Complex Boosts: A Hermitian Clifford Algebra Approach

The aim of this paper is to study complex boosts in complex Minkowski space-time that preserves the Hermitian norm. Starting from the spin group Spin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${^+(2n, 2m, \mathbb{R})}$$\end{document} in the real Minkowski space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{2n,2m}}$$\end{document} we construct a Clifford realization of the pseudo-unitary group U(n,m) using the space-time Witt basis in the framework of Hermitian Clifford algebra. Restricting to the case of one complex time direction we derive a general formula for a complex boost in an arbitrary complex direction and its KAK-decomposition, generalizing the well-known formula of a real boost in an arbitrary real direction. In the end we derive the complex Einstein velocity addition law for complex relativistic velocities, by the projective model of hyperbolic n-space.


Introduction
Lorentz boosts are linear transformations of space-time that preserve the space-time interval between any two events in Minkowski space. They are very important in many fields of mathematics and physics when relativistic effects come into play. In the real case, Lorentz boosts are elements of the Lorentz group SO (3,1), which are rotation-free and preserve the indefinite norm ||x|| 2 − t 2 , with x ∈ R 3 and t ∈ R.
The generalization of real boosts to complex boosts requires the study of the unitary group U(n, 1). This is just the group of isometries of the (n + 1)−dimensional complex space C n+1 which preserves the Hermitian indefinite norm ||z|| 2 − ||T || 2 , with z ∈ C n and T ∈ C. One of the first papers studying the real group structure of the complex Lorentz group in four dimensions was the paper of Barut [1]. Real and complex boosts in arbitrary pseudo-Euclidean spaces were discussed in [19], where the law of composition of generalized velocities (subluminal and superluminal velocities) was found. Real Lorentz transformation groups in arbitrary pseudo-Euclidean Advances in Applied Cliff ord Algebras 340 M. Ferreira and F. Sommen A dv. Appl. Clifford Algebras spaces where also presented in Eq.(8.14e) of Ref. [24] using the language of gyrogroups, established by A. Ungar [25,26]. Using this new formalism A. Ungar studied abstract real and complex Lorentz transformations and its associated gyrogroups in a series of papers [21,22,23,18]. In these papers it is shown the strong connection between boosts and gyrogroups in the real and complex cases, through the study of the automorphisms of the unit ball (real or complex).
In the real Minkowski-space time R n,1 a boost in an arbitrary direction ω ∈ S n−1 can be described using the universal real Clifford algebra R n,1 by the spin element s ω = cosh(α/2) + ω sinh(α/2), where α ∈ R and is the vector that spans the time axis. In [11] it was shown that there is a bijection between hyperbolic rotations generated by s ω and relativistic velocity additions (non-standard velocities, coordinate velocities, and proper velocities), giving rise to three different models of hyperbolic geometry (Poincaré, Klein, and Hyperbola models). Thus, not only the infinitesimal generators of the Lorentz group are important, but also the formula of a boost in an arbitrary direction is of foremost importance for the construction of concrete examples of gyrogroups and the study of the relativistic velocities. The formula of the complex boost in an arbitrary complex direction constructed in this paper allows the derivation of the complex Einstein relativistic velocity addition by the projection of its spin action on Minkowski space to the complex unit ball. It turns out that this transformation belongs to the automorphism group of the complex unit ball considered by Rudin in [16]. Our work has also several applications in harmonic analysis, quantum phase-space analysis, coherent states and wavelets (c.f. [13,14,15,10]). For instance, in [10] the author used the automorphisms of the unit ball to construct a family of spherical continuous wavelet transforms on the unit sphere in R n . Thus, the results of this paper are of interest for people working in physics and also mathematics.
In the last years Hermitian Clifford analysis has emerged as a refinement of Clifford analysis but also as an independent theory. While Clifford analysis focuses essentially on the study of the null-solutions of the Dirac operator on R n , called monogenic functions, Hermitian Clifford analysis focuses on the study of Hermitian monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, which are null solutions of two complex mutually adjoint Dirac operators. In the real case, the Dirac operator is invariant under the orthogonal group SO(n) which is double covered by the group Spin(n), while in the complex case, the two Hermitian Dirac operators are invariant under the unitary group U(n). A vast literature on such function theories is available, see e.g. [4,6,12,17,2,3,5].
Clifford analysis has also been investigated in real Minkowski spacetimes R n,1 or R n,m , m > 1. In [9] it was developed a function theory for Clifford algebra valued null solutions for the Dirac operator on the hyperbolic unit ball, the so-called hyperbolic monogenics. In this case the invariance group is the proper real Lorentz group Spin + (n, 1) (see [9] and the vast Vol. 23 (2013) Complex Boosts: A Hermitian Cliff ord Algebra Approach 341 literature therein). Our results can lead to the construction of a function theory for Hermitian hyperbolic monogenic functions on the complex projective model, generalizing the results in the real case (see e.g. [9]). It is well-known that a very large class of Lie groups can be described as spin groups (see [8], [7]). Therefore, Clifford algebras or geometric algebras are a very powerful mathematical tool for the study of Lie groups. In this paper we construct a Clifford realization of the pseudo-unitary group U(n, m) as a subgroup of the real orthogonal group Spin + (2n, 2m, R). The paper is organized as follows. In Section 2 we define the space-time Witt basis for working in the Hermitian space H n,m and we establish all the algebraic relations and properties needed for our constructions. In Section 3 we study the spin group Spin + (2n, 2m, R) in the real Minkowski space R 2n,2m and we construct a Clifford realization of the pseudo-unitary group U(n, m) using the space-time Witt basis in the framework of Hermitian Clifford algebra. We compute all the complex infinitesimal transformations (holomorphic, antiholomorphic and non-holomorphic transformations) in the Hermitian space H n,m . Hereafter, in Section 4 we construct the holomorphic, anti-holomorphic and non-holomorphic complex boosts in an arbitrary complex direction for the case of one complex time direction. Each of these boosts turn out to be the composition of two specific real boosts. We show also the Cartan or KAK−decomposition of such complex boosts. Finally, in Section 5 we will derive the complex Einstein velocity addition for complex relativistic velocities by the projective model of hyperbolic n−space, which belongs to the automorphism group of the complex unit ball in C n .

The Pseudo-Hermitian Space H n,m
We will denote by H n,m the standard pseudo-Hermitian space of type (n, m) which corresponds to the standard complex space C n+m , of complex dimension p = n + m, endowed with the non-degenerate sesquilinear Hermitian form, called the standard scalar product, defined by We consider that H n,m is identified with (R 2p , J), where R 2p=2n+2m is the real vector space subordinate to H n,m and J is the R−linear mapping fixing the complex structure. Since we want to incorporate complex space and complex time in this abstract setting we will consider z=(z 1 , . . . , z n , t 1 , . . . , t m ) a vector in C n+m with z j = x j + iy j ∈ C, j = 1, . . . , n and t r = u r + iv r ∈ C, r = 1, . . . , m. Then, z can be identified with the vector (x 1 , . . . , x n , y 1 , . . . , y n , u 1 , . . . , u m , v 1 , . . . , v m ) ∈ R 2p . The vector space R 2p=2n+2m turns out to be a pseudo-Euclidean space of signature (2n, 2m).
Let us consider {e j , ξ r , j = 1, . . . 2n, r = 1, . . . , 2m} an orthonormal basis of the real Minkowski space-time R 2n,2m , endowed with a non-degenerate A dv. Appl. Clifford Algebras real quadratic form of signature (2n, 2m), and let R 2n,2m be the associated real Universal Clifford algebra. The non-commutative multiplication in R 2n,2m is governed by the rules e j e k + e k e j = −2δ jk , ξ r ξ s + ξ s ξ r = 2δ rs , e j ξ r + ξ r e j = 0, (2.2) for j, k = 1, . . . , n, and r, s = 1, . . . , m. In particular, e 2 j = −1, j = 1, . . . , n and ξ 2 r = 1, r = 1, . . . , m. With these elements we construct the space-time Witt basis and Here, the symbol † stands for the Hermitian conjugation, which is the composition of the usual conjugation on the Clifford algebra R 2n,2m defined by a → a, ab = ba, a + b = a + b, e j = −e j , ξ r = −ξ r , 1 = 1 and the complex conjugation A → A c for A ∈ C 2p , where C 2p denotes the complexification of the Clifford algebra R 2n,2m . The elements of the spacetime Witt basis satisfy the following Grassmannian and duality identities: for j, k = 1, . . . , n and r, s = 1, . . . , m. In particular, Thus, every X ∈ R 2n,2m is written in the Witt basis as (2.21) Using the Witt basis elements we can define two complex Grassmann algebras (see [5]): The projection of the Clifford vector Z − Z † onto these complex algebras can be made by introducing the primitive (anti-)idempotent element which satisfies I † = I, I 2 = (−1) m I, and the conversion relations (2.23) Given two Hermitian vector variables Z 1 , Z 2 we can define the dot and wedge product by The following lemmas generalize Lemmas 1 and 2 presented in [5].

The Pseudo-Unitary Group U(n, m)
The pseudo-unitary group U(n, m) is the group of holomorphic transformations preserving the Hermitian form (2.1). It is well-known that U(n, m) = SO + (2n, 2m, R) ∩ Sp(2(n + m), R) i.e., U(n, m) is both a real subgroup of the pseudo-orthogonal group SO + (2n, 2m, R) and of the sympletic group Sp(2(n + m), R). In this section we will consider the group Spin + (2n, 2m, R), the double covering group of SO + (2n, 2m, R) to construct a representation of the unitary group U(n, m). The group Spin + (2n, 2m, R) can be described by where Γ + (2n, 2m, R) is the even Clifford group in R 2n,2m . Usually, the Lie algebra spin + (2n, 2m, R) is the real algebra spanned by the bivectors Vol. 23 (2013) Complex Boosts: A Hermitian Cliff ord Algebra Approach 345 generating space rotations, time rotations, and space-time rotations, or boosts, in R 2n,2m . It is easy to see that the dimension of spin + (2n, 2m, R) is n(2n − 1) + m(2m − 1) + 4nm = (n + m)(2(n + m) − 1). When we want to exploit complex symmetries of spaces of even real dimension it is more appropriate to split the vector basis of R 2n,2m into in order to identify real and imaginary axes. Therefore, we can write another basis for the Lie algebra of Spin + (2n, 2m, R), more suited for our purposes.
Henceforward, we shall refer to elements (3.4)-(3.8) as complex space bivectors, elements (3.9)-(3.13) as complex time bivectors, and elements (3.14)-(3.17) as complex space-time bivectors since they will generate complex space rotations, complex time rotations, and complex-time rotations or complex boosts, respectively, as we will see in this section.

Commutation relations between complex space bivectors:
Commutation relations between complex time bivectors: Commutation relations between complex space-time bivectors:

Commutation relations between complex space and time bivectors:
Commutation relations between complex space and space-time bivectors: Commutation relations between complex time and space-time bivectors: From these commutation relations, a subalgebra of spin + (2n, 2m, R) can be identified.
Proof. Since the Lie bracket is closed under these elements they define a subalgebra of spin + (2n, 2m, R).
The Lie subalgebra defined in Lemma 3.3 defines a Lie group of dimension (n + m) 2 isomorphic to the unitary group U(n, m). Before we realize this let us compute the spin actions generated by the elements (3.4)-(3.17) of the Lie algebra spin + (2n, 2m, R). For s ∈ Spin + (2n, 2m, R) its spin-1 representation is given by h(s) : X → sXs, X ∈ R 2n,2m , which preserves the multi-structure of R 2n,2m. . The spin elements s ∈ Spin + (2n, 2m, R) associated to (3.4)-(3.8) are obtained by exponentiation of the elements of the Lie algebra spin + (2n, 2m, R) : Their complex form is obtained by passing to the Witt basis. Using (2.15) and (2.16) we obtain the following complex transformations: Vol. 23 (2013) Complex Boosts: A Hermitian Cliff ord Algebra Approach 351

352
M. Ferreira and F. Sommen A dv. Appl. Clifford Algebras The complex transformations (3.47)-(3.60) preserve the Hermitian norm and they can be divided into two classes: the holomorphic transformations (a group in itself) and the non-holomorphic transformations. Indeed, multiplying at left the spin actions (3.47)-(3.60) with the (anti-)primitive idempotent I (projection onto CΛ n,m ), we immediately see that transformations Proof. First we see that all the real bivectors are linearly independent and their number equals (n + m) 2 − 1, which is exactly dim(su(n, m))= dim(SU(n, m)). Furthermore, the spin elements associated to these bivectors satisfy the condition Is = I.

Complex Boosts in an Arbitrary Complex Direction
We have seen that elements (3.28)-(3.31) are the Lorentz boosts generators in R 2n,2m , yielding complex Lorentz transformations. In this section we restrict ourselves to R 2n,2 , the case of one complex time dimension, and we compute the formula for a general complex boost in an arbitrary complex direction.
In the real case it is well-know that a real boost in the real Minkowski spacetime R n,1 is parameterized by a direction ω ∈ S n−1 (S n−1 is the unit sphere in R n ) and a hyperbolic angle α ∈ R, by the formula where ξ is the Clifford basis element which spans the time axis and satisfies ξ 2 = 1. The spin element (4.1) corresponds to the exponentiation of the element ωξ, which belongs to the Lie algebra spin + (n, 1). Therefore, we have s ω,α = e ωξ = e (w1e1+...+wnen)ξ = e w1e1ξ+...+wnenξ (4.2) i.e., the real boost s ω,α in an arbitrary direction ω ∈ S n−1 appears as the exponentiation of the real linear combination of the boosts e 1 ξ, . . . , e n ξ in spin + (n, 1). For a real space-time vector x + tξ where x ∈ R n and t ∈ R, the spin action induced by s ω,α is given by In this section we will derive a similar formula for the complex case, by studying all possible linear combinations of the complex boost bivectors in the Lie algebra spin + (2n, 2, R), which are given by where λ j , λ n+j ∈ R, for some j = 1, . . . , n. The last equality is valid since [(λ j e j − λ n+j e n+j )ξ 1 , (λ j e n+j + λ n+j e j )ξ 2 ] = 0 and thus, the exponential law holds. The elements s 1 and s 2 are defined by and s 2 = cosh α 2 + (λ j e n+j + λ n+j e j )ξ 2 sinh α 2 (4.10) which belong to Spin + (2n, 2, R) if and only if λ 2 j + λ 2 n+j = 1. This is the normalization condition that we have to impose. By straightforward computations, the spin actions induced by s 1 and s 2 in an arbitrary space-time vector X ∈ R 2n,2 are given in real coordinates by and +λ n+j u 2 sinh α)e j +((cosh α−1)(λ 2 j y j +λ j λ n+j x j )+λ j u 2 sinh α)e n+j +u 1 ξ 1 + (u 2 cosh α + sinh α(λ n+j x j + λ j y j ))ξ 2 . (4.12) Finally, the composition of s 1 and s 2 gives us the spin action sXs: (x j e j +y j e n+j )+((cosh α−1)x j +sinh α(λ j u 1 +λ n+j u 2 ))e j + ((cosh α − 1)y j + sinh α(λ j u 2 − λ n+j u 1 ))e n+j Writing (4.13) in terms of the Witt basis using (2.15) and (2.16), we obtain the following complex transformation: with w j = λ j + iλ n+j such that λ 2 j + λ 2 n+j = 1, and T = u 1 + iu 2 . Since λ n+j ∈ R is arbitrary we can replace λ n+j by −λ n+j in (4.14). Considering Vol. 23 (2013) Complex Boosts: A Hermitian Cliff ord Algebra Approach 355 also ω j = (0, . . . , w j , . . . , 0) ∈ C n , which satisfies ||ω j || 2 = 1, we finally obtain the spin action with z = (z 1 , . . . , z n ) ∈ C n , and z, ω j the usual Hermitian inner product on C n . Formula (4.15) corresponds to the action of a complex boost in the complex direction ω j = (0, . . . , w j , . . . , 0) ∈ S, where S denotes the complex unit sphere in C n . Replacing λ n+j by −λ n+j in (4.8) the spin element can be written in Hermitian form as To have a boost in an arbitrary direction of the complex sphere we can consider the linear combination of all boosts of types I and II or we can simply consider rotation arguments. Let ω = (w 1 , . . . , w n ) ∈ S be an arbitrary direction in C n . Then it is always possible to find s * ∈ U(n, 0) such that ω = s * w j s * , where ω j = (0, . . . , w j , . . . , 0) ∈ S. Therefore, by performing the action s * s ωj ,α s * we will arrive at the formula of the complex boost s ω,α in an arbitrary direction ω ∈ S, which is given by (4.17) Its action on X ∈ R 2n,2 is given by Multiplying at left the spin action (4.18) with the primitive (anti-)idempotent I, we obtain a holomorphic transformation in the variables z j , j = 1, . . . , n, and T, whereas, the multiplication of I at right gives an anti-holomorphic transformation. We summarize our results in the next theorem.