other, which is not possible in three dimensions, where two planes always intersect in a line.

In Appendix D, it is shown that for quaternions one needs two distinct representations of vector-type rotations, $\alpha q{\alpha }^{\ast }$ and $\alpha q\alpha$, each corresponding to $SO(2,1)$-subgroups of $SO(2,2)$. Thus, the four components of a split quaternion $q^a$ ($a=0,1,2,3$) cannot be treated as a single 4-vector of $SO(2,2)$. The $\alpha q\alpha$-type $SO(2,1)$-transformations contain two independent hyperbolic rotations (boosts) of x and y axes towards the second time-like direction $\lambda$ of split quaternions (2 + 2) space, and compact rotations in time-like $(\lambda -t)$-plain, while the products $\alpha q{\alpha }^{\ast }$, forming automorphism group of split quaternions $SO(2,1)$, give successfully representation of rotations in three dimensional Minkowski space-time with two space-like and one time-like coordinates $(x,y,t)$ and can be used to define physical 3-vectors.

So, the standard representation of quaternion rotations $\alpha q{\alpha }^{\ast }$, which generate transformations of only vector part of split quaternions,

$$V_q=q_1{e}_1+q_2{e}_2+q_3{e}_3(e_1^2=e_2^2=-e_3^2=1),$$

(18)

naturally separates the Lorentz-like $SO(2,1)$ subgroup from the full group of symmetry of the split quaternion norms, $SO(2,2)$. Then, in field theory applications in three-dimensional Minkowski space-time, the second $SO(2,1)$ subgroup that generates rotations involved the second time-like coordinate, can be interpreted as corresponding to the internal symmetries of the model.

Note that to define coordinate transformations as quaternionic products, all four elements of a split quaternion (7) are needed, i.e., one should not remove the scalar part $q_0$ (see Appendix D). In general, interpretation of scalar parts of quaternions, which is not affected by $\alpha q{\alpha }^{\ast }$-type rotations, causes difficulties. Hamilton himself without notable success tried to interpret the scalar part of quaternions as an extra-spatial unit [25,26]. Later, when inspired by quaternions, vector algebra was introduced in physics, the scalar part of quaternions even was omitted. Introduction of vectors was successful, however because of removing of the scalar part of quaternions the division operation is not defined for vectors. There was lost also the property of quaternions, that they are rotation generators and expresses not only the final state achieved after a rotation, but the direction in which this rotation has been performed. It is this direction of rotation that the standard matrix representation of the rotation group fails to give.

2.2. Maximal Velocity and Uncertainty Principle

Analysis of the spinor-type one-side boosts of quaternionic intervals (14), can help in physical interpretation of the scalar parts of split quaternions $\lambda$. From definitions (A77) one notices that for the boosts along the positive x-direction, i.e., when $x^{\prime }$ increases (or the x-component of the momentum, $p_x$, increases), the scalar part ${\lambda }^{\prime }$ decreases and vice versa. In quantum mechanics the quantity with the dimension of length which is inversely proportional to the momentum is particle’s wavelength. So, in geometrical application it is natural to interpret the scalar part of the split quaternion (14) as the wavelength describing the inertial properties of particle’s reference frame.

Suppose the split quaternion (14) is used to describe motion of a particle with the wavelength $\lambda$ along the x-axis with the velocity $v_x$ in a laboratory coordinate system. Denoting the wavelength in particles own system by $\mathsf{\Lambda }$, from the condition of invariance of the intervals,

$$ds=d\lambda \left(1+\frac{dy}{d\lambda }{e}_2\right)+dt\left(1-v_x{e}_2\right)e_3=d\mathsf{\Lambda }+d\tau e_3,$$

(19)

where $\tau$ is the proper time of the particle, one obtains the two conditions:

$$\left|\frac{d\tau }{dt}\right|=\sqrt{1-v_x^2},\left|\frac{d\mathsf{\Lambda }}{d\lambda }\right|=\sqrt{1-\frac{d{y}^2}{d{\lambda }^2}}.$$

(20)

From these conditions follows the relations:

$$\left|v_x\right|<1,\left|\frac{dy}{d\lambda }\right|<1,$$

(21)

which must be obeyed simultaneously. This means that, together with speed of light $c=1$, there must exist the second fundamental constant (which can be extracted from $\lambda$) characterizing this critical property of the algebra.

In (2 + 2) space there are two different light-cones with two class of critical rotations and there must exist two fundamental constants characterizing this property of algebra.

It is known that the parameter with required properties is the reduced Planck constant, ℏ, which relates particles wavelength to its momentum,

$$\lambda =\frac{\hslash }{p}.$$

(22)

So, in the approach here considered, two fundamental physical constants, c and ℏ, have the algebraical origin and correspond to two kinds of critical signals in (2 + 2) space of split quaternions [8].

Furthermore, it was shown that for the boosts with the positive velocity, when $p_x$ increases ($p_y$ decreases), the quantity $\lambda$ decreases, i.e., $d\lambda <d\mathsf{\Lambda }$, and vice versa. So, for the change in the scalar part of the split quaternion, $\Delta \lambda$, when the particle momentum in x-direction increases, one can write

$$\Delta \lambda \sim -\frac{\hslash }{\Delta p_y},$$

(23)

where $p_y$ is the y-component of the momentum. When the variety of wave lengths becomes overall shorter, the overall magnitude of the variety of momenta must become greater, i.e., the shorter the wavelength, the higher will be its frequency and hence carry a greater amount of momentum. Similar relation can be obtained for the boost along the y-axis.

Inserting Equation (23) into Equation (21), one can conclude that the uncertainty principle,

$$\Delta x\Delta p_x\gtrsim \hslash ,$$

(24)

probably has the same geometrical meaning as the existence of the maximal velocity c [8].

In the formalism here applied, one can reach the relation (24) using also stochastic approach [27]. The trajectory $X^n(t)$ ($n=x,y$) of a classical test particle in (2 + 1) space, in general, depends on the second time-like coordinate $\lambda$. Then in the (2 + 2) space of split quaternions one gets a diffusion-like process:

$$d{X}^n=v^{n}dt+\frac{\partial x^n}{\partial \lambda }d\lambda (n=x,y),$$

(25)

where $X^n$ is the random 2-position. The 2-velocity, $v^n=\partial x^n/\partial t$, and diffusion parameters toward the x and y directions, $\partial x^n/\partial \lambda$, in Equation (25) are restricted by condition (21).

The additional time-like coordinate of quaternionic (2 + 2) space, $\lambda$, can be imagined to represent a Wiener process (the most common example of a Wiener process is Brownian motion) if is characterized by the following properties:

• at a starting moment can be set to zero together with t;
• has independent increments for every $t>0$;
• is normally distributed with mean 0 and have some finite variance;
• has continuous paths in t.

These requirements are indeed satisfied for physical processes in (2 + 1)-sub-subspace of the (2 + 2) space of split quaternions. In Appendix D it is shown that any rotation in the spatial $(x-y)$-plain (represented with one-side multiplications by a time-like unit split quaternions) is accompanied with the compact rotations (represented with the harmonic functions) in the temporal $(t-\lambda )$-plain. Thus, the variable $\lambda$ in (2 + 1) space has continues path in t and its mean value is zero for sufficiently large time intervals. On the other hand, to a quaternionic boost, e.g., towards x, it is corresponded non-compact rotation in $(y-\lambda )$-plain with the restriction $dy/d\lambda <1$, which follows from the positive norm condition (21). So, in (2 + 1) space a normal distribution of the continuous variable $\lambda$ with the mean value 0 and the variance 1 can be assumed.

For a process described by the stochastic differential Equation (25) the probability density obeys the Fokker–Planck equation. Using a stationary solution to this equation one can define the standard deviations, ${\sigma }_X$ and ${\sigma }_P$, of the random coordinate X and momentum P. Then, the Cauchy-Schwartz inequality,

$${\sigma }_X{\sigma }_P\ge |Cov(X,P)|,$$

(26)

leads to the standard uncertainty relation (24) [28].

3. Quaternions and Triality

Lie groups corresponding to division algebras have the property of having vector, spinor and conjugate spinor representations of the same (real) dimensions, a property known as triality [10,29,30]. By split quaternions the basic operations involving vectors, spinors and scalars can be described. These include an operation that takes two spinors, $\xi$ and $\chi$ (defined in Appendix F using zero divisors of the algebra Appendix E), and forms a vector $A=\xi \overline{\chi }$, and an operation that takes a vector A and a spinor $\xi$ and forms a spinor $\chi =A\xi$.

3.1. Vector and Spinor Transformations

In quaternionic (2 + 2) space the vector (for example the interval (14)) and the first and second kind spinors can be parameterize using the quaternionic basis (A1),

$$s=x^a{e}_a,\xi ={\xi }^a{e}_a,\chi ={\chi }^a{e}_a.$$

(27)

Thus, all three spinors are realized as split quaternion and their rotations will take a multiplicative form, different in each case. One can check that their indices can be raised/lowered by the matrix (A47) and that, in all three cases, the generators can be chosen to be real antisymmetric matrices. These are consequences of the triality property, which implies that one can just use a single index a for all three representations, which will be distinguished only by the symbols, s, $\xi$ and $\chi$.

Note that the $(2\times 2)$ matrices

$${\gamma }^a=e^a\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

(28)

satisfy the Clifford algebra

$${\gamma }^a{\gamma }^b+{\gamma }^b{\gamma }^a=2{\eta }^{ab},$$

(29)

so rotations of spinors of the first and second kinds, $\xi$ and $\chi$, can be written by two sets of matrices $e^a$ and $e^a{}^{\ast }$,

$$\begin{array}{c}\hfill {\xi }^{\prime }=e^{\frac{1}{2}{ϵ}^{ab}{S}_{ab}}\xi =\xi +\frac{1}{4}{ϵ}^{ab}{e}_a{e}_b^{\ast }\xi ,\\ \hfill {\chi }^{\prime }=\chi e^{\frac{1}{2}{ϵ}^{ab}{S}_{ab}}=\chi +\frac{1}{4}{ϵ}^{ab}{e}_a^{\ast }{e}_b\chi ,\end{array}$$

(30)

where $S_{ab}$ are defined in (A54).

Furthermore, one can write the $SL(2,R)$-matrix representation of the Lorentz-type transformations (A49), $L_{nm}$ ($n,m=x,y,t$), in the (2 + 1)-subspace $(x,y,t)$ of the vector (14):

$$\begin{array}{cc}\hfill x^{\prime }{}^n=L_m^n{x}^m\iff s^{\prime }& =e^{\frac{1}{2}{ϵ}^{mn}{(L_{mn})}_a^b}{x}^a{e}_b\hfill \\ \hfill & =s+\frac{1}{4}{ϵ}^{mn}\left({\delta }_m^b{\delta }_{na}-{\delta }_n^b{\delta }_{ma}\right)x^a{e}_b\hfill \\ \hfill & =s+\frac{1}{4}{ϵ}^{mn}\left[e_m(e_n^{\ast }{e}_a)-e_a(e_m^{\ast }{e}_n)\right]x^a\hfill \\ \hfill & =s+\frac{1}{4}{ϵ}^{mn}\left(e_m{e}_n^{\ast }s-s{e}_m^{\ast }{e}_n\right).\hfill \end{array}$$

(31)

3.2. Triality Algebra

The triality algebra of split quaternions can be defined as:

$${\alpha }_1(pq)={\alpha }_2(p)q+p{\alpha }_3(q),$$

(32)

where p and q denote some split quaternions and ${\alpha }_1$, ${\alpha }_2$ and ${\alpha }_3$ are unit quaternions that generate $SO(2,1)$ rotations.

In the formulations of $SO(2,2)$ vector A, covariant spinor $\xi$ and contravariant spinor $\chi$ representations in terms of quaternions (see Appendix F), the relationships between the three may be expressed without gamma matrices. For example, a covariant and a contravariant spinor can be used to form a vector

$$A=\xi {\chi }^{\ast },$$

(33)

or a contravariant spinor can be made from a vector and a covariant spinor

$$\chi =A^{\ast }\xi ,$$

(34)

or a covariant spinor can be made from a vector and a contravariant spinor

$$\xi =A\chi .$$

(35)

Supposing that all three objects A, $\xi$ and $\chi$ are transformed by a priori unrelated unit quaternions ${\alpha }_1$, ${\alpha }_2$ and ${\alpha }_3$ and still insisting that A is made from $\xi$ and $\chi$, these transformations would have to be related. This relation is exactly the condition that defines triality (32) and leads to the conclusion that the triality transformations is the largest group that preserves (33). However, these transformations are exactly of the form used in supersymmetry (see, for example [31]), so it is only natural that the overall symmetry of these theories is given by the triality algebras.

If A, $\xi$ and $\chi$ are quaternionic vector and covariant and contravariant spinors, respectively, then, the triality construction

$$\mathrm{Tri}\equiv \chi A\xi$$

(36)

is $SO(2,1)$ invariant. This can be checked explicitly by applying to (36) spinor and vector transformation rules (see Appendix D and Appendix F),

$${\chi }^{\prime }=\chi {\alpha }^{\ast },{\xi }^{\prime }=\alpha \xi ,A^{\prime }=\alpha A{\alpha }^{\ast }.$$

(37)

where $\alpha$ is a unit quaternion, i.e., $\alpha {\alpha }^{\ast }=1$.

The construction of division algebras from trialities has tantalizing links to physics. In the Standard Model of particle physics, all particles (other than the Higgs boson) transform either as vectors or spinors, and the interaction between matter and the forces is described by a trilinear map (36), involving two spinors and one vector. Moreover, split normed algebras naturally introduce pseudo-Euclidean spaces which are needed to describe physical spinors and vectors that are associated with Lorentz-type groups.

4. Quaternionic Analyticity and (2 + 1) Electrodynamics

In this section, the quaternionic analyticity condition, that is generalization of the Cauchy–Riemann equations from complex analysis, is derived. Then it is shown that this condition, written for the triality invariant element of split quaternions, leads to the system of equations for quaternionic vector and spinors, which is equivalent to the ordinary Dirac–Maxwell system in three-dimensional Minkowski space-time [1].

4.1. Quaternionic Gradient Operator

In physical applications it is important to define the quaternionic gradient operator. Although there have been some derivations of this operator in the literature with different level of details (see, [32,33,34,35,36,37,38,39,40,41] and references therein), it is still not fully clear how this operator can be written in the most general case and how it can be applied to various quaternion-valued functions.

Employ analogy with complex analysis, one can define the split quaternionic derivative operator in the form [35,36,37],

$$\frac{d}{ds}=\frac{1}{2}\left({\partial }_{\lambda }+e_1{\partial }_x+e_2{\partial }_y+e_3{\partial }_t\right),$$

(38)

with ${\partial }_a\equiv \partial /{\partial }_a$, such that its action upon interval quaternion (14) is equal to one,

$$\frac{ds}{ds}=\frac{1}{2}\left(1+e_1^2+e_2^2+e_3^2\right)=1,$$

(39)

while if applied to conjugated interval element,

$$s^{\ast }=\lambda -x{e}_1-y{e}_2-t{e}_3,$$

(40)

it gives zero,

$$\frac{d{s}^{\ast }}{ds}=\frac{1}{2}\left(1-e_1^2-e_2^2-e_3^2\right)=0.$$

(41)

Similarly, the conjugated gradient can be defined by the operator

$$\frac{d}{d{s}^{\ast }}=\frac{1}{2}\left({\partial }_{\lambda }-e_1{\partial }_x-e_2{\partial }_y-e_3{\partial }_t\right),$$

(42)

which annihilates s. Thus, from the definitions of quaternionic gradients, (38) and (42), one finds:

$$\frac{d{s}^{\ast }}{ds}=\frac{ds}{d{s}^{\ast }}=0.$$

(43)

From these relations it is clear that the interval (15) is a constant function for the restricted left quaternionic gradient operators,

$$\begin{array}{c}\hfill \frac{d}{ds}\left(s^{\ast }s\right)=\left(\frac{d{s}^{\ast }}{ds}\right)s=0,\\ \hfill \frac{d}{d{s}^{\ast }}\left(s{s}^{\ast }\right)=\left(\frac{ds}{d{s}^{\ast }}\right)s^{\ast }=0.\end{array}$$

(44)

4.2. Analyticity Condition

The main obstacle in physical applications of quaternions is that the real-valued functions of quaternion variables $\mathsf{\Phi }(q)$ are not analytic according to quaternion analysis [32,33,34,35,36,37,38,39,40,41]. To bypass the issue of non-existent derivatives of real functions of quaternion variables, current applications typically rewrite $\mathsf{\Phi }(q)$ in terms of the four real components ${\varphi }^a(q^a)$ ($a=0,1,2,3$) of four quaternion variables:

$$\mathsf{\Phi }(s,s^{\ast })={\varphi }_{\lambda }+e_1{\varphi }_x+e_2{\varphi }_y+e_3{\varphi }_t,$$

(45)

and take the real derivatives with respect to $q^a$ [37]. Thus, analogously to the Cauchy–Riemann equations from complex analysis,

$${\partial }_{z^{\ast }}f(z,z^{\ast })=0,(z=x+iy)$$

(46)

the Cauchy–Riemann–Fueter condition of analyticity for quaternionic functions of quaternionic variables can be written as [35,36,37],

$$\frac{d\mathsf{\Phi }(s,s^{\ast })}{d{s}^{\ast }}=\frac{1}{2}\left({\partial }_{\lambda }-e_1{\partial }_x-e_2{\partial }_y-e_3{\partial }_t\right)\mathsf{\Phi }=0.$$

(47)

This statement, that quaternionic functions should be independent of the variable $s^{\ast }$, represents the condition that quaternionic derivative be independent of direction along which it is evaluated.

Note that the simple case of the quaternionic analyticity condition for interval vectors (43), due to the existence of the relations of the type (41), holds only for split quaternions. This justifies in split quaternionic calculus the use of the simple gradient operators (38) introduced in [35,36]. For ordinary quaternions, even the polynomial functions do not satisfy these Cauchy–Riemann–Fueter conditions and several generalizations are necessary [37,38,39,40,41].

The coordinate transformations, e.g., $q^{\prime }=\alpha q$, in general do not preserve the property of analyticity, since unit split quaternions representing rotations and boosts are not analytic functions. Thus, to construct analytic and invariant structures (Lagrangians, Superpotentials, etc.), combinations of spinor-like and vector-like split quaternions are needed. For instance, if the transformation laws of $\chi$, $\xi$ and A have the forms,

$${\chi }^{\prime }=\chi {\alpha }^{\ast },{\xi }^{\prime }=\alpha \xi ,A^{\prime }=\alpha A{\alpha }^{\ast },$$

(48)

the invariant constructions are products of the type $\chi \xi$ and $\chi A\xi$. Indeed,

$$\begin{array}{cc}\hfill {(\chi \xi )}^{\prime }& ={\chi }^{\prime }{\xi }^{\prime }=\chi {\alpha }^{\ast }\alpha \xi =\chi \xi ,\hfill \\ \hfill {(\chi A\xi )}^{\prime }& =\chi {\alpha }^{\ast }\alpha A{\alpha }^{\ast }\alpha \xi =\chi A\xi .\hfill \end{array}$$

(49)

Then, using validity of the distribution law for quaternionic gradient operators (38) and (42) [35,36,37], one can write Cauchy–Riemann–Fueter analyticity condition (47) for the triality invariant construction (36) in the form:

$$\frac{d(\chi A\xi )}{d{s}^{\ast }}=\frac{d\chi }{d{s}^{\ast }}A\xi +\chi \frac{dA}{d{s}^{\ast }}\xi +\chi A\frac{d\xi }{d{s}^{\ast }}=0.$$

(50)

This condition is equivalent to the system of equations for quaternionic vector and spinors of the first and second kind (covariant and contravariant),

$$\frac{dA}{d{s}^{\ast }}=0,\frac{d\chi }{d{s}^{\ast }}=0,\frac{d\xi }{d{s}^{\ast }}=0.$$

(51)

Below, it is shown that this system can be reduced to the equations of (2 + 1) electrodynamics, i.e., to the system of standard Dirac and Maxwell equations in three-dimensional Minkowski space-time.

4.3. Quaternionic Dirac Equation

Let us demonstrate that the algebraic Cauchy–Riemann–Fueter condition (51) for the covariant spinor $\xi$ can be understood as the Dirac equation. Analogous logic can be applied for the case of the second kind (contravariant) spinor $\chi$ to obtain Dirac equation for conjugated spinors.

Consider the first kind of spinor, ${\xi }^{+}$ (A128),

$${\xi }^{+}=(q_0-i{q}_3)D^{+}+(q_2+i{q}_1)G^{-},$$

(52)

expressed in terms of the idempotent and nilpotent elements in $SU(1,1)$ representation (A92), where $q^n$ represents real functions of $\lambda$, x, y and t. One can replace

$${\xi }^{+}(\lambda ,x,y,t)\to e^{m\lambda }{\xi }^{+}(x,y,t),$$

(53)

where m is a real parameter. Then, the condition (51) for the covariant spinor $\xi$ takes the form:

$$\left({\partial }_{\lambda }-e_1{\partial }_x-e_2{\partial }_y-e_3{\partial }_t+m\right){\xi }^{+}(x,y,t)=0.$$

(54)

Next let us assume that the derivative of the covariant split quaternion ${\xi }^{+}(x,y,t)$ by the extra time-like coordinate $\lambda$ generates the supersymmetric (triality) transformation:

$${\partial }_{\lambda }{\xi }^{+}=B{\chi }^{+}{}^{\ast },$$

(55)

where B is some vector-type split quaternion, while ${\chi }^{+}{}^{\ast }$ is the conjugated contravariant quaternionic spinor (A129) in $SU(1,1)$ representation:

$${\chi }^{+}{}^{\ast }=(q_0-i{q}_3)D^{+}-(q_2+i{q}_1)G^{-}.$$

(56)

Then the quaternionic Cauchy–Riemann condition (54) obtains the form:

$$\left(e_1{\partial }_x+e_2{\partial }_y+e_3{\partial }_t-m\right){\xi }^{+}-B{\chi }^{+}{}^{\ast }=0,$$

(57)

or in matrix notation,

$$\left[\left(\begin{array}{cc}-i{\partial }_t+m& {\partial }_y-i{\partial }_x\\ {\partial }_y+i{\partial }_x& i{\partial }_t+m\end{array}\right)+\left(\begin{array}{cc}{B}_0-i{B}_3& B_2-i{B}_1\\ B_2+i{B}_1& B_0+i{B}_3\end{array}\right)\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\right]\left(\begin{array}{c}{q}_0-i{q}_3\\ q_2+i{q}_1\end{array}\right)=0.$$

(58)

On the other hand, the (2 + 1)-dimensional Dirac equation for a massive particle has the form,

$$\left[i{\gamma }^k({\partial }_k+i{A}_k)-m\right]\mathsf{\Psi }(t,x,y)=0(k=1,2,3),$$

(59)

where i denotes the standard complex unit, $A_k$ is the vector-potential and ${\gamma }^3$ plays the role of the ${\gamma }^0$ matrix. As the (2 + 1) gamma matrices the split quaternionic basis elements can be used in $SU(1,1)$-matrix representation,

$$i{\gamma }^1=e_1=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),i{\gamma }^2=e_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),i{\gamma }^3=e_3=\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right),$$

(60)

and identify the complex Dirac spinors $\mathsf{\Psi }(t,x,y)$ with the complex $SU(1,1)$ representation of the quaternionic spinor ${\xi }^{+}(t,x,y)$. Then, the matrix form of (2 + 1) Dirac Equation (59) obtains the form:

$$\left[\left(\begin{array}{cc}-i{\partial }_t+m& {\partial }_y-i{\partial }_x\\ {\partial }_y+i{\partial }_x& i{\partial }_t+m\end{array}\right)+\left(\begin{array}{cc}{A}_t& A_x+i{A}_y\\ -A_x+i{A}_y& -A_t\end{array}\right)\right]\left(\begin{array}{c}{q}_0-i{q}_3\\ q_2+i{q}_1\end{array}\right)=0,$$

(61)

where $q^n(t,x,y)$ denote the four real components of the complex Dirac spinor ${\xi }^{+}(t,x,y)$.

Than assuming

$$B_0=A_t,B_1=A_y,B_2=-A_x,B_3=0,$$

(62)

the quaternionic analyticity conditions (58) becomes equivalent to the complex (2 + 1) Dirac Equation (61) [1].

Note that Dirac’s theory in (2 + 1) space has some novel features. For example, there exists two inequivalent representations of Dirac gamma matrices, without which parity operation and its conservation would not have been possible. Further, when an external gauge field is introduced, it induces an ‘anomalous’ current which can be related to the anomalous divergence of an axial current and to the topological Chern-Simons charge.

4.4. (2 + 1) Maxwell Fields

Firs let us remind ordinary Maxwell’s equations in three-dimensional Minkowski space-time. To write the full set of Maxwell’s equations in (2 + 1) spaces with the signature

$${\eta }_{km}=\mathrm{diag}(+1,+1,-1)(k,m=x,y,t),$$

(63)

let us define the three potential $A^k=(A^x,A^y,A^t)$ and the Faraday tensor,

$$F^{km}={\partial }^k{A}^m-{\partial }^m{A}^k,$$

(64)

which has only three independent components. The relation between the vector potential and the magnetic and electric fields can be done through

$$H={\partial }_x{A}_y-{\partial }_y{A}_x,E_x={\partial }_x{A}_t-{\partial }_t{A}_x,E_y={\partial }_y{A}_t-{\partial }_t{A}_y.$$

(65)

Note that in (2 + 1) space, Faraday’s tensor has only three independent components and the magnetic field is no longer a vector—it becomes a (pseudo) scalar field. Indeed, in a world where all electric effects are confined to two planes, the magnetic field would be along the perpendicular direction. One must notice also that the dual tensor,

$${\tilde{F}}^k={ϵ}^{klm}{F}_{lm},$$

(66)

will become a vector.

In covariant form, Maxwell’s equations in (2 + 1) spaces are then

$${\partial }_k{F}^{mk}=j^m,{\partial }_k{\tilde{F}}^k=0,$$

(67)

where $j_t$ is the surface charge and $j_x$ and $j_y$ are surface currents. One can then work out the differential Maxwell’s equations in (2 + 1) spaces in terms of the fields,

$$\begin{array}{cc}\hfill {\partial }_x{E}_x+{\partial }_y{E}_y& =j_t,{\partial }_t{E}_x-{\partial }_{y}H=-j_x,\hfill \\ \hfill {\partial }_x{E}_y-{\partial }_y{E}_x+{\partial }_{t}H& =0,{\partial }_t{E}_y+{\partial }_{x}H=-j_y.\hfill \end{array}$$

(68)

One can see that there is no equivalent to the three-dimensional magnetic Gauss’ law.

Electrodynamics become complete with the Lorentz force law $f^k=F^{km}{v}_m$,

$$f_x=e(E_x+v_{y}H),f_y=e(E_y-v_{x}H),$$

(69)

where e is the charge and $v_m=(v_x,v_y,0)$ represents components of the velocity in the $(x-y)$-plain.

An immediate consequence of (2 + 1) Maxwell’s equations in is the existence of electromagnetic waves in two spatial dimensions,

$$\begin{array}{c}\hfill ({\partial }_x^2+{\partial }_y^2-{\partial }_t^2)E_x={\partial }_x{j}_t+{\partial }_t{j}_x,\\ \hfill ({\partial }_x^2+{\partial }_y^2-{\partial }_t^2)E_y={\partial }_y{j}_t+{\partial }_t{j}_y,\\ \hfill ({\partial }_x^2+{\partial }_y^2-{\partial }_t^2)H={\partial }_y{j}_x-{\partial }_x{j}_y.\end{array}$$

(70)

In the Lorenz gauge,

$${\partial }_k{A}^k=0,$$

(71)

Equation (70) are the wave equations with sources for the components of $A^k$,

$$({\partial }_x^2+{\partial }_y^2-{\partial }_t^2)A_k=-j_k.$$

(72)

Short discussions about the electrodynamics in (2 + 1) space can be found in [42,43,44,45], where it is noted that the model faces a few issues:

• The Coulomb force in (2 + 1) space must change, the electric field of a point charge now falls off as the inverse of the distance which entails a logarithmic electrostatic potential. This dramatically alters the phenomenology, since the attractive potential between opposite charges becomes confining, i.e., an infinite amount of energy would be required to extract the electron from the hydrogen atom, for example.
• Part of the vector calculus must change. The absence of a right-hand rule is obvious and the magnetic field must be qualitatively different; it turns out that it cannot be a vector any more—it becomes a scalar field.
• One of the main new features is connected to the retarded potentials. The reason for this is directly linked to the Huygens principle, which states that every point on a wave front is itself the source of (spherical) waves and relies on the fact that all waves propagate with a single speed. In (2 + 1) space, however, a solution to the wave equation can be understood as a superposition of waves travelling with speeds ranging from zero to the maximum value c, with which the first wave front travels.

4.5. First-Order Maxwell System

Let us write down the split quaternion that contains the electromagnetic potentials as

$$A=A_0+e_1{A}_x+e_2{A}_y+e_3{A}_t,$$

(73)

where $A_x$, $A_y$ and $A_t$ are components of the (2 + 1) vector and $A_0$ corresponds to extra degree of freedom in the quaternionic algebra. Using quaternionic gradient operator (42) the Cauchy–Riemann–Fueter analyticity condition (51) for the vector potential (73) can be written in the form:

$$\left({\partial }_{\lambda }-e_1{\partial }_x-e_2{\partial }_y-e_3{\partial }_t\right)A=\left[{\partial }_{\lambda }A-\left(e_1{\partial }_x+e_2{\partial }_y+e_3{\partial }_t\right)A_0-F\right]=0.$$

(74)

Here F denotes the quaternionic electro-magnetic field,

$$F=\left(e_1{\partial }_x+e_2{\partial }_y+e_3{\partial }_t\right)\left(e_1{A}_x+e_2{A}_y+e_3{A}_t\right)=-{\partial }_k{A}^k-e_1{E}_y+e_2{E}_x+e_{3}H,$$

(75)

where H, $E_x$ and $E_y$ are components of the magnetic and electric fields (65).

As in Equation (53), the variable $\lambda$ is separated in the scalar part of the vector-potential (73) (with the unit ‘charge’, $m=1$),

$${\partial }_{\lambda }{A}_0=A_0.$$

(76)

Similar to the case of spinors (55), it is also assumed here that the derivative of the vector part of (73) by the extra time-like coordinate $\lambda$ generates the supersymmetric (triality) transformations ($A\to \xi \chi$),

$${\partial }_{\lambda }\left(e_1{A}_x+e_2{A}_y+e_3{A}_t\right)=e_1{j}_x+e_2{j}_y-e_3{j}_t,$$

(77)

where $j_k$ are the components of the current vector, which is constructed by the covariant and contravarint spinors $j\sim \xi \chi$.

Using Equations (76) and (77) the Cauchy–Riemann–Fueter analyticity condition (74) can be written as the first-order Maxwell system [46,47]:

$$\begin{array}{cc}\hfill F& =A_0,\hfill \\ \hfill \left(e_1{\partial }_x+e_2{\partial }_y+e_3{\partial }_t\right)A_0& =-e_1{j}_x-e_2{j}_y+e_3{j}_t.\hfill \end{array}$$

(78)

Utilize the auxiliary function $A_0$ this system can be transferred to the single second-order Maxwell’s equation in (2 + 1) space,

$$\left(e_1{\partial }_x+e_2{\partial }_y+e_3{\partial }_t\right)F=-e_1{j}_x-e_2{j}_y+e_3{j}_t.$$

(79)

Indeed, the left side of this equation under the Lorenz gauge,

$${\partial }_k{A}^k={\partial }_x{A}_x+{\partial }_y{A}_y-{\partial }_t{A}_t=0,$$

(80)

in $SU(1,1)$-matrix representation has the form:

$$\begin{array}{c}\hfill \left(\begin{array}{cc}-i{\partial }_t& ({\partial }_y-i{\partial }_x)\\ ({\partial }_y+i{\partial }_x)& i{\partial }_t\end{array}\right)\left(\begin{array}{cc}-iH& (E_x+i{E}_y)\\ (E_x-i{E}_y)& iH\end{array}\right)\\ \hfill =\left({\partial }_y{E}_x-{\partial }_x{E}_y-{\partial }_{t}H\right)e_0+\left({\partial }_t{E}_x-{\partial }_{y}H\right)e_1+\left({\partial }_t{E}_y+{\partial }_{x}H\right)e_2+\left({\partial }_x{E}_x+{\partial }_y{E}_y\right)e_3.\end{array}$$

(81)

Equating to zero the coefficients in front of the four quaternionic basis units, one obtains the system of four real equations which are identical to the system of the Maxwell Equation (68) [1].

Note that the algebra of quaternions was applied to the study of Maxwell’s equations starting from the work of Maxwell himself [48]. Maxwell’s equations also were obtained long ago using Cauchy–Riemann-type analyticity conditions for functions of real quaternions [49]. However, in previous studies, first-order quaternionic equations were considered for the fields (65) and not second-order equations for vector potentials (78). A review of different applications of quaternionic analysis to the Maxwell equations can be found in [50]. Furthermore, usually it is considered sourceless quaternionic Maxwell’s equation, while in this paper, coupled quaternionic Dirac–Maxwell system is obtained receiving Dirac’s current from triality transformations.

5. Conclusions

Spinors and vectors in (2 + 2) space of split quaternions were studied and connections of their algebraic and physical properties were noted.

We think that some properties of physical models, such as invariance of space-time intervals, non-commutativity and half-angle spinor representations, are encoded in the structure of split quaternions. In the approach considered, two fundamental physical constants (light speed and Planck’s constant) have similar geometrical meanings and appear from the positive definiteness of quaternionic norms.

Quaternionic representation of rotations naturally separates two $SO(2,1)$ subgroups of the full group of symmetry of the norms of split quaternions, $SO(2,2)$. One of them represents symmetries of the three-dimensional Minkowski space-time, while the extra $SO(2,1)$ rotations by the second time-like coordinate can be viewed as internal symmetries of the model.

It is shown that the quaternion analyticity condition, analog of the Cauchy–Riemann equations from complex analysis, applied to the triality invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that derivatives by the extra time-like coordinate of quaternionic (2 + 2) space generate triality (supersymmetric) rotations of vectors and spinors, the analyticity equations is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

We should state that we do not know of any physical system which can be described by the (2 + 1) electrodynamics considered in this paper. Rather, the main purpose was to consider split quaternionic analysis as a toy-model on the way of developing the realistic field theoretical models using split octonions, in the spirit of [2,3,4,5,6,7].