1.Introduction

Complex numbers are frequently employed in physics even in areas where the quantities of physical interest are real. For instance, the standard spherical harmonics, Ylm, are complex-valued functions that are commonly used in electromagnetism, in spite of the fact that the potentials and electromagnetic fields are real. In some cases, the complex numbers appear in expressions for real-valued functions. Take, for example, the integral representations

for the Legendre polynomials and the Bessel functions of integral order, respectively.

There exist other lesser known types of numbers, somewhat similar to the complex ones, called double and dual numbers (although other names are also employed), which are also useful in various applications (see, e.g., Refs. [¹-¹⁰] and the references cited therein). The double and the dual numbers (introduced by Clifford in 1873 [⁵]) are characterized by the presence of “imaginary units” j and ε, respectively, such that j² = 1 and ε2=0. Most of the existing applications in physics for these numbers are restricted to the double numbers (see, e.g., [¹-⁴, ⁶, ⁷], in many cases looking for new theories that might substitute the established ones. As shown in Refs. [⁸-¹⁰], the double and the dual numbers are useful in the solution of well-established systems of real differential equations.

At first sight, it would seem that the use of a unit, j, whose square is +1 is hardly necessary or useful (for instance, it is not required in the solution of algebraic equations with real coefficients), but the examples given in Refs.[¹-⁴, ⁶-¹⁰], together with those presented below, show that having a second unit, besides the real number 1, is indeed very useful.

It is well known that SU(2), the group formed by the 2×2 complex unitary matrices with determinant equal to 1, is homomorphic to SO(3), the group of rotations in the three-dimensional Euclidean space; this homomorphism is useful and physically relevant. For instance, under a rotation, the wavefunctions (spinors) of spin-1/2 particles in non-relativistic quantum mechanics, are transformed by means of a SU(2) matrix. In this paper we explicitly show that the analogs of SU(2), with double or dual numbers as entries, in place of complex numbers, are homomorphic to SO(2,1) (the group of Lorentz transformations in two spatial dimensions) or the group of rigid motions of the Euclidean plane, respectively. In fact, as shown below, the three cases (complex, double and dual) can be treated simultaneously and the corresponding two-component spinors can be defined in a unified manner.

We also show that, making use of the double numbers, we can find a generating function for certain functions analogous to the associated Legendre functions, which arise in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

In Sec. 2 we study the special unitary groups formed by 2×2 matrices whose entries are complex, double or dual numbers; we show that they are homomorphic to SO(3), SO(2,1) or SE(2) (the group of rigid motions of the Euclidean plane that preserve the orientation), respectively, and we introduce the corresponding two-component spinors.

In Sec. 3 we study separable solutions of the Laplace equation in the Minkowski (2+1) space, and we find that some of the separated equations coincide with some of the separated equations found in the solution by separation of variables of the Laplace equation in the three-dimensional Euclidean space in spheroidal and toroidal coordinates. In Sec. 4 we show that, making use of the double numbers, one obtains a generating function for the special functions analogous to the associated Legendre functions encountered at Sec. 3. The basic rules for the use of the double and the dual numbers can be found in Refs. [⁸, ¹⁰]; a rigorous and fairly complete discussion of their algebraic properties can be found, e.g., in Refs. [⁴-⁷].

2.Special unitary groups

In Sec. 3.1 of Ref. the Hamiltonian

was considered. Here m and ω are constants, and h may be the usual imaginary unit, i; the hypercomplex unit j (which satisfies the condition j² = 1); or the hypercomplex unit ε (which satisfies ε2=0). (That is, h² is equal to -1, +1, or 0, respectively, so that, in all cases, H is real.) This Hamiltonian is invariant under the group of 2×2 matrices of the form

where a,b,c,d are real numbers satisfying the condition

The matrices of the form (2), fulfilling Eq. (3), form a group with the usual matrix multiplication. In what follows, this group will be denoted by SU(2)_h. When h = i this group is the usual SU(2) group which, as is well known, is homomorphic to the rotation group in three dimensions, SO(3).

The invariance of the Hamiltonian (1) under the group SU(2)_h is evident if one expresses H in the form

where Ψ is the two-component spinor

and the Hermitian adjoint of a matrix is defined as the transpose of the conjugate matrix (in all cases, the conjugate of a + hb is defined by a+hb¯=a-hb). Indeed, as a consequence of (3), the matrix (2) satisfies U†U=I and therefore (4) is invariant under the transformation Ψ↦UΨ.

The group SU(2)_h is a (real) Lie group of dimension three (the matrix (2) depends on four parameters, which are restricted by one equation) and a basis for its Lie algebra is given by the matrices

The matrices σi are antihermitian because σi†=-σi. The trace of each matrix (6) is zero and these matrices form a basis for the real vector space formed by the 2×2 antihermitian matrices with trace equal to zero. The commutators of these matrices are given by [σi,σj]=cijkσk, with sum over repeated indices, where the structure constants cijk are determined by

(note that all of them are real, despite the fact that the matrices (6) are not all real).

The products of the matrices (6) can be expressed in the compact form

where I is the unit 2×2 matrix and the gij are the entries of the diagonal 3×3 matrix

Thus, the matrix (gij) is singular only if h = ε. With the aid of the matrices (6) we can construct a homomorphism between SU(2)_h and a subgroup of SL(3,R) (the group of 3×3 real matrices with determinant equal to 1). In fact, if g∈SU(2)h, then the product gσig-1 is also a traceless antihermitian matrix. (In fact, tr(gσig-1)=trσi=0 and (gσig-1)†=(g-1)†σi†g†=g(-σi)g-1=-gσig-1, since the elements of SU(2)_h are unitary matrices.) Hence, there exist real numbers aij such that

The mapping g↦(aji) given by Eq. (10) is a group homomorphism. In fact, if g' is another element of SU(2)_h then there exists a matrix (bji) such that g'σig'-1=bijσj and therefore,

thus showing that the product of the matrix (aij) by (bij) corresponds to the product gg'. ( aij is the entry at the j-th row and i-th column of the matrix (aij).)

In the cases where h is equal to i or to j, the matrices (aji) preserve the metric tensor (gij) (see Eq. (11), below). Indeed, from Eqs. (8) and (10) we have

which must coincide with (using Eqs. (10) and (8) again)

By virtue of the linear independence of the set {I,σ1,σ2,σ3}, this amounts to

and

Multiplying both sides of Eq. (12) by asngmn we obtain

or, using Eq. (11),

We introduce the real constants

and from Eqs. (7) and (9) we find that csij=-2h2εsij. Therefore, if h≠ε, Eq. (13) is equivalent to εnklasnaikajl=εsij, which means that det(aji)=1. An explicit computation shows that also in the case where ${\rm h} = \varepsilon$, det(aji)=1 (see Eq. (14) below).

Thus, if h = i or h = j, the matrix (aij) is orthogonal or pseudo-orthogonal; respectively, that is, (aij) belongs to SO(3) or to SO(2,1). In the remaining case, where h=ε, condition (3) reads a2+c2=1, and therefore we can parameterize a and c in the form a=cosθ/2, c=sinθ/2, then, a straightforward computation shows that if g∈SU(2)h has the form (2), making use of (6) and (10),

which represents an orientation-preserving rigid motion of the Euclidean plane. (If x,y are the coordinates of a point of the Euclidean plane with respect to a set of Cartesian axes, then x',y' given by

are the coordinates with respect to the same axes of the point obtained by rotating the plane through an angle θ in the clockwise direction about the origin and then translating the points of the plane by the vector (x0,y0).)

3.The Laplace equation in the Minkowski (2+1) space

In this section we shall consider the Laplace equation for the Minkowski (2+1) space, which is a three-dimensional space with a metric tensor given, e.g., by

in terms of an appropriate coordinate system (similar to the Cartesian coordinate systems of the Euclidean space). Instead of the coordinates (x,y,z) appearing in Eq. (20), we can make use of the local coordinates (r,θ,ϕ) defined by

in terms of which the metric tensor (20) takes the form

This last expression shows that the coordinates (r,θ,ϕ) can be regarded as orthogonal, so that the standard formula for the Laplace operator in orthogonal coordinates is applicable, taking care of the minus signs. The Laplace equation in this case (which is just the wave equation in two spatial dimensions) is given by

or, equivalently,

Equation (24) admits separable solutions R(r)Θ(θ)Φ(ϕ), where Θ(θ) has to satisfy the equation

and l,m are separation constants. This equation appears in the solution by separation of variables of the Laplace equation in the three-dimensional Euclidean space in prolate spheroidal equations (with l being an integer) (see, e.g., 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. (8.10.11) or Ref. [¹⁴], Sec. IV).

Now, in place of (21), we define the local coordinates (r,θ,ϕ) by

and we find that the metric tensor (20) takes the form

Hence, the Laplace equation (23) is now given by

Equation (28) admits separable solutions R(r)Θ(θ)Φ(ϕ), where Θ(θ) has to satisfy the equation

and l,m are separation constants. Equation (29) coincides with one of the separated equations obtained in the solution by separation of variables of the Laplace equation in the three-dimensional Euclidean space, in oblate spheroidal coordinates (see, e.g., Ref. [¹³], Eq. (8.6.13) or Ref. [¹⁴], Table 1.07).

Thus, even though the Minkowski (2+1) space may not seem as interesting as the three-dimensional Euclidean space or the standard Minkowski ( (3+1)) space, as we have shown, the solution of the differential equation (23) is relevant to the solution of the Laplace equation in the three-dimensional Euclidean space.

4.Generating functions

In this section we shall show that one can obtain solutions of Eqs. (25) and (29) by means of generating functions. The starting point is the one employed in Ref. (15): if the metric tensor of the space has the form ds2=gijdxidxj, with sum over repeated indices, i,j,…=1,2,…,p, and the components gij are constant, then the function (k1x1+k2x2+⋯+kpxp)l is a solution of the Laplace equation if and only if the constantsk1,k2,…,kp satisfy the condition

where (gij) is the inverse of the matrix (gij). The proof is a straightforward computation, taking into account that, in these coordinates, the Laplacian is given by ∇2u=gij(∂/∂xi)(∂u/∂xj).

5.Final remarks

The real groups found in Sec. 2 are related by means of a contraction in the sense defined in Ref.[¹⁶]; however, as we have shown, the use of the complex, double and dual numbers allows us to study these groups simultaneously, without having to take limits.

An advantage of the use double and the dual numbers is that their basic algebraic rules are the same as those of the real or the complex numbers. This means that we can do many computations that are equally applicable to complex, double or dual quantities.