Double and Dual Numbers. SU(2) Groups, Two-Component Spinors and Generating Functions

RESEARCH Revista Mexicana de F´ısica 66 (4) 418–423 JULY-AUGUST 2020

Double and dual numbers. SU(2) groups, two-component spinors and generating functions

G.F. Torres del Castillo Instituto de Ciencias, Universidad Autonoma´ de Puebla, 72570 Puebla, Pue., Mexico´ K.C. Gutierrez-Herrera´ Facultad de Ciencias F´ısico Matematicas,´ Universidad Autonoma´ de Puebla, 72570 Puebla, Pue., Mexico.´ Received 15 March 2020; accepted 27 March 2020

We explicitly show that the groups of 2 × 2 unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to SO(2, 1) or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two- component spinors. We show that with the aid of the double numbers we can ﬁnd generating functions for separable solutions of the Laplace equation in the (2 + 1) Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

Keywords: Double numbers; dual numbers; unitary groups; spinors; Minkowski (2 + 1) space; Laplace’s equation; spheroidal coordinates; toroidal coordinates.

Mostramos expl´ıcitamente que los grupos de matrices unitarias 2 × 2 con determinante igual a 1 cuyas entradas son numeros´ dobles o numeros´ duales son homomorfos a SO(2, 1) o al grupo de movimientos r´ıgidos del plano euclideano, respectivamente, e introducimos los espinores de dos componentes correspondientes. Mostramos que con la ayuda de los numeros´ dobles podemos hallar funciones generatrices para soluciones separables de la ecuacion´ de Laplace en el espacio de Minkowski (2 + 1), las cuales contienen funciones especiales que tambien´ aparecen en la solucion´ de la ecuacion´ de Laplace en el espacio euclideano tridimensional, en coordenadas esferoidales y toroidales.

Descriptores: Numeros´ dobles; numeros´ duales; grupos unitarios; espinores; espacio de Minkowski (2 + 1); ecuacion´ de Laplace; coordenadas esferoidales; coordenadas toroidales.

PACS: 02.30.Gp; 02.30.Jr DOI: https://doi.org/10.31349/RevMexFis.66.418

1. Introduction applications in physics for these numbers are restricted to the double numbers (see, e.g., [1–4, 6, 7]), in many cases looking Complex numbers are frequently employed in physics even for new theories that might substitute the established ones. in areas where the quantities of physical interest are real. As shown in Refs. [8–10], the double and the dual numbers For instance, the standard spherical harmonics, Ylm, are are useful in the solution of well-established systems of real complex-valued functions that are commonly used in electro- differential equations. magnetism, in spite of the fact that the potentials and electro- magnetic fields are real. In some cases, the complex numbers At first sight, it would seem that the use of a unit, j, whose appear in expressions for real-valued functions. Take, for ex- square is +1 is hardly necessary or useful (for instance, it is ample, the integral representations not required in the solution of algebraic equations with real coefficients), but the examples given in Refs. [1–4, 6–10], to- Z2π 1 gether with those presented below, show that having a second P (cos θ) = (cos θ + i sin θ cos w)l dw, l 2π unit, besides the real number 1, is indeed very useful. 0 SU(2) Zπ It is well known that , the group formed by the 1 2 × 2 complex unitary matrices with determinant equal to J (x) = ei(x sin t−nt) dt, n 2π 1, is homomorphic to SO(3), the group of rotations in the −π three-dimensional Euclidean space; this homomorphism is for the Legendre polynomials and the Bessel functions of in- useful and physically relevant. For instance, under a rota- tegral order, respectively. tion, the wavefunctions (spinors) of spin-1/2 particles in non- There exist other lesser known types of numbers, some- relativistic quantum mechanics, are transformed by means of what similar to the complex ones, called double and dual a SU(2) matrix. In this paper we explicitly show that the numbers (although other names are also employed), which analogs of SU(2), with double or dual numbers as entries, are also useful in various applications (see, e.g., Refs. [1–10] in place of complex numbers, are homomorphic to SO(2, 1) and the references cited therein). The double and the dual (the group of Lorentz transformations in two spatial dimen- numbers (introduced by Clifford in 1873 [5]) are character- sions) or the group of rigid motions of the Euclidean plane, ized by the presence of “imaginary units” j and ε, respec- respectively. In fact, as shown below, the three cases (com- tively, such that j2 = 1 and ε2 = 0. Most of the existing plex, double and dual) can be treated simultaneously and the DOUBLE AND DUAL NUMBERS. SU(2) GROUPS, TWO-COMPONENT SPINORS AND GENERATING FUNCTIONS 419 corresponding two-component spinors can be defined in a where Ψ is the two-component spinor unified manner. µ ¶ µ ¶ 1 We also show that, making use of the double numbers, ψ px + hmωx Ψ ≡ 2 ≡ (5) we can find a generating function for certain functions anal- ψ py + hmωy ogous to the associated Legendre functions, which arise in the solution of the Laplace equation in the three-dimensional and the Hermitian adjoint of a matrix is defined as the trans- Euclidean space, in spheroidal and toroidal coordinates. pose of the conjugate matrix (in all cases, the conjugate of a + hb is defined by a + hb = a − hb). Indeed, as a conse- In Sec. 2 we study the special unitary groups formed by † 2 × 2 matrices whose entries are complex, double or dual quence of (3), the matrix (2) satisfies U U = I and therefore numbers; we show that they are homomorphic to SO(3), (4) is invariant under the transformation Ψ 7→ UΨ. SO(2, 1) or SE(2) (the group of rigid motions of the Eu- The group SU(2)h is a (real) Lie group of dimension clidean plane that preserve the orientation), respectively, and three (the matrix (2) depends on four parameters, which are we introduce the corresponding two-component spinors. restricted by one equation) and a basis for its Lie algebra is In Sec. 3 we study separable solutions of the Laplace given by the matrices equation in the Minkowski (2 + 1) space, and we find that µ ¶ µ ¶ h 0 0 h some of the separated equations coincide with some of the σ ≡ , σ ≡ , 1 0 −h 2 h 0 separated equations found in the solution by separation of µ ¶ variables of the Laplace equation in the three-dimensional 0 1 σ ≡ . (6) Euclidean space in spheroidal and toroidal coordinates. In 3 −1 0 Sec. 4 we show that, making use of the double numbers, one obtains a generating function for the special functions anal- † The matrices σi are antihermitian because σi = −σi. The ogous to the associated Legendre functions encountered at trace of each matrix (6) is zero and these matrices form a Sec. 3. The basic rules for the use of the double and the basis for the real vector space formed by the 2 × 2 antiher- dual numbers can be found in Refs. [8, 10]; a rigorous and mitian matrices with trace equal to zero. The commutators fairly complete discussion of their algebraic properties can k of these matrices are given by [σi, σj] = cijσk, with sum be found, e.g., in Refs. [4–7]. k over repeated indices, where the structure constants cij are determined by

2. Special unitary groups 3 2 1 2 c12 = 2h , c23 = −2, c31 = −2 (7) In Sec. 3.1 of Ref. [8] the Hamiltonian (note that all of them are real, despite the fact that the matri- 1 ¡ ¢ h2 ¡ ¢ ces (6) are not all real). H = p 2 + p 2 − mω2 x2 + y2 , (1) 2m x y 2 The products of the matrices (6) can be expressed in the compact form was considered. Here m and ω are constants, and h may be the usual imaginary unit, i; the hypercomplex unit j (which σ σ = g I + 1 ck σ , (8) satisfies the condition j2 = 1); or the hypercomplex unit ε i j ij 2 ij k (which satisfies ε2 = 0). (That is, h2 is equal to −1, +1, or where I is the unit 2 × 2 matrix and the g are the entries of 0, respectively, so that, in all cases, H is real.) This Hamil- ij the diagonal 3 × 3 matrix tonian is invariant under the group of 2 × 2 matrices of the form µ ¶ 2 2 a + hb c + hd (gij) = diag (h , h , −1). (9) U = , (2) −c + hd a − hb Thus, the matrix (gij) is singular only if h = ε. With the where a, b, c, d are real numbers satisfying the condition aid of the matrices (6) we can construct a homomorphism between SU(2) and a subgroup of SL(3, R) (the group of 2 2 2 2 2 h a + c − h (b + d ) = 1. (3) 3 × 3 real matrices with determinant equal to 1). In fact, if −1 g ∈ SU(2)h, then the product gσig is also a traceless an- The matrices of the form (2), fulfilling Eq. (3), form a group −1 tihermitian matrix. (In fact, tr (gσig ) = tr σi = 0 and with the usual matrix multiplication. In what follows, this −1 † −1 † † † −1 −1 (gσig ) = (g ) σi g = g(−σi)g = −gσig , since group will be denoted by SU(2)h. When h = i this group is the elements of SU(2)h are unitary matrices.) Hence, there SU(2) j the usual group which, as is well known, is homomor- exist real numbers a such that phic to the rotation group in three dimensions, SO(3). i The invariance of the Hamiltonian (1) under the group −1 j gσig = ai σj. (10) SU(2)h is evident if one expresses H in the form i 1 The mapping g 7→ (aj) given by Eq. (10) is a group homo- H = Ψ†Ψ, (4) 0 2m morphism. In fact, if g is another element of SU(2)h then

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i 0 0−1 j there exists a matrix (bj) such that g σig = bi σj and of the Euclidean plane with respect to a set of Cartesian axes, therefore, then x0, y0 given by ¡ ¢       0 0 −1 0 0−1 −1 j −1 0 (gg )σi(gg ) = g g σig g = gbi σjg x cos θ sin θ x0 x  0      j k k j y = − sin θ cos θ y0 y = bi aj σk = (aj bi )σk, 1 0 0 1 1 j j thus showing that the product of the matrix (ai ) by (bi ) cor- are the coordinates with respect to the same axes of the point 0 j responds to the product gg .(ai is the entry at the j-th row j obtained by rotating the plane through an angle θ in the clock- and i-th column of the matrix (ai ).) wise direction about the origin and then translating the points In the cases where h is equal to i or to j, the matrices (ai ) j of the plane by the vector (x0, y0).) preserve the metric tensor (gij) (see Eq. (11), below). Indeed, from Eqs. (8) and (10) we have 2.1. Two-component spinors −1 1 k m g(σiσj)g = gijI + c a σm, 2 ij k Traditionally, spinors are associated with the orthogonal or which must coincide with (using Eqs. (10) and (8) again) pseudo-orthogonal groups, SO(n) or SO(p, q), and the most important examples are related to the SO(3) group (e.g., in −1 −1 −1 k l g(σiσj)g = gσig gσjg = ai σkajσl the description of the spin for a spin-1/2 particle) and the SO(3, 1) group (e.g., in the spinor formalism employed in = akal (g I + 1 cmσ ). i j kl 2 kl m special or general relativity and in the Dirac equation for the By virtue of the linear independence of the set electron). The basic (one-index) spinors form representation spaces for the spin groups, which are covering groups of the {I, σ1, σ2, σ3}, this amounts to orthogonal or pseudo-orthogonal groups. In the standard ap- k l proach, the spin groups are represented by complex matrices ai ajgkl = gij (11) (belonging, e.g., to SU(2) or to SL(2, C)) and the one-index and spinors have complex components. k l m k m ai ajckl = cij ak . (12) As we shall show below, this can be modiﬁed in two n ways: instead of spinors with complex components, we can Multiplying both sides of Eq. (12) by a gmn we obtain s consider spinors whose components are double or dual num- n k l m k m n as gmnai ajckl = cij ak as gmn bers and, instead of orthogonal or pseudo-orthogonal groups, we can consider “inhomogeneous” groups, speciﬁcally, the or, using Eq. (11), group of rigid motions of the Euclidean plane (formed by ro-

m n k l k tations and translations on the Euclidean plane). gmncklas ai aj = gkscij . (13) A two-component spinor will be represented by a column We introduce the real constants of the form µ ¶ µ ¶ ψ1 a + hb c ≡ g ck Ψ = ≡ , (15) sij ks ij ψ2 c + hd 2 and from Eqs. (7) and (9) we find that csij = −2h εsij. n k l just as in Eq. (5), where a, b, c, d are real numbers. Under the Therefore, if h 6= ε, Eq. (13) is equivalent to εnklas ai aj = i change of spinor frame given by g ∈ SU(2)h, the compo- εsij , which means that det(aj) = 1. An explicit computation i nents (15) transform according to shows that also in the case where h = ε, det(aj) = 1 (see Eq. (14) below). j Ψ 7→ gΨ. (16) Thus, if h = i or h = j, the matrix (ai ) is orthogonal j or pseudo-orthogonal; respectively, that is, (ai ) belongs to Hence, Ψ†Ψ is invariant under these transformations. (Note SO(3) or to SO(2, 1). In the remaining case, where h = ε, † 1 1 2 2 2 2 that Ψ Ψ = ψ ψ +ψ ψ is always real, but only in the case condition (3) reads a + c = 1, and therefore we can param- where h = i it is positive definite.) eterize a and c in the form a = cos θ/2, c = sin θ/2, then, A non-zero spinor Ψ defines a vector belonging to a real a straightforward computation shows that if g ∈ SU(2) has h vector space of dimension three with components R given the form (2), making use of (6) and (10), i by   † cos θ sin θ 2(b sin θ/2 − d cos θ/2) hRi = Ψ σiΨ i = 1, 2, 3. (17) j   (ai )= − sin θ cos θ 2(b cos θ/2 + d sin θ/2) , (14) † 0 0 1 In fact, since Ψ σiΨ is a 1 × 1 matrix, the conjugate of the right-hand side of Eq. (17) is equal to which represents an orientation-preserving rigid motion of ¡ † ¢† † † † the Euclidean plane. (If x, y are the coordinates of a point Ψ σiΨ = Ψ σi Ψ = −Ψ σiΨ,

Rev. Mex. F´ıs.66 (4) 418–423 DOUBLE AND DUAL NUMBERS. SU(2) GROUPS, TWO-COMPONENT SPINORS AND GENERATING FUNCTIONS 421 which shows that the numbers Ri are indeed real. Making components; the basic transformation rule (16) determines use of the explicit expression of the matrices σi [see Eqs. the transformation of a spinor with any number of indices or, (6)], one finds that equivalently, of any spin (see, e.g., Ref. [11]). We close this section with some remarks of a more formal 1 1 2 2 1 2 R1 = ψ ψ − ψ ψ ,R2 + hR3 = 2ψ ψ . (18) nature. By contrast with the real and the complex numbers, which are fields with the usual operations of sum and prod- According to Eqs. (17) and (10), under (16) the compo- uct, the double and the dual numbers are only commutative nents Ri transform as rings with identity. The standard definition of a vector space makes use of a field of scalars; its analog in the case of a ring hR 7→ (gΨ)†σ (gΨ)=Ψ†g−1σ gΨ=Ψ†a˜jσ Ψ=hãjR , i i i i j i j is called a module (see, e.g., Ref. [12]). j j where (ãi ) is the inverse of the matrix (ai ). Then, de- ij noting by (g ) the inverse of the matrix (gij ) (which ex- 3. The Laplace equation in the Minkowski (2+ ists only when h is i or j), from Eq. (11) it follows that 1) space ij g RiRj is invariant. In fact, when h = i we find that ij † 2 g RiRj = −(Ψ Ψ) , while in the case where h = j, In this section we shall consider the Laplace equation for the ij † 2 g RiRj = (Ψ Ψ) . Minkowski (2+1) space, which is a three-dimensional space In the case where h = ε, from Eq. (14) we see that the with a metric tensor given, e.g., by 2 × 2 block at the upper left corner of the matrix (aj) rep- i ds2 = −dx2 + dy2 + dz2, (20) resents an ordinary rotation about the origin on the plane 2 2 R3 = 0, hence, under these transformations (R1) + (R2) in terms of an appropriate coordinate system (similar to the 2 2 † 2 is invariant and one finds that (R1) + (R2) = (Ψ Ψ) . Cartesian coordinate systems of the Euclidean space). In- Owing to the specific form of the matrices (2) (the entries stead of the coordinates (x, y, z) appearing in Eq. (20), we at the second row are, up to a sign, the conjugates of those at can make use of the local coordinates (r, θ, φ) defined by the first row) one readily finds that the components of µ ¶ x = r sinh θ cosh φ, −ψ2 Ψˆ ≡ (19) ψ1 y = r sinh θ sinh φ, (21) z = r cosh θ, transform in the same manner as the components of Ψ. The ˆ two-component spinor Ψ is the mate of Ψ as defined in Ref. in terms of which the metric tensor (20) takes the form [11] for the case where h = i, and differs by a constant factor from the definition of the mate of a spinor in a space with in- ds2 = −r2dθ2 + dr2 + r2 sinh2 θ dφ2. (22) definite metric given there. In the applications of the spinors in quantum mechanics (where h = i), the spinor Ψˆ represents This last expression shows that the coordinates (r, θ, φ) can a state with the spin in the opposite direction to that corre- be regarded as orthogonal, so that the standard formula for sponding to Ψ. (In the Bloch sphere, Ψ and Ψˆ correspond to the Laplace operator in orthogonal coordinates is applicable, diametrically opposite points.) taking care of the minus signs. The Laplace equation in this As an illustration of the differences between the three case (which is just the wave equation in two spatial dimen- types of numbers considered here, we look for spinors Ψ sions) is given by which are proportional to their mates (with the proportional- ∂2u ∂2u ∂2u − + + = 0 (23) ity factor being a complex, double, or dual number according ∂x2 ∂y2 ∂z2 to the case at hand): Ψˆ = λΨ. According to Eq. (19), we have −ψ2 = λψ1, ψ1 = λψ2, which leads to λλ = −1. or, equivalently, µ ¶ µ ¶ This condition cannot be satisfied in the case of the complex 1 ∂ ∂u 1 ∂ ∂u − sinh θ + r2 or the dual numbers; however, in the case of the double num- r2 sinh θ ∂θ ∂θ r2 ∂r ∂r bers it has the general solution λ = ±j ejα, with α ∈ R arbitrary (note that j j = (−j) j = −1), furthermore, in that 1 ∂2u + = 0. (24) case, Ψ†Ψ = 0. r2 sinh2 θ ∂φ2 The spinor formalism can be employed in the study of Equation (24) admits separable solutions R(r)Θ(θ)Φ(φ), differential geometry (see, e.g., Ref. [11]) and, according to where Θ(θ) has to satisfy the equation the results of this section, it is possible to develop a unified µ ¶ formalism applicable to three-dimensional Riemannian man- 1 d dΘ sinh θ ifolds of any signature. sinh θ dθ dθ The one-index spinors are the basic objects from which · 2 ¸ vectors and tensors of any rank can be constructed; a field m − l(l + 1) + 2 Θ = 0, (25) of an arbitrary spin can be expressed in terms of its spinor sinh θ

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ij and l, m are separation constants. This equation appears where (g ) is the inverse of the matrix (gij ). The proof in the solution by separation of variables of the Laplace is a straightforward computation, taking into account that, equation in the three-dimensional Euclidean space in pro- in these coordinates, the Laplacian is given by ∇2u = late spheroidal equations (with l being an integer) (see, e.g., gij(∂/∂xi)(∂u/∂xj). Ref. [13], Eq. (8.6.7) or Ref. [14], Table 1.06) and in toroidal coordinates (where l is a half-integer) (see, e.g, Ref. [13], Eq. 4.1. Generating solutions of Eq. (25) (8.10.11) or Ref. [14], Sec. IV). In the case of the three-dimensional Euclidean space, with Now, in place of (21), we define the local coordinates 2 2 2 (r, θ, φ) by gij = δij , condition (30) reads (k1) +(k2) +(k3) = 0, and therefore one is led to make use of complex numbers [15], but x = r cosh θ cosh φ, in the case of the Minkowski (2+1) space, in the coordinates (x, y, z) appearing in Eq. (20), condition (30) takes the form y = r cosh θ sinh φ, (26) 2 2 2 − (k1) + (k2) + (k3) = 0, (31) z = r sinh θ, which can be satisfied by nonzero real numbers k1, k2, k3. and we find that the metric tensor (20) takes the form However, as we shall see, it is convenient to make use of double numbers: ds2 = −dr2 + r2dθ2 + r2 cosh2 θ dφ2. (27) (k1, k2, k3) = (j cosh v, −j sinh v, 1), (32) Hence, the Laplace equation (23) is now given by where v is an auxiliary parameter. Since, by definition, µ ¶ µ ¶ 1 ∂ ∂u 1 ∂ ∂u j2 = 1, the components (32) satisfy condition (31) for all − r2 + cosh θ r2 ∂r ∂r r2 cosh θ ∂θ ∂θ values of v. Then, making use of Eqs. (21) and the fact that ejv = cosh v + j sinh v, we have 1 ∂2u + = 0. (28) ¡ r2 cosh2 θ ∂φ2 k1x + k2y + k3z = r j cosh v sinh θ cosh φ ¢ Equation (28) admits separable solutions R(r)Θ(θ)Φ(φ), − j sinh v sinh θ sinh φ + cosh θ £ ¤ where Θ(θ) has to satisfy the equation = r cosh θ + j sinh θ(cosh φ cosh v − sinh θ sinh v) µ ¶ · ¸ £ ¤ 1 d dΘ m2 = r cosh θ + j sinh θ cosh(φ − v) cosh θ − l(l+1)− Θ=0, (29) cosh θ dθ dθ 2 £ cosh θ 1 −j(φ−v) = r cosh θ + 2 j sinh θ e and l, m are separation constants. Equation (29) coincides ¤ 1 j(φ−v) with one of the separated equations obtained in the solu- + 2 j sinh θ e , (33) tion by separation of variables of the Laplace equation in the which shows that k1x + k2y + k3z depends on φ and v only three-dimensional Euclidean space, in oblate spheroidal co- through the difference φ − v. ordinates (see, e.g., Ref. [13], Eq. (8.6.13) or Ref. [14], Table With the aid of (33), one can convince oneself that, for 1.07). l l = 0, 1, 2,... , the expression (k1x + k2y + k3z) must be Thus, even though the Minkowski (2 + 1) space may not of the form seem as interesting as the three-dimensional Euclidean space l or the standard Minkowski ((3+1)) space, as we have shown, X (k x+k y +k z)l = j|m| rlf (θ) ejmφ e−jmv. (34) the solution of the differential equation (23) is relevant to the 1 2 3 lm m=−l solution of the Laplace equation in the three-dimensional Eu- l clidean space. Since (k1x + k2y + k3z) must be a solution of the Laplace equation in the Minkowski (2 + 1) space and the parameter v is arbitrary, it follows that each term of (34) is a separable 4. Generating functions solution of the Laplace equation in the Minkowski (2 + 1) In this section we shall show that one can obtain solutions space. In particular, this means that flm(θ) is a solution of |m| of Eqs. (25) and (29) by means of generating functions. The Eq. (25). (The factor j is included in order for flm(θ) to starting point is the one employed in Ref. [15]: if the metric be a real-valued function.) 2 i j Thus, setting v = 0 in Eq. (34), we obtain the generating tensor of the space has the form ds = gij dx dx , with sum over repeated indices, i, j, . . . = 1, 2, . . . , p, and the compo- function 1 2 ¡ ¢l nents gij are constant, then the function (k1x +k2x +···+ cosh θ + 1 j sinh θ e−jφ + 1 j sinh θ ejφ p l 2 2 kpx ) is a solution of the Laplace equation if and only if the Xl constants k1, k2, . . . , kp satisfy the condition |m| jmφ = j flm(θ) e . (35) ij g kikj = 0, (30) m=−l

Rev. Mex. F´ıs.66 (4) 418–423 DOUBLE AND DUAL NUMBERS. SU(2) GROUPS, TWO-COMPONENT SPINORS AND GENERATING FUNCTIONS 423

(In the case of the spherical harmonics, considered in Ref. where each term of the sum on the right-hand side is a separa- [15], it was useful to keep the factor analogous to e−jmv, in ble solution of the Laplace equation in the Minkowski (2+1) order to obtain integral expressions for the spherical harmon- space and the functions hlm(θ) are solutions of Eq. (29). ics and the Legendre polynomials.) As a simple example, we have ¡ ¢ 5. Final remarks 1 −jφ 1 jφ 2 1 2 −2jφ cosh θ+ 2 j sinh θ e + 2 j sinh θ e = 4 sinh θ e The real groups found in Sec. 2 are related by means of a −jφ 2 1 2 + j sinh θ cosh θ e + cosh θ + 2 sinh θ contraction in the sense deﬁned in Ref. [16]; however, as we have shown, the use of the complex, double and dual num- + j sinh θ cosh θ ejφ + 1 sinh2 θ e2jφ, 4 bers allows us to study these groups simultaneously, without which shows that the functions f2,±2(θ), f2,±1(θ), and having to take limits. 2 f20(θ), are proportional to sinh θ, sinh θ cosh θ, and An advantage of the use double and the dual numbers is 2 2 2 cosh θ + sinh θ, respectively. that their basic algebraic rules are the same as those of the real or the complex numbers. This means that we can do 4.2. Generating solutions of Eq. (29) many computations that are equally applicable to complex, double or dual quantities. Following essentially the same steps as in the preceding sub- section, with (k1, k2, k3) given again by (32), but using now the coordinates (26) we have Acknowledgements £ k x + k y + k z = r sinh θ + 1 j cosh θ e−j(φ−v) 1 2 3 2 The authors acknowledge one of the referees for useful com- ¤ 1 j(φ−v) ments and for pointing out Refs. [1, 3, 6, 7] to them. + 2 j cosh θ e (36) and, therefore, for l = 0, 1, 2,... , we have ¡ ¢ l 1 −jφ 1 jφ l r sinh θ + 2 j cosh θ e + 2 j cosh θ e Xl |m| l jmφ = j r hlm(θ) e , (37) m=−l

1. A. Crumeyrolle, Variet´ es´ differentiables´ à coordonnees´ hy- Fis. E 65 (2019) 152, https://doi.org/10.31349/ percomplexes. Application à une geom´ etrisation´ et à une RevMexFisE.65.152. gen´ eralisation´ de la theorie´ d’Einstein-Schrodinger,¨ Ann. Fac. 9. G. F. Torres del Castillo, Differentiable Manifolds: A Theo- Sci. Toulouse 26 (1962) 105, https://doi.org/10. retical Physics Approach, 2nd ed. (Birkhauser,¨ Cham, 2020), 5802/afst.505. https://doi.org/10.1007/978-3-030-45193-6. 2. A. Crumeyrolle, Construction d’une nouvelle theorie´ unitaire 10. G. F. Torres del Castillo, Applications of the double and the en mecanique´ relativiste. Theorie´ unitaire hypercomplexe ou dual numbers. The Bianchi models, Rev. Mex. Fis. E 17 (2020) complexe hyperbolique, Ann. Fac. Sci. Toulouse 29 (1965) 53, (to be published). https://doi.org/10.5802/afst.518. 11. G. F. Torres del Castillo, 3-D Spinors, Spin-Weighted Functions 3. P. F. Kelly and R. B. Mann, Ghost properties of al- and their Applications (Birkhauser,¨ Basel, 2003), https: gebraically extended theories of gravitation, Class. Quan- //doi.org/10.1007/978-0-8176-8146-3. tum Grav. 3 (1986) 705, https://doi.org/10.1088/ 12. S. Roman, Advanced Linear Algebra, 3rd ed. (Springer-Verlag, 0264-9381/3/4/023. New York, 2008), Chap. 4, https://doi.org/10.1007/ 4. F. Antonuccio, Semi-complex Analysis and Mathematical 978-0-387-72831-5. Physics, e-Print: arXiv:grqc/ 9311032. 13. N. N. Lebedev, Special Functions and their Applications 5. B. Rosenfeld and B. Wiebe, Geometry of Lie Groups (Springer, (Dover, New York, 1972). Boston, 1997), Chap. 1, https://doi.org/10.1007/ 14. P. Moon and D. E. Spencer, Field Theory Handbook, 2nd ed. 978-1-4757-5325-7. (Springer-Verlag, Berlin, 1988), https://doi.org/10. 6. F. P. Schuller, Born-Infeld Kinematics, Ann. Phys. 299 (2002) 1007/978-3-642-83243-7. 174, https://doi.org/10.1006/aphy.2002.6273 . 15. G. F. Torres del Castillo, A generating function for the spheri- 7. P. O. Hess, M. Schafer,¨ and W. Greiner, Pseudo-Complex Gen- cal harmonics in p dimensions, Rev. Mex. Fis. 59 (2013) 248, eral Relativity (Springer, Cham, 2016), Chap. 1, 7, https: https://rmf.smf.mx/ojs. //doi.org/10.1007/978-3-319-25061-8 . 16. E. Inon¨ u¨ and E. P. Wigner, On the Contraction of Groups and 8. G. F. Torres del Castillo, Some applications in classical me- Their Representations, Proc. Natl. Acad. Sci. 39 (1953) 510, chanics of the double and the dual numbers, Rev. Mex. https://doi.org/10.1073/pnas.39.6.510.

Rev. Mex. F´ıs.66 (4) 418–423