A new approach to the representation of the Lorentz group on the spinors

Yaakov Friedman

Jerusalem College of Technology E-mail: friedman@avoda.jct.ac.il

and

Bernard Russo

University of California,Irvine E-mail: brusso@math.uci.edu

(DEDICATED TO PROFESSOR HOLGER PETERSSON ON THE OCCASION OF HIS 60TH BIRTHDAY)

We propose a triple product representation of the canonical anticommutation relations which does not make use of the associative Clifford algebra. Imposition of these commutation relations on the natural basis of C^{n} defines a triple product making C^{n} into a Cartan factor of type 4 that we denote by S^{n}.

This Jordan structure is used to represent the Lorentz group on S^{3} and S^{4}. The irreducible representation on S^{3} corresponds to the relativistic transformations of the electromagnetic field. The irreducible spin 1 representation on S^{4} extends the Lorentz space-time transformation. By taking the self-adjoint part of this representation with respect to the spin conjugation, a reducible spin \frac{1}{2} representation on S^{4} results. The latter is shown to induce two spin 1 representations in the space of determinant preserving maps on S^{4} showing that the same spin factor could be used to represent the two types of elementary particles: bosons and fermions.

Key Words: JB^{∗}-triple,canonical anticommutation relations,triple product,tripotent,Cartan factor,spin factor,Lorentz group,space-time transformation,relativity,electromagnetism,determinant preserving map,Jordan triple system

0. INTRODUCTION

Why are physicists interested in group representation theory? Because a determination of all unitary representations of the inhomogeneous Lorentz group is equivalent to a determination of all possible relativistic wave equations ([20, p. 27]). Moreover, the formalism of relativistic quantum theory is based on a unitary (possibly reducible) representation of the restricted Poincaré group. The representation provides the quantum mechanical Hilbert space, and the infinitesimal generators of its Lie algebra are interpreted as the basic physical observables (energy, momentum, angular momentum) ([13, p. 160]). The irreducible representations of the Lorentz group were first determined by E. Wigner in the 1930s.

Geometric characterizations, with quantum mechanical significance, of the state spaces of Jordan algebras [2] and of C^{∗}-algebras [1] have been known for some time. These characterizations are expressed axiomatically in the category of ordered Banach spaces. However, a geometric model for quantum mechanics based on the premise that states are just the unit vectors of a normed space was initiated in [8] and culminated in the characterization of the state spaces of atomic JB^{∗}-triples in [11]. The introduction of [8] explains why insistence on a binary product leads naturally to ordered spaces and why the resulting triple product in the non-ordered context would have to be conjugate linear in one of its variables.

An important example which corresponds to “two state” physical systems gives rise to the JB^{∗}-triple called a Cartan factor of type 4, or a spin factor. JB^{∗}-triples occur in the study of bounded symmetric domains in finite and infinite dimensions. In recent years, bounded symmetric domains and JB^{∗}-triples have appeared naturally in physics, for example in transmission line theory and special relativity, see [5] and [6]. In this paper we show how the JB^{∗}-triple structure occurs naturally in the representation theory of the Lorentz group.

This paper is organized as follows. The main results are Theorems 3.1, 3.2, 3.3, 4.1. With the exception of a review of some physical concepts in subsection 1.1, sections 1–3 are concerned with JB^{∗}-triples. The definition of JB^{∗}-triple and background on the connection with bounded symmetric domains is given in subsection 1.2. In section 2 we introduce a triple product structure on C^{n} and show that it satisfies the algebraic part of the definition of JB^{∗}-triple. Then we construct two linear Lie groups, as well as their Lie algebras, which are derived from this Jordan structure. These spaces will be used in the constructions of the representations of the Lorentz group in section 4. We shall see that although these finite dimensional representations cannot be unitary, they do have an algebraic invariant, namely the determinant. In section 3, using the determinant, a simple characterization of the tripotents (the basic building blocks of the triple product) is given. Also, a crucial decomposition, interesting in its own right, which we call the facial decomposition (Theorem 3.1), of any element into a linear combination of two maximal tripotents, is used to define the norm which together with the triple product makes C^{n} into a JB^{∗}-triple (Theorem 3.2) denoted S^{n}. We also use the facial decomposition to obtain an analog of the polar decomposition for an arbitrary element (Theorem 3.3), which we believe has physical significance.

Section 4 contains the construction of the representations corresponding to the relativistic transformations of the electromagnetic field and to the Lorentz space-time transformation. The latter is a spin 1 representation. The Jordan theoretic relation between the spin 1 and spin \frac{1}{2} representations is described in Theorem 4.1.

1. PRELIMINARIES

1.1. Lorentz group in relativity and in electromagnetic fields The Lorentz group and its representations pervade any advanced text-

book of physics. Most often the Lorentz group acts on four-vectors, which form a four dimensional real space. Such vectors may represent space-time, energy-momentum, scalar and vector potentials of the electromagnetic field and other physical quantities. The Lorentz group also introduces nonlinear maps of the ball in IR^{3} of all possible velocities, and acts on the intensities of the electromagnetic field, which are a pair of three dimensional vectors. In Relativistic Quantum Mechanics, this group is represented on Dirac bispinors which are complex four tuples and are used to represent the state of an electron. For easy reference in section 4, we will now recall some of these facts, which can be found in [15].

The space-time is a four dimensional real Euclidean space with one coordinate representing time and three coordinates representing position. A reference frame is an orthonormal basis in the space-time. A reference system is called inertial if a moving body which is not acted on by external forces moves with constant velocity, that is, is freely moving. The principle of relativity states that all laws of nature are identical in all inertial frames of reference, that is, the equations expressing these laws are invariant with respect to appropriate transformations of coordinates and time from one system to another. The principle of relativity of Einstein leads to relativistic mechanics, in which the interval between two events is identical for all inertial frames.

The interval between two events (t_{1}, x_{1}, y_{1}, z_{1}) and (t_{2}, x_{2}, y_{2}, z_{2}) is defined as

s = [c^{2}(t_{2} − t_{1})^{2} − (x_{2} − x_{1})^{2} − (y_{2} − y_{1})^{2} − (z_{2} − z_{1})^{2}]^{1/2} .

If we set

x_{0} = ct, x_{1} = x, x_{2} = y, x_{3} = z,

then the interval becomes

s = [△x_{0}^{2} − △x_{1}^{2} − △x_{2}^{2} − △x_{3}^{2}]^{1/2} .

The group generated by all possible transformations from one inertial frame to another is called the Poincaré group. This group contains the subgroups of spatial and time translations and the Lorentz group. The Lorentz group consists of all space-time transformations preserving the interval and the origin of the reference frame.

The Lorentz group is a six dimensional Lie group. It contains as a subgroup the group of spatial rotations, which is isomorphic to the group SO(3) consisting of all 3 by 3 orthogonal real matrices with determinant one. The rotation group is a three dimensional Lie group. The remaining three dimensions of the Lorentz group come from the boosts representing transformations from one inertial system to another that is moving in parallel to the first one with constant velocity V.

The Lie group of rotations consists of spatial rotations about some axis in space by an angle φ. For example the rotation matrix R_{1} about the x-axis by an angle t in coordinates x_{0}, x_{1}, x_{2}, x_{3} is,

R_{1}(t) = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos t & sin t \\ 0 & 0 & − sin t & cos t \end{matrix} ] . (1)

The infinitesimal generator of R_{1}(t), denoted by J_{1}, represents the angular momentum and in the above coordinates is given by

J_{1} = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ] . (2)

Similarly, the infinitesimal generators of rotations about the y-axis and z-axis are

J_{2} = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ] , J_{3} = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ] . (3)

The Lie algebra of the Lie group SO(3), is linearly spanned by these three matrices, which satisfy the commutation relations

[J_{1}, J_{2}] = −J_{3}, [J_{2}, J_{3}] = −J_{1}, [J_{3}, J_{1}] = −J_{2}. (4)

To determine the Lorentz transformations from one inertial reference system to another obtained by boosts, consider the boost in the x-direction. It is easy to show ([15, (4.3),p.10]) that the linear transformation which preserve the intervals for this case leads to

x_{0} = \frac{x′_{0} + \frac{V}{c} x′_{1}}{\sqrt{1 − \frac{V^{2}}{c^{2}}}} , x_{1} = \frac{x′_{1} + \frac{V}{c} x′_{0}}{\sqrt{1 − \frac{V^{2}}{c^{2}}}} , x_{2} = x′_{2}, x_{3} = x′_{3},

where V is the velocity of the second reference frame in direction of the x-axis with respect to the first. Let K_{1} be the infinitesimal generator corresponding to this boost. Then the matrix of K_{1} in the above mentioned coordinates is

K_{1} = [ \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ] .

Similarly, the infinitesimal generators K_{2}, K_{3} of the Lorentz boosts in the y and z directions are given by

K_{2} = [ \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ] , K_{3} = [ \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ] .

Direct calculation show that the remaining commutation relations of the Lie algebra (the real span of J_{k}, K_{k} for k = 1, 2, 3) of the Lorentz group are

[J_{1}, K_{1}] = 0, [J_{1}, K_{2}] = −K_{3}, [J_{1}, K_{3}] = K_{2}, (5)
[J_{2}, K_{1}] = K_{3}, [J_{2}, K_{2}] = 0, [J_{2}, K_{3}] = −K_{1}, (6)
[J_{3}, K_{1}] = −K_{2}, [J_{3}, K_{2}] = K_{1}, [J_{3}, K_{3}] = 0, (7)
[K_{1}, K_{2}] = J_{3}, [K_{2}, K_{3}] = J_{1}, [K_{3}, K_{1}] = J_{2}. (8)

The above action of the Lorentz group is called the realization on the four-vectors. Transformation from one inertial system to another induces also a nonlinear transformation on the ball of possible velocities. These transformations are affine and preserve segments. We will discuss them in a forthcoming paper.

The equations of motion of a charge in a given electromagnetic field involve the electric field intensity E and the magnetic field intensity H.

The transformation of the electric field intensity from one inertial system to another one moving with velocity V in the x-direction is given by ([15, (24.2),p.62])

E_{x} = E′_{x}, E_{y} = \frac{E′_{y} + \frac{V}{c} H′_{z}}{\sqrt{1 − \frac{V^{2}}{c^{2}}}} , E_{z} = \frac{E′_{z} − \frac{V}{c} H′_{y}}{\sqrt{1 − \frac{V^{2}}{c^{2}}}} ,

with similar equations for the magnetic field, where E has coordinates E_{x}, E_{y}, E_{z}.

If we let F denote E + iH, it is a fact that the complex valued quantity

F^{2} = F_{x}^{2} + F_{y}^{2} + F_{z}^{2} = ∥E∥^{2} − ∥H∥^{2} + 2iE · H (9)

is invariant under the transition from one inertial frame to another. The above transformations of electric and magnetic fields take the form

F_{x} = F′_{x}, F_{y} = F′_{y} cosh ψ − iF′_{z} sinh ψ, F_{z} = iF′_{y} sinh ψ + F′_{z} cosh ψ,

and are thus described by the matrix

[ \begin{matrix}1 & 0 & 0 \\ 0 & cosh ψ & i sinh ψ \\ 0 & −i sinh ψ & cosh ψ \end{matrix} ] , (10)

where tanh ψ = V/c. Thus, the natural model for the electromagnetic field intensity is the vector F in C^{3} and the Lorentz group is represented by the linear maps preserving F^{2}.

1.2. JB^{∗}-triples

Great insights have been obtained in the last 30 years by stripping a C^{∗}- algebra either of its ^{∗}-operation (leading to the theory of nonself-adjoint operator algebras) or of its multiplication (leading to the theory of operator spaces). One way to think of a JB^{∗}-triple is to consider it as a C^{∗}-algebra (or a Jordan C^{∗}-algebra) which has been stripped of its order structure. Since squares of self-adjoint elements are positive, the (binary) multiplication must be discarded in favor of a ternary product, necessarily nonassociative. The abstract structure obtained in this way has been used for example to classify all bounded symmetric domains in finite (and to some extent infinite) dimensions and to give the complete structure of the Toeplitz C^{∗}-algebra on the Hardy space of a bounded symmetric domain. Moreover, many of the landmark results of the C^{∗}-theory (Gelfand-Naimark theorem, state space characterizations, automatic continuity of derivations, Grothendieck inequality, to name a few) had their analogs established in the last 15 years by using techniques from affine geometry and holomorphy to overcome the lack of an order structure.

A symmetric domain is an open connected subset D of a complex (finite or infinite dimensional) Banach space with the following property: for each point of D, there is an involutive holomorphic automorphism of D which has that point as an isolated fixed point. This is the same as the domain having the property at a single point, and being homogeneous (the group of all holomorphic automorphisms acts transitively). For example, in the complex plane, the unit disk, and hence by the Riemann mapping theorem, every simply connected proper planar domain, is a symmetric domain.

For a bounded symmetric domain D in C^{n}, the set G = aut D of complete holomorphic vector fields on D is a real Lie algebra making the group G = Aut (D) of all holomorphic automorphisms of D into a real Lie group. From this the irreducible bounded symmetric domains in C^{n} were classified. These results were accomplished in the 1930s by E. Cartan and H. Cartan.

In 1969, M. Koecher established a correspondence between bounded symmetric domains in C^{n} and finite dimensional JB^{∗}-triples; for a thorough discussion of this see the lecture notes of Loos [16]. Unlike the Lie theoretic methods, the Jordan machinery could be extended so as to give results in infinite dimensions. I mention only the following fundamental result of Kaup from 1983 [14]: every bounded symmetric domain in a complex Banach space is holomorphically equivalent to the open unit ball of some other Banach space (namely, a JB^{∗}-triple).

A forerunner of Kaup’s theorem was proved by Harris in 1973 [12], thereby providing a link between infinite dimensional holomorphy and functional analysis: the open unit ball of a C^{∗}-algebra is a bounded symmetric domain, a transitive family of automorphisms being provided by the Möbius transformations

z ↦ (I − aa^{∗})^{−1/2}(z + a)(I + a^{∗}z)^{−1}(I − a^{∗}a)^{1/2}.

Since this formula involves only the symmetrized triple product

\{abc\} = \frac{ab^{∗}c + cb^{∗}a}{2}, (11)

the result holds for norm closed subspaces of C^{∗}-algebras stable for the map (a, b, c) ↦ {abc}, equivalently a ↦ aa^{∗}a. Such spaces were called J^{∗}-algebras by Harris but are now called JC^{∗}-triples.

In 1985, Friedman and Russo ([7]) showed that JB^{∗}-triples occur naturally in functional analysis by proving that the image of a contractive projection on a C^{∗}-algebra is a JB^{∗}-triple. This result complemented earlier results of Effros-Stormer and Effros-Choi which required the projection to be positive and completely positive, respectively, and it led to its extension by Kaup to arbitrary JB^{∗}-triples. The latter result played an important role in the structure theory of JB^{∗}-triples, for which we refer to [18] for a summary. We simply recall here the Gelfand-Naimark theorem for JB^{∗}-triples [9], which asserts that every JB^{∗}-triple is isometrically isomorphic to a subtriple of a direct sum of Cartan factors. The earlier Gelfand Naimark type theorem for JB-algebras was achieved in 1978 by Alfsen, Shultz, and Stormer; and it extended the finite dimensional result which is due to Jordan, von Neumann, and Wigner in 1934.

There are six types of Cartan factors. The types 1,2, and 3 are very easy to describe. In finite dimensions, they consist respectively of the rectangular m by n complex matrices, the k by k anti-symmetric complex matrices, and the l by l symmetric complex matrices. Here, m ≥ 1, n ≥ 1, k ≥ 3, l ≥ 2. The types 5 and 6 are exceptional of dimensions 16 and 27 respectively, and we will not be concerned with them. The Cartan factor of type 4, sometimes called a spin factor will be a key ingredient of the rest of this paper. Cartan factors of types 1-4 are JC^{∗}-triples, and the above descriptions of them are valid in infinite dimensions.

We introduce now some facts about spin factors which will help us to motivate our paper (cf [4]). An even dimensional spin factor is a linear space with a basis G = {v_{j}, ṽ_{j} : j ∈ J} and a (conjugate linear) conjugation ^{♯} defined by v_{i}^{♯} = ṽ_{i} (and ṽ_{i}^{♯} = v_{i}). Let u_{i} be either v_{i} or ṽ_{i}. Define the triple product on the space by extending it linearly (conjugate linear in one of the variables) from the basis elements as follows. The only non-zero triple products of the basis elements for any i, j ∈ J with i ≠ j are the following

(i) {u_{i}u_{i}u_{i}} = u_{i}, meaning that u_{i} is a tripotent;

(ii) \{u_{i}u_{i}u_{j}\} = \frac{1}{2}u_{j}, meaning that u_{i} and u_{j} are co-orthogonal (colinear in the literature)

(iii) \{u_{i}u_{j}u_{i}^{♯}\} = −\frac{1}{2}u_{j}^{♯}, meaning that u_{i}, u_{j}, u_{i}^{♯}, u_{j}^{♯} form an odd quadrangle.

Note that from the above definition, it follows that {u_{i}u_{i}^{♯}u_{k}} = 0 for any u_{k}, meaning that the pairs v_{i}, ṽ_{i} are orthogonal. Also, each u_{i} is a minimal tripotent, meaning that {u_{i}u_{j}u_{i}} = 0 for all j ≠ i. We shall also make use of maximal tripotents, which are defined by {www} = w and {wwa} = a for every a ∈ G. For example, for each a ∈ G, a+a^{♯} is a maximal tripotent. A basis of the above type for a spin factor is called a spin grid.

We give two examples of spin grids. The second will be important in subsection 4.4. The triple product used is given by (11).

Example 1.1. [2 by 2 matrices]

v_{1} = [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ]; ṽ_{1} = [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]; v_{2} = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ]; ṽ_{2} = [ \begin{matrix}0 & 0 \\ −1 & 0 \end{matrix} ] .

Example 1.2. [4 by 4 anti-symmetric matrices]

v_{1} = [ \begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ]; v_{2} = [ \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ]; v_{3} = [ \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ];

ṽ_{1} = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ]; ṽ_{2} = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]; ṽ_{3} = [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ] .

We note that in both Examples 1.1 and 1.2, there is an unexpected minus sign. This is due to the fact that in order for all the quadrangles (v_{i}, v_{j}, ṽ_{i}, ṽ_{j}) to “glue” together correctly, as noted above, they must be odd quadrangles. Note also that in Examples 1.1 and 1.2, the conjugation ♯ is different from the usual adjoint for matrices, behaving more like the Hodge operator for differential forms.

A spin grid not only determines the triple product, but it can also be used to define an inner product by declaring the spin grid to be an orthonormal basis, that is, ⟨a|b⟩ = ∑ a_{j}\overline{b}_{j} + ∑ ã_{j}\overline{b̃}_{j} if a = ∑ a_{j}v_{j} + ∑ ã_{j}ṽ_{j} and b = ∑ b_{j}v_{j} + ∑ b̃_{j}ṽ_{j}. Then the triple product, inner product, and conjugation are related by

2{abc} = ⟨a|b⟩c + ⟨c|b⟩a − ⟨a|c^{♯}⟩b^{♯}. (12)

A useful tool for studying spin factors is the so-called determinant, which is defined by

det a = \frac{1}{2}⟨a|a^{♯}⟩. (13)

For example, the spectral theorem says that if det a ≠ 0, then a = s_{1}e+s_{2}f, where e and f are orthogonal minimal tripotents, s_{1} ≥ s_{2} > 0 and ∥a∥ = max{s_{1}, s_{2}} = s_{1}. Moreover, for the dual norm, we have

sup_{∥b∥≤1} |⟨a|b⟩| = s_{1} + s_{2}. (14)

The definition of JB^{∗}-triple will now be given. A complex Banach space U is said to be a JB^{∗}-triple if it is equipped with a continuous triple product (a, b, c) ↦ {abc} mapping U × U × U to U such that

(i) {abc} is linear in a and c and conjugate linear in b; (ii) {abc} is symmetric in the outer variables: {abc} = {cba}; (iii) for any x ∈ U, the operator Δ(x) from U to U defined by Δ(x)y =

{xxy} is hermitian (that is, exp itΔ(x) is an isometry for all real t) with non-negative spectrum;

(iv) the triple product satisfies the following identity,

Δ(x){abc} = {Δ(x)a, b, c} − {a, Δ(x)b, c} + {a, b, Δ(x)c}. (15)

(v) ∥{xxx}∥ = ∥x∥^{3}.

The identity (15) can be linearized to

{x, y, {abc}} = {{xya}, b, c} − {a, {yxb}, c} + {a, b, {xyc}}.

Conditions (i), (ii), and (iv) are the defining properties for a (complex) Jordan triple system, which is a purely algebraic object. The building blocks of a Jordan triple system are the tripotents (that is, elements e with {eee} = e) and their corresponding Peirce decompositions. There are important relations between pairs, triples, and quadruples of tripotents (orthogonal, colinear, governing, trangle, quadrangle). These relations are fundamental tools in the classification of atomic JB^{∗}-triples [4] as well as in their abstract counterparts, the facially symmetric spaces [11].

2. TRIPLE PRODUCT REPRESENTATION OF CAR

The classical definition of the canonical anticommutation relations (CAR) involves a sequence p_{k} of elements of an associative algebra which satisfy the relations

p_{l}p_{k} + p_{k}p_{l} = 2δ_{kl}.

This trivially implies that

p_{l}p_{k}p_{l} = −p_{k} for k ≠ l; and p_{k}p_{k}p_{l} = p_{l} for any k, l.

The last relations can be called “the triple canonical anticommutation relations” (TCAR for short).

We note now that the TCAR occur naturally in a Cartan factor of type 4. Let G = {v_{j}, ṽ_{j} : j ∈ J} be a spin grid (see section 1.2) for a spin factor of dimension n which we assume finite for simplicity of notation. Define a basis of maximal tripotents w_{1}, …, w_{n} by

w_{2j} = v_{j} + ṽ_{j}, w_{2j−1} = i(v_{j} − ṽ_{j}). (16)

Then a direct calculation shows that ⟨w_{p}|w_{q}⟩ = 2δ_{pq}, w_{p}^{♯} = w_{p} and

{w_{l}w_{k}w_{k}} = {w_{k}w_{k}w_{l}} = w_{l}; {w_{l}w_{k}w_{l}} = −w_{k} + 2δ_{kl}w_{l}

for any 1 ≤ k, l ≤ n, and

{w_{k}w_{l}w_{m}} = 0 if k, l, m are distinct.

This motivates the definitions in the next subsection.

2.1. A triple product structure on C^{n}

Now let C^{n} be n dimensional complex Euclidean space with the usual inner product ⟨·|·⟩, Euclidean norm ∥ · ∥_{2}, and natural basis

e_{1} = (1, 0, …, 0), e_{2} = (0, 1, …, 0), ⋯, e_{n} = (0, …, 0, 1).

We define a triple product on C^{n} by use of the TCAR for the elements of the natural basis and extend it to the whole space to be linear and symmetric in the outer variables and conjugate linear in the middle variable. Precisely, define for any 1 ≤ k, l ≤ n

{e_{l}e_{k}e_{k}} = e_{l}; {e_{k}e_{k}e_{l}} = e_{l}; {e_{l}e_{k}e_{l}} = −e_{k} + 2δ_{kl}e_{l}; (17)

and

{e_{k}e_{l}e_{m}} = 0 if k, l, m are distinct. (18)

Then for arbitrary a = (a_{1}, …, a_{n}), b = (b_{1}, …, b_{n}), c = (c_{1}, …, c_{n}) ∈ C^{n} define

{abc} = ∑_{j,k.l} a_{j}b̄_{k}c_{l}{e_{j}e_{k}e_{l}}.

It is trivial to check that

{abc} = ⟨a|b⟩c + ⟨c|b⟩a − ⟨a|c̄⟩b̄. (19)

where \overline{a} = (\overline{a}_{1}, …, \overline{a}_{n}). This should be compared with (12).

We now define a T CAR basis to be any linearly independent set u_{1}, ⋯, u_{n} in C^{n} which satisfies (17) and (18), where the triple product is defined by (19). It follows that such a basis is an orthonormal basis in C^{n}. Of course, an example of such a basis is the natural basis.

The straightforward proof of the following lemma will be omitted.

Lemma 2.1. An orthonormal basis u_{1}, ⋯, u_{n} is a TCAR basis if and only if u_{j} = ⟨u_{k}|ū_{k}⟩ū_{j} for all k, j. In particular, ⟨u_{j}|ū_{k}⟩ = 0 for all k ≠ j. Moreover, there is a λ ∈ C, |λ| = 1 such that ⟨u_{k}|ū_{k}⟩ = λ and u_{k} = λū_{k}.

Corollary 2.1. An orthonormal set which satisfies the TCAR properties can be extended to a TCAR basis.

Proof. Let u_{j}, 1 ≤ j ≤ m satisfy TCAR. There is a λ with |λ| = 1 such that for all j, u_{j} = λ\overline{u}_{j}. Thus u_{j} = μx_{j} where μ^{2} = λ and x_{1}, …, x_{m} is an orthonormal set of vectors with real coordinates. Now complete x_{1}, …, x_{m} to an orthonormal basis x_{1}, …, x_{n} of IR^{n} and set u_{j} = μx_{j} for m+1 ≤ j ≤

n.

The space C^{n} with the triple product defined by (19) will be denoted by S^{n}. For any x, y ∈ S^{n} define a linear map D(x, y) by

D(x, y)z = {xyz} for any z ∈ S^{n}.

By straightforward calculation, using (19),

D(x, y){abc} = {D(x, y)a, b, c} − {a, D(x, y)b, c} + {a, b, D(x, y)c}(20)

which shows that S^{n} is a Jordan triple system, that is, satisfies the “algebraic part” of the definition of JB^{∗}-triple ((i),(ii) and (iv)). The analytic part, that is, conditions (iii) and (v) will be verified in section 3 after constructing a norm different from the Hilbert space norm used so far.

2.2. The linear automorphism group Taut (S^{n}) and its Lie

algebra

The following proposition will be used to define a Lie group of linear transformations which will be instrumental in the construction of the representations in section 4.

Proposition 2.1. Let T be a linear map of S^{n} into S^{n}. The following are equivalent:

(a) T is invertible and satisfies

T {abc} = {T aT bT c}. (21)

(b) If {u_{1}, …, u_{n}} is any TCAR basis, then so is {T u_{1}, …, T u_{n}}.

(c) The matrix of T in the natural basis has the form λU where λ ∈ C, |λ| = 1 and U is a real orthogonal matrix.

Proof. [(a)⇒(b)] Let {u_{j}} be a TCAR basis and let w_{j} = T u_{j}. Since T preserves the triple product, the w_{j} satisfy (17)-(18). We have already remarked that this implies that ⟨w_{j}|w_{k}⟩ = δ_{jk}.

[(b)⇒(c)] Since T takes an orthonormal basis to an orthonormal basis, it preserves the inner product. Since {T e_{j}} is a TCAR basis, by Lemma 2.1 there is a μ ∈ C, |μ| = 1 such that \overline{T e_{k}} = μT e_{k}. Define λ = μ̄^{1/2} and U = λ̄T or T = λU. Then

\overline{Ue_{k}} = \overline{λ̄T e_{k}} = λ\overline{T e_{k}} = λμT \overline{e_{k}} = λ̄T (e_{k}) = Ue_{k}.

This implies that the matrix of U in the natural basis is a real unitary matrix and thus orthogonal.

[(c)⇒(a)] Since the matrix for U is real, \overline{Ua} = U\overline{a}, so (21) follows.

A linear map T on S^{n} satisfying one of the equivalent conditions of the above Proposition will be called a triple automorphism of S^{n}. We denote by Taut (S^{n}) the Lie group of all such triple automorphisms. From Proposition 2.1 it follows that that this group is the product of O(n), the orthogonal group of dimension n, and the group U(1) of rotations in the complex plane. Thus, the real dimension of Taut (S^{n}) is n(n − 1)/2 + 1.

To describe the Lie algebra taut (S^{n}) of the group Taut (S^{n}) of triple automorphisms, note that the infinitesimal generators of such automorphisms are triple derivations, that is, linear maps δ on S^{n} that satisfy

δ{abc} = {δa, b, c} + {a, δb, c} + {a, b, δc}.

For any u_{k}, u_{l} of a TCAR basis with k ≠ l we have D(u_{k}, u_{l}) = −D(u_{l}, u_{k}) and thus from (20)

D(u_{k}, u_{l}){a, b, c} = {D(u_{k}, u_{l})a, b, c}+{a, D(u_{k}, u_{l})b, c}+{a, b, D(u_{k}, u_{l})c}

implying that D(u_{k}, u_{l}) ∈ taut (S^{n}). From (17) and (18) it follows that the matrix of D(u_{k}, u_{l}) with respect to the basis u_{1}, …, u_{n} is a basic real antisymmetric matrix. Since a real linear combination of triple derivations is also a triple derivation, the operator δ defined on S^{n} by

δ = ∑_{k<l} d_{kl}D(u_{k}, u_{l}) with d_{kl} ∈ IR

is represented by an n × n real antisymmetric matrix and δ ∈ taut (S^{n}).

Note that by properties of the triple product, it follows that the operator diI is also a triple derivation for any real d. Counting dimensions we arrive at the following proposition.

Proposition 2.2. The Lie algebra taut (S^{n}) is the direct sum of the algebra of real antisymmetric n × n matrices and IRi, that is,

taut (S^{n}) = {δ = diI + ∑_{k<l} d_{kl}D(u_{k}, u_{l}) : d, d_{kl} ∈ IR}.

This description of taut (S^{n}) is clearly independent of the choice of TCAR basis, up to isomorphism of Lie algebras.

2.3. The determinant preserving group Dinv (S^{n}) and its Lie

algebra

The finite dimensional representations that we are going to construct in section 4 cannot be unitary because of the non-compactness of the Lorentz group. However, there is another algebraic invariant which will be preserved by our representations, and this invariant will be related to the relativistic invariant of intensity in an electromagnetic field. We introduce that invariant, called determinant, in this section. It will be used also in section 3 to classify the tripotents of S^{n} and to prove the important facial decomposition.

For any a ∈ S^{n} define a notion of determinant by

det a = ⟨a|ā⟩ = ∑ a_{j}^{2}.

After we have proved Theorem 3.2, we shall see that this definition agrees with the determinant in a spin factor, given by (13), and coincides with the ordinary determinant of a matrix in Examples 1.1 and 1.2.

Define the group Dinv (S^{n}) to be the set of all invertible linear maps of S^{n} which preserve the determinant. Note that SO(n) ⊂ Dinv (S^{n}) ⊂ GL(S^{n}) and if we denote the Lie algebra of Dinv (S^{n}) by dinv (S^{n}), we have so(n) ⊂ dinv (S^{n}) ⊂ gl(S^{n}). By general principles, if g(t) is a one-parameter subgroup of Dinv (S^{n}), then X := g′(0) ∈ dinv (S^{n}) and g(t) = exp tX. Conversely, if X ∈ dinv (S^{n}), then exp tX is a one-parameter subgroup of Dinv (S^{n}). That is, dinv (S^{n}) consists of all infinitesimal generators of one-parameter subgroups of Dinv (S^{n}).

Note that for λU ∈ Taut (S^{n}), ⟨λUa|\overline{λUa}⟩ = λ^{2}⟨a|\overline{a}⟩ so that

Taut (S^{n}) ∩ Dinv (S^{n}) = SO(n).

Proposition 2.3. If g(t) is a one-parameter subgroup in Dinv (S^{n}) then its infinitesimal generator X (= g′(0) ∈ dinv (S^{n})) is an anti-symmetric matrix with respect to any TCAR basis of S^{n}. Hence dinv (S^{n}) ⊂ A_{n}(C), where A_{n}(C) denotes the Cartan factor of type 2 (n by n complex antisymmetric matrices).

Proof. Differentiating the equation

⟨g(t)a|\overline{g(t)a}⟩ = ⟨a|\overline{a}⟩

at t = 0 yields ⟨Xa|\overline{a}⟩ = 0 for every a ∈ S^{n} and by polarization ⟨Xb|\overline{a}⟩ + ⟨Xa|\overline{b}⟩ = 0 for every a, b ∈ S^{n}.

If X has the matrix [x_{jk}] with respect to a TCAR basis u_{1}, …, u_{n}. Then

⟨Xu_{j}|u_{k}⟩ + ⟨Xu_{k}|u_{j}⟩ = 0,

so X is anti-symmetric.

We shall see later (in subsection 4.2) that dinv (S^{4}) = A_{4}(C) (= S^{6}).

The problem of characterizing the linear maps which preserve the determinant has a long history going back to Frobenius in 1897 and culminating with a comprehensive study of the problem when the field is replaced by a commutative ring [21]. Frobenius showed that the linear maps of the n by n complex matrices which preserve determinant are of one of the forms A ↦ P AQ or A ↦ P A^{t}Q, where A^{t} is the transpose of A and P, Q are matrices with det P Q = 1. It would be interesting to find a characterization of determinant preserving linear maps on S^{n}. It would also be interesting to explore the analog in our setting of the pseudo-orthogonal groups SO(n, C) of determinant 1 linear maps of C^{n} which preserve the form z_{1}^{2} + ⋯ + z_{n}^{2}.

3. THE JB^{∗}-triple STRUCTURE OF S^{N}

In this section we construct the norm which makes S^{n} with the triple product (19) into a JB^{∗}-triple. To do this we need some decomposition theorems which are interesting in their own right. Moreover, the fact that S^{n} is a JB^{∗}-triple will be useful in our forthcoming study of nonlinear representations of the Lorentz group. Only Proposition 3.3 will be used explicitly in section 4.

3.1. Facial Decomposition

We will describe first the tripotents u, meaning elements u ∈ S^{n} satisfying u = {uuu}, that are the building blocks of the triple product.

If det a ≠ 0 the argument of the determinant is defined as

ard a = \frac{det a}{| det a|}

and ard a = 1 if det a = 0. Note that the elements with zero determinant are called null-vectors in the literature.

From (19) an element u is a tripotent in one of the following two cases:

det u = 0 and ⟨u|u⟩ = 1/2,

in which case u is called a minimal tripotent; or

ū = \overline{(ard u)}u and ⟨u|u⟩ = 1,

in which case u is called a maximal tripotent.

Note that it follows from (19) that for any maximal tripotent u we have D(u, u) = I and from (17) that each element u_{j} of a TCAR basis is a maximal tripotent. On the other hand, for a minimal tripotent v, we have D(v, v)v̄ = 0 and both v + v̄ and v − v̄ are maximal tripotents.

Lemma 3.1. Let w_{1}, w_{2} be a pair of maximal tripotents in S^{n}. The following are equivalent.

(a) w_{1}, w_{2} are linearly independent and {w_{1}w_{2}w_{1}} = w_{2}

(b) ard w_{1} = −ard w_{2} and ⟨w_{1}|w_{2}⟩ = 0

(c) w_{1}, iw_{2} are linearly independent and safisfy TCAR.

Proof. [(a)⇒(b)] From \overline{w}_{j} = \overline{ard w_{j}}w_{j}, and 2⟨w_{1}|w_{2}⟩w_{1} − ⟨w_{1}|\overline{w}_{1}⟩\overline{w}_{2} = w_{2} we obtain

2⟨w_{1}|w_{2}⟩w_{1} − (⟨w_{1}|\overline{ard w_{1}}w_{1}⟩\overline{ard w_{2}} + 1)w_{2} = 0.

[(b)⇒(c)] Since w_{1} and iw_{2} are maximal tripotents, we need only to show that {w_{1}, iw_{2}, w_{1}} = −iw_{2} and {iw_{2}, w_{1}, iw_{2}} = −w_{1}. For example

\{iw_{2}, w_{1}, iw_{2}\} = −\{w_{2}w_{1}w_{2}\} = ⟨w_{2}|\overline{w}_{2}⟩\overline{w}_{1}
= ard w_{2}\overline{ard w_{1}}w_{1} = −w_{1}.

[(c)⇒(a)] {w_{1}w_{2}w_{1}} = i{w_{1}, iw_{2}, w_{1}} = i(−iw_{2}) = w_{2}.

A pair of maximal tripotents w_{1}, w_{2} will be called a complementary pair if one of the equivalent conditions in Lemma 3.1 is satisfied.

Theorem 3.1 (Facial Decomposition). For any element a ∈ S^{n} there exist a unique complementary pair w_{1}, w_{2} of maximal tripotents and a unique pair of non-negative real numbers α_{1}, α_{2} satisfying

α_{1} ≥ α_{2} ≥ 0

and

a = α_{1}w_{1} + α_{2}w_{2}. (22)

The formula (22) will be called the “facial decomposition” of a. This terminology will be justified later in this section.

Proof. For any element a ∈ S^{n} with det a ≠ 0, if λ is a complex number with |λ| = 1, then ard (λa) = λ^{2}ard a. Thus, if μ = (ard a)^{1/2}, then the element b defined by a = μb has a nonnegative determinant. If we decompose b as

b = x + iy x, y ∈ span_{IR}{e_{j}},

then (compare to (9))

det b = ∥x∥_{2}^{2} − ∥y∥_{2}^{2} + 2i⟨x|y⟩, (23)

and the condition det b ≥ 0 implies

⟨x|y⟩ = 0 and ∥x∥_{2} ≥ ∥y∥_{2}. (24)

Therefore, in the above notation we have

a = μ(x + iy) with μ = (ard a)^{1/2}, x, y ∈ span_{IR}{e_{j}} (25)

and x, y satisfying (24).

For any element a ∈ S^{n} satisfying (25) we have

a = μ(∥x∥_{2} \frac{x}{∥x∥_{2}} + ∥y∥_{2}i\frac{y}{∥y∥_{2}} ) = α_{1}w_{1} + α_{2}w_{2},

where the positive real numbers α_{1} = ∥x∥_{2} , α_{2} = ∥y∥_{2} and the elements w_{1} = μ\frac{x}{∥x∥_{2}} , w_{2} = iμ\frac{y}{∥y∥_{2}} are complementary maximal tripotents with ard a = ard w_{1}. It is readily seen that w_{1} and w_{2} satisfy the condition (b) for complementary pair.

Let us now prove the uniqueness of the facial decomposition. If a = α_{1}w_{1} + α_{2}w_{2} = β_{1}v_{1} + β_{2}v_{2}, with w_{1}, w_{2} and v_{1}, v_{2} complementary pairs of maximal tripotents and α_{1} ≥ α_{2} ≥ 0 and β_{1} ≥ β_{2} ≥ 0, there are, by Lemma 2.1, constants λ, μ of absolute value one with w_{1} = λ\overline{w}_{1}, w_{2} = −λ\overline{w}_{2} and v_{1} = μ\overline{v}_{1}, v_{2} = −μ\overline{v}_{2}. From det a = ⟨a|\overline{a}⟩ we obtain (α_{1}^{2} − α_{2}^{2})λ = (β_{1}^{2} − β_{2}^{2})μ, and from ∥a∥_{2}^{2} = ⟨a|a⟩ we obtain α_{1}^{2} + α_{2}^{2} = β_{1}^{2} + β_{2}^{2} and so α_{j} = β_{j} and λ = μ.

Now from

a = α_{1}w_{1} + α_{2}w_{2} = α_{1}v_{1} + α_{2}v_{2}

and

\overline{a} = α_{1}\overline{λ}w_{1} − α_{2}\overline{λ}w_{2} = α_{1}\overline{λ}v_{1} − α_{2}\overline{λ}v_{2},

we obtain

w_{1} = \frac{a + λ\overline{a}}{2α_{1}}, w_{2} = \frac{a − λ\overline{a}}{2α_{2}}, (26)

and the same formulas hold for v_{1} and v_{2}.

3.2. The JB^{∗}-triple norm of S^{n}; Polar Decomposition As consequences of the facial decomposition, we can now define a norm

which will make S^{n} into a JB^{∗}-triple, and prove a polar decomposition for arbitrary elements of the JB^{∗}-triple S^{n}.

The norm is defined for any element a ∈ S^{n} with facial decomposition (22) by

∥a∥ = α_{1} + α_{2}. (27)

We will show first that

∥{aaa}∥ = (α_{1} + α_{2})^{3} = ∥a∥^{3}, (28)

which is the analog of the defining identity ∥a^{∗}a∥ = ∥a∥^{2} for a C^{∗}-algebra. Thus we can refer to this norm as the “operator norm”.

As in the proof of Theorem 3.1, ā = μ̄(x − iy) = μ̄^{2}(α_{1}w_{1} − α_{2}w_{2}), ⟨a|a⟩ = α_{1}^{2} + α_{2}^{2}, and ⟨a|ā⟩ = μ^{2}(α_{1}^{2} − α_{2}^{2}).

From (19) it follows that

{aaa} = (α_{1}^{3} + 3α_{2}^{2}α_{1})w_{1} + (α_{2}^{3} + 3α_{1}^{2}α_{2})w_{2},

which implies (28).

We still need to show that the function ∥ · ∥ is a norm. It is obvious from the uniqueness of the facial decomposition that

∥λa∥ = |λ|∥a∥ (29)

for every λ ∈ C. The proof of the triangle inequality is more difficult. An elementary proof requires the solution of a optimization problem for which a direct proof seems intractible. Instead, we shall prove the following lemma using properties of spin factors.

Lemma 3.2. The function ∥ · ∥ defined by (27) satisfies the triangle inequality, and is therefore a norm on S^{n}.

Proof. Let C denote the Cartan factor of type 4 and dimension n. Assume for simplicity of notation that n is even and let v_{1}, ṽ_{1}, …, v_{m}, ṽ_{m} be a spin grid for C (see subsection 1.2). Let ϕ : C → S^{n} be defined by

ϕ(a) = \frac{1}{2}(a_{1}+ã_{1}, −i(a_{1}−ã_{1}), a_{2}+ã_{2}, −i(a_{2}−ã_{2}), …, a_{m}+ã_{m}, −i(a_{m}−ã_{m})),

for a = ∑ a_{j}v_{j} + ∑ ã_{j}ṽ_{j} ∈ C.

One checks easily that

• ⟨ϕ(a)|ϕ(a)⟩ = \frac{1}{2}⟨a|a⟩

• ⟨ϕ(a)|\overline{ϕ(a)}⟩ = ∑ a_{j}ã_{j}

• ϕ(a^{♯}) = \overline{ϕ(a)}.

Hence, from (12) and (19), ϕ({aaa}) = {ϕ(a), ϕ(a), ϕ(a)} (and by polarization, ϕ is an isomorphism: ϕ({abc}) = {ϕ(a), ϕ(b), ϕ(c)}).

Since the function ∥ϕ(a)∥ is continuous, there is a constant k < ∞ such that sup{∥ϕ(a)∥ : ∥a∥ ≤ 1} ≤ k, and by (29), ∥ϕ(a)∥ ≤ k∥a∥. We now have ∥ϕ(a)∥^{3} = ∥{ϕ(a)ϕ(a)ϕ(a)}∥ = ∥ϕ({aaa})∥ ≤ k∥{aaa}∥ = k∥a∥^{3}. By iteration ∥ϕ(a)∥ ≤ ∥a∥, and by symmetry ∥ϕ^{−1}(b)∥ ≤ ∥b∥. Finally ∥a+b∥ = ∥ϕ^{−1}(a)+ϕ^{−1}(b)∥ ≤ ∥ϕ^{−1}(a)∥+∥ϕ^{−1}(b)∥ = ∥a∥+∥b∥.

Theorem 3.2. S^{n} is a JB^{∗}-triple isomorphic to a Cartan factor of type 4.

Proof. Referring to the definition of JB^{∗}-triple given earlier, it remains to show that the operator Δ(a) is Hermitian and positive. In the first place, if a = α_{1}w_{1} + α_{2}w_{2} is the facial decomposition of a, then the matrix of Δ(a) with respect to a TCAR basis w_{1}, iw_{2}, u_{3}, ⋯, u_{n} containing w_{1}, iw_{2} is

Δ(a) = [ \begin{matrix}α_{1}^{2} + α_{2}^{2} & 2iα_{1}α_{2} & 0 & ⋯ & 0 \\ −2iα_{1}α_{2} & α_{1}^{2} + α_{2}^{2} & 0 & ⋯ & 0 \\ 0 & 0 & α_{1}^{2} + α_{2}^{2} & ⋯ & 0 \\ · & · & · & · & · \\ 0 & 0 & 0 & ⋯ & α_{1}^{2} + α_{2}^{2} \end{matrix} ] ,

and this matrix has eigenvalues α_{1}^{2} + α_{2}^{2}, (α_{1} + α_{2})^{2}, and (α_{1} − α_{2})^{2}.

Next let T denote the operator exp itΔ(a) for real t. Since itΔ(a) is a derivation of the triple product, T is an automorphism. Thus, as in the proof of Lemma 3.2, from {T xT xT x} = T {xxx} we obtain

∥T x∥^{3} = ∥{T xT xT x}∥ = ∥T {xxx}∥ ≤ ∥T ∥∥x∥^{3},

so that by iteration ∥T x∥ ≤ ∥T ∥^{3^{−k}}∥x∥ and ∥T ∥ ≤ 1. Since T^{−1} is also an automorphism, ∥T^{−1}∥ ≤ 1, so T is an isometry, which proves that Δ(a) is hermitian.

Now that we know that ∥·∥ is a norm, we can use the facial decomposition to prove the polar decomposition.

Theorem 3.3 (Polar Decomposition). For any a ∈ S^{n} with facial decomposition a = α_{1}w_{1} +α_{2}w_{2}, there exists a positive Hermitian operator T on S^{n} such that a = T (w_{1}) and ∥T ∥ = ∥a∥. If w is any maximal tripotent and T is any Hermitian operator such that a = T w, then w = w_{1}.

Proof. For any a ∈ S^{n} with the facial decomposition (22),

a = (α_{1}I + α_{2}D(w_{1}, w_{2}))w_{1},

which we will call the canonical polar decomposition of a. We shall show that the operator H := α_{1}I + α_{2}D(w_{1}, w_{2}) is an Hermitian operator with positive spectrum and with norm ∥a∥. Note that H = i(−iα_{1}I + α_{2}D(w_{1}, iw_{2})) = iδ with δ ∈ taut (S^{n}).

The matrix of H with respect to a TCAR basis w_{1}, iw_{2}, u_{3}, …, u_{n} which extends w_{1}, iw_{2} is

H = [ \begin{matrix}α_{1} & iα_{2} & 0 & ⋯ & 0 \\ −iα_{2} & α_{1} & 0 & ⋯ & 0 \\ 0 & 0 & α_{1} & ⋯ & 0 \\ · & · & · & · & · \\ 0 & 0 & 0 & ⋯ & α_{1} \end{matrix} ] , (30)

which has eigenvalues α_{1}, α_{1} ± α_{2}.

To show H is a hermitian operator with respect to the norm ∥ · ∥, it is required to show that exp itH is an isometry for all real t. As is well known for JB^{∗}-triples, this is the same as exp itH being a (triple) automorphism. But we have already observed above that iH ∈ taut (S^{n}), and so exp itH ∈ Taut (S^{n}).

Finally, by the well known inequality for JB^{∗}-triples ∥{abc}∥ ≤ ∥a∥∥b∥∥c∥ ([9]), if x ∈ S^{n}, then ∥H(x)∥ = ∥α_{1}x + α_{2}{w_{1}w_{2}x}∥ ≤ (α_{1} + α_{2})∥x∥. Since H(w_{1}) = a, ∥H∥ = ∥a∥. This proves the first statement. (Alternatively, ∥H∥ = ∥a∥ follows from that fact, due to Sinclair, that the norm of a Hermitian operator equals its spectral radius, [3, p.57])

We now prove the second statement. Since T is hermitian, iT ∈ taut (S^{n}), so by Proposition 2.2,

T = i ∑_{j<k} d_{jk}D_{jk} − dI for some d_{jk}, d ∈ IR, (31)

where D_{jk} = D(u_{j}, u_{k}) and u_{1}, u_{2}, …, u_{n} is any TCAR basis with w = u_{1}. In particular

a = T u_{1} = −du_{1} − i(∑_{2}^{n} d_{1j}u_{j}).

With \overline{u}_{j} = μu_{j} and \overline{w}_{1} = λw_{1}, \overline{w}_{2} = −λw_{2},

\overline{λ}(α_{1}^{2} − α_{2}^{2}) = det a = \overline{μ}(d^{2} − ∑_{2}^{n} d_{1j}^{2}).

Therefore \overline{λ}μ = 1 and

α_{1}^{2} − α_{2}^{2} = d^{2} − ∑_{2}^{n} d_{1j}^{2}.

Similary, using ⟨a|a⟩, we obtain

α_{1}^{2} + α_{2}^{2} = d^{2} + ∑_{2}^{n} d_{1j}^{2},

so that α_{1} = −d and α_{2} = (∑_{2}^{n} d_{1j}^{2})^{1/2}.

From (26)

w_{1} = \frac{(1 + \overline{λ}μ)(−du_{1}) − (1 − \overline{λ}μ)i ∑_{2}^{n} d_{1j}u_{j}}{2α_{1}} = u_{1},

so w_{1} = w.

Although the phase w_{1} in the polar decomposition is unique, the positive Hermitian operator T with T w_{1} = a is not. Note that the matrix of any such T with respect to the TCAR basis u_{1}, …, u_{n} is, by (31),

[ \begin{matrix}α_{1} & id_{12} & id_{13} & ⋯ & id_{1n} \\ −id_{12} & α_{1} & id_{23} & ⋯ & id_{2n} \\ · & · & · & · & · \\ −id_{1n} & −id_{2n} & −id_{3n} & ⋯ & α_{1} \end{matrix} ]

whereas the matrix of α_{1}I + α_{2}D(w_{1}, w_{2}) is (30), where α_{2} = (∑_{2}^{n} d_{1j}^{2})^{1/2}.

For example, for n = 3, the eigenvalues of T are α_{1}, α_{1} ± \sqrt{α_{2}^{2} + d_{23}^{2}}. Thus T is positive if and only if α_{2}^{2} + d_{23}^{2} ≤ α_{1}^{2}. On the other hand, by Sinclair’s theorem quoted above, ∥T ∥ = ∥a∥ if and only if d_{23} = 0. Thus the operators

T_{θ} = [ \begin{matrix}α_{1} & iα_{2} cos θ & iα_{2} sin θ \\ −iα_{2} cos θ & α_{1} & 0 \\ −iα_{2} sin θ & 0 & α_{1} \end{matrix} ]

are precisely the ones that occur in the possible polar decompositions of the element a with facial decomposition (22). The canonical polar decomposition occurs for θ = 0.

We have already mentioned that every spin factor is a JC^{∗}-triple, that is, is isomorphic to a subtriple of B(H) for some Hilbert space H. In particular, the spin factors of dimensions 3,4, and 6 are isomorphic to the symmetric 2 by 2 complex matrices, the 2 by 2 complex matrices, and the antisymmetric 4 by 4 complex matrices, respectively, with the triple product (11). It is instructive to compare our constructions on C^{n}, that is, triple product, norm, inner product, and conjugations, as well as the facial and polar decompositions with their counterparts in these examples. For instance, in dimension 4, the isomorphism is

[ \begin{matrix}a_{1} & a_{2} \\ −ã_{2} & ã_{1} \end{matrix} ] ↦ \frac{1}{2}(a_{1} + ã_{1}, −i(a_{1} − ã_{1}), a_{2} + ã_{2}, −i(a_{2} − ã_{2})), (32)

the inner product corresponds to tr (AB^{∗})/2, and in the facial decomposition A = α_{1}W_{1} + α_{2}W_{2} of a matrix, U := W_{1}^{∗}W_{2} is a self adjoint unitary matrix with AA^{∗} = (α_{1}I + α_{2}U)^{2}.

In particular, S^{3} is isomorphic to the JB^{∗}-triple S_{2}(C) of all symmetric 2 by 2 complex matrices and S^{4} is isomorphic to the JB^{∗}-triple M_{2}(C) of all 2 by 2 complex matrices. In the latter case, a matrix a has a left and right polar decomposition a = w|a| = |a^{∗}|w when considered as an element in M_{2}(C). Any T occuring in the polar decomposition of a considered as an element of S^{4} has the form (by another Theorem of Sinclair) T = L_{b} + R_{c} (left and right multiplication) where b, c are self-adjoint matrices and it follows that any pair (b, c) with c = w^{∗}(|a^{∗}| − b)w gives rise to a Hermitian operator T with T (w) = a. On the other hand, a close look at the isomorphism (32) of S^{4} with M_{2}(C) shows that the canonical polar decomposition corresponds to the right polar decomposition a = |a^{∗}|w.

3.3. The geometric meaning of the norm

Denote the unit ball of the JB^{∗}-triple S^{n} with respect to the operator norm by BS^{n}, namely

BS^{n} = {a ∈ S^{n} : ∥a∥ < 1}.

Thus BS^{n} is a Cartan domain of type 4, called the the Lie ball and any isometry of S^{n} is a triple product automorphism.

From (28) it follows that the norm of any tripotent is one and from (22) that any element in the interior of the unit ball is a convex combination of zero and two complementary maximal tripotents. Thus, (22) could be considered as the analog of the Russo-Dye decomposition [19]. Moreover, an element of norm one lies on the line segment connecting w_{1} and w_{2} in the boundary of the unit ball. This is why we call (22) the facial decomposition of a, cf. [10, Section 1].

For a ∈ S^{n} let ℜa = (a+\overline{a})/2 and ℑa = (a−\overline{a})/2i and define a “1-norm” ∥a∥_{1} as follows:

∥a∥_{1} = ∥ℜa∥_{2} + ∥ℑa∥_{2}.

Since the 1-norm is the ℓ^{1} norm of two copies of IR^{n}, it defines a norm on S^{n} as a real vector space (but not as a complex one). The relation of this norm to the operator norm is given in the following proposition.

Proposition 3.1. For any a ∈ S^{n}

∥a∥ = min{∥λa∥_{1} : λ ∈ C, |λ| = 1}.

Proof. For some φ ∈ IR and orthogonal vectors x, y ∈ IR^{n}, ℜa = cos φx − sin φy and ℑa = sin φx + cos φy. Since ⟨x|y⟩ = 0,

∥λa∥_{1}^{2} = ∥x∥_{2}^{2} + ∥y∥_{2}^{2}
+ 2\sqrt{∥x∥_{2}^{2}∥y∥_{2}^{2} + sin^{2} 2φ(∥x∥_{2}^{4} + ∥y∥_{2}^{4})/4}
≥ (∥x∥_{2} + ∥y∥_{2})^{2} = ∥a∥^{2}.

Equality is attained for μ = (ard a)^{1/2} as in (25).

3.4. The trace norm unit ball

We may consider any element f ∈ S^{n} as a functional by

f(a) = ⟨a|f⟩ for any a ∈ S^{n},

with the norm ∥ · ∥_{∗}. called the trace norm, defined by

∥f∥_{∗} = sup{|f(a)| : a ∈ BS^{n}}.

Proposition 3.2. If the facial decomposition of f is f = α_{1}w_{1} + α_{2}w_{2} then

∥f∥_{∗} = α_{1}.

Proof. Let ϕ be the isomorphism of C onto S^{n} used in the proof of Lemma 3.2. Recall that ϕ takes the elements v_{j} + ṽ_{j}, i(v_{j} − ṽ_{j}) into the (natural) TCAR basis e_{1}, …, e_{n} of S^{n}. Let u_{1}, …, u_{n} be a TCAR basis extending w_{1}, iw_{2} and let ψ be the isomorphism of C onto S^{n} which maps v_{j} + ṽ_{j}, i(v_{j} − ṽ_{j}) onto u_{1}, …, u_{n}. We shall show that

ψ^{∗}(f) = \frac{α_{1} − α_{2}}{2} v_{1} + \frac{α_{1} + α_{2}}{2} ṽ_{1} (33)

and therefore by (14),

∥f∥_{∗} = ∥ψ^{∗}(f)∥C^{∗} = \frac{α_{1} − α_{2}}{2} + \frac{α_{1} + α_{2}}{2} = α_{1}.

To prove (33), write a = ∑ a_{j}v_{j} +∑ ã_{j}ṽ_{j} so that ψ(a) = a_{1}(u_{1}−iu_{2})/2+ ã_{1}(u_{1} + iu_{2})/2 + · · ·. Now f = α_{1}u_{1} − iα_{2}u_{2} and therefore

f(ψ(a)) = α_{1} \frac{a_{1} + ã_{1}}{2} − α_{2} \frac{a_{1} − ã_{1}}{2} . (34)

On the other hand, writing ψ^{∗}(f) = ∑ c_{j}v_{j} +∑ c̃_{j}ṽ_{j} for suitable c_{j}, c̃_{j} ∈ C, we have

ψ^{∗}(f)(a) = ∑ c_{j}a_{j} + ∑ c̃_{j}ã_{j}. (35)

Since (34) and (35) hold for all a,

c_{1} = \frac{α_{1} − α_{2}}{2} , c_{2} = c_{3} = ⋯ = 0

and

c̃_{1} = \frac{α_{1} + α_{2}}{2} , c̃_{2} = c̃_{3} = ⋯ = 0.

If we denote the unit ball in S^{n} with respect to the trace norm by S, it is obvious from the facial decomposition that the boundary of S is the union of the sets

\begin{matrix}G_{u} = \{u + αw : 0 ≤ α ≤ 1; \{www\} = w; ⟨u|w⟩ = 0; ard w = − ard u\}, \\ (36) \end{matrix}

where u varies over all maximal tripotents.

Proposition 3.3. The sets G_{u} are norm exposed faces of S, that is, G_{u} = F_{u} where

F_{u} = {a ∈ S^{n} : ⟨u|a⟩ = 1 = ∥a∥_{∗}},

and are therefore n − 1 dimensional unit balls. For n = 4 this implies that the boundary of S is the union of real 3-dimensional balls.

Proof. We prove that G_{u} = F_{u}. If a = u + αw ∈ G_{u} then clearly ⟨u|a⟩ = 1 = ∥a∥_{∗}, so that a ∈ F_{u}. Conversely, let a ∈ F_{u} have facial decomposition a = w_{1} +α_{2}w_{2}. Then a ∈ F_{w_{1}} and the intersection F_{u} ∩F_{w_{1}} is a face of rank one or two containing a. In the latter case, F_{u} = F_{w_{1}} and w_{1} = u so that a = w_{1} + α_{2}w_{2} = u + α_{2}w_{2} ∈ G_{u}. In the former case the intersection consists of a alone so that a is an extreme point of S and hence a minimal tripotent. Thus ⟨a|a⟩ = 1/2 and det a = 0. But det a = λ(1 − α_{2}^{2}) for some complex number λ of absolute value one, so α_{2} = 1. But then ⟨a|a⟩ = 2, a contradiction. So F_{u} ⊂ G_{u}, as stated.

4. SPIN 1 AND SPIN \frac{1}{2} REPRESENTATIONS OF THE LORENTZ GROUP

All of our representations will be defined on the Lie algebras so(3) and sl(2, C) so by the simple connectedness of SU(2) and SL(2, C) we can pass freely to representations of the groups. The corresponding representations of the groups SO(3) and the Lorentz group may therefore be double valued.

4.1. Lorentz group representations in dinv (S^{3}) Consider the rotation group SO(3) consisting of all 3 by 3 orthogonal

real matrices with determinant one. As was mentioned earlier, this group coincides with Taut (S^{3}) ⋂ Dinv (S^{3}). The Lie algebra of the Lie group SO(3), which is linearly spanned by the three 3 by 3 matrices J′_{1}, J′_{2}, J′_{3} obtained from (2) and (3) by omitting the first rows and columns, and satisfying the commutation relations (4) can be represented in taut (S^{3}) by the operators

σ(J′_{1}) = D(u_{2}, u_{3}), σ(J′_{2}) = D(u_{3}, u_{1}), σ(J′_{3}) = D(u_{1}, u_{2}),

that satisfy the same commutation relations, that is, σ extends to a Lie algebra homomorphism of so(3) into taut (S^{3}). Here of course, u_{1}, u_{2}, u_{3} is any fixed TCAR basis of S^{3}. Note that the corresponding representation at the group level is the identity map, if elements of taut (S^{3}) are replaced by their matrices with respect to u_{1}, u_{2}, u_{3}. For example the rotation matrix R_{1} = exp tσ(J′_{1}) generated by J′_{1} is the 3 by 3 matrix obtained from (1) by omitting the first row and column. Note that since \overline{R_{1}a} = R_{1}\overline{a}, R_{1} ∈ Dinv (S^{3}).

We shall now extend σ to a linear representation of the Lorentz group. Let J_{1}, J_{2}, J_{3}, K_{1}, K_{2}, K_{3} be the standard infinitesimal generators of the

Lorentz group, the J_{j} corresponding to the rotation group, and the K_{j} corresponding to the boosts (these are denoted by a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} in [17]).

The commutation relations between J_{1}, J_{2}, J_{3} are given by (4). The other commutation relations are (5)-(8).

It is trivial to check that if J_{1}, J_{2}, J_{3} are any three elements of a complex Lie algebra satisfying (4), and if we set K_{j} = iJ_{j} for j = 1, 2, 3, then (5)-(8) hold.

We use this fact to define a three dimensional linear representation σ̃ of the Lorentz Lie algebra into dinv (S^{3}), extending σ, via:

σ̃(K_{k}) = iσ(J_{k}),

for k = 1, 2, 3.

Proposition 4.1. σ̃ is an irreducible representation with range in dinv (S^{3}).

To see that the image of σ̃ lies in dinv (S^{3}), note that for example the one parameter group (boost) B_{1}(t) = exp tσ̃(K_{1}) is the same as the 3 by 3 matrix (10). Thus, although B_{1}(t) cannot be a unitary matrix, it does preserve the determinant of elements of S^{3}. This connection with (10) and the connection of (23) with (9) shows the relevance of our construction with the natural model for the electromagnetic field intensity.

4.2. Spin 1 representations of the Lorentz group in dinv (S^{4})

In the previous subsection, we constructed a three-dimensional linear representation σ of the rotation group, and extended it to the full Lorentz group in such a way that it acts by determinant preserving linear maps on S^{3}. We shall now construct a linear irreducible four-dimensional representation of the Lorentz group which also extends σ and show that it is equivalent to the standard representation of the Lorentz group on spacetime.

We shall again let J_{1}, J_{2}, J_{3}, K_{1}, K_{2}, K_{3} be the standard infinitesimal generators of the Lorentz Lie algebra. We also let u_{0}, u_{1}, u_{2}, u_{3} denote a TCAR basis for S^{4} and define a representation π from the Lorentz Lie algebra to operators on S^{4}. For notation’s sake, we shall use D_{jk} to denote D(u_{j}, u_{k}), for j, k ∈ {0, 1, 2, 3}, j ≠ k. Note that here for example D(u_{2}, u_{3}) is acting on the space spanned by u_{0}, u_{1}, u_{2}, u_{3} whereas in the previous section it was acting on the span of u_{1}, u_{2}, u_{3}. We define π by

π(J_{1}) = D_{23}, π(J_{2}) = D_{31}, π(J_{3}) = D_{12},
π(K_{1}) = iD_{01}, π(K_{2}) = iD_{02}, π(K_{3}) = iD_{03}.

Proposition 4.2. π is an irreducible representation with range in dinv (S^{4}).

We omit the proof as it is a straightforward calculation. We just note that for example if T = exp tπ(K_{1}), then with respect to the basis {u_{0}, u_{1}, u_{2}, u_{3}},

T = [ \begin{matrix}cosh t & i sinh t & 0 & 0 \\ −i sinh t & cosh t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ]

so that if a = ∑_{0}^{3} a_{j}u_{j}, then ⟨T a|\overline{T a}⟩ = ⟨a|\overline{a}⟩.

Note that π(J_{k}) and π(K_{k}) for k = 1, 2, 3 are elements of dinv (S^{4}) which is a subset of A_{4}(C) by Proposition 2.3. But A_{4}(C) is the complex span of the D_{jk} and as we have just seen D_{jk} and iD_{jk} belong to dinv (S^{4}). Therefore dinv (S^{4}) = A_{4}(C).

To understand this representation, we will consider its action on the positive cone Λ = ∪_{t>0}tF_{u_{0}} generated by the face F_{u_{0}}.

Proposition 4.3. Let Λ = ∪_{t>0}tF_{u_{0}}. Then

(a) Λ = {su_{0} + αw : s ∈ R^{+}, αw = xiu_{1} + yiu_{2} + ziu_{3}, x, y, z ∈ R, x^{2} + y^{2} + z^{2} < s^{2}, 0 ≤ α ≤ 1}.

(b) π is an extension of the space-time representation of the Lorentz group on the four-vectors.

Proof. From Proposition 3.3 we may use the description of F_{u_{0}} given in (36). If u_{0} +αw ∈ F_{u_{0}}, then w = ∑_{1}^{3} a_{j}u_{j} where a_{j} = x_{j} +iy_{j} ∈ C satisfies ∑_{1}^{3} |a_{j}|^{2} = ⟨w|w⟩ = 1. Now ard w = −ard u_{0} implies det w = − det u_{0} and so λ ∑_{1}^{3} a_{j}^{2} = −λ for a suitable λ of absolute value 1. Thus ∑_{1}^{3} a_{j}^{2} = −1 so that a_{j} = iy_{j} is purely imaginary with ∑_{1}^{3} y_{j}^{2} = 1. Since

Λ = {tu_{0} + tα ∑_{1}^{3} iy_{j}u_{j} : y_{j} ∈ IR, ∑_{1}^{3} y_{j}^{2} = 1, t > 0},

(a) follows.

By taking s = ct, where c is the speed of light in vacuum we can identify the cone Λ with the “light cone”. A direct calculation shows that Λ is invariant under the representation π. For example with T = exp tπ(K_{1})

T (s, ix, iy, iz) = (cosh t s + sinh t x, −i sinh t s + i cosh t x, iy, iz).

Since this representation preserves the determinant of any element in S^{4} and since for the elements of the cone Λ, the determinant coincides with the space-time interval, (b) follows.

4.3. Spin \frac{1}{2} representations in dinv (S^{4})

The elements D_{jk} from dinv (S^{4}), defined in the previous subsection, form a spin grid (as described in subsection 1.2) generating the spin factor of dimension 6. The sharp operation on the grid (see Example 1.2) of this factor is thus given by

D_{01}^{♯} = D_{23}, D_{02}^{♯} = D_{31}, D_{03}^{♯} = D_{12}, D_{23}^{♯} = D_{01}, D_{31}^{♯} = D_{02}, D_{12}^{♯} = D_{03}

and is extended to be conjugate linear on the whole factor.

For the representation π, constructed in the previous subsection, π(J_{k}) and π(K_{k}) are minimal tripotents of dinv (S^{4}). We now use the conjugation ♯ to construct, from π, two representations π^{+} and π^{−} of the Lorentz group using maximal tripotents. The representation π^{+} is going to be the selfadjoint part of π with respect to ♯ on J_{1}, J_{2}, J_{3} and the skew-adjoint part of π with respect to ♯ on K_{1}, K_{2}, K_{3}. That is,

π^{+}(J_{k}) = \frac{1}{2}(π(J_{k}) + π(J_{k})^{♯}), π^{+}(K_{k}) = \frac{1}{2}(π(K_{k}) − π(K_{k})^{♯}).

Note that π^{+}(K_{k}) = iπ^{+}(J_{k}) for k = 1, 2, 3.

For example, the matrices of π^{+}(J_{1}), π^{+}(J_{2}) and π^{+}(J_{3}) in the basis u_{0}, u_{1}, u_{2}, u_{3} are extensions of the anti-symmetric matrices J′_{1}, J′_{2}, J′_{3} (see subsection 4.1) to a multiple of 4 by 4 orthogonal matrices as follows:

π^{+}(J_{1}) = \frac{1}{2} [ \begin{matrix}0 & 1 0 0 \\ \begin{matrix}−1 \\ 0 \\ 0 \end{matrix} & J′_{1} \end{matrix} ] = \frac{1}{2} [ \begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ];

π^{+}(J_{2}) = \frac{1}{2} [ \begin{matrix}0 & 0 1 0 \\ \begin{matrix}0 \\ −1 \\ 0 \end{matrix} & J′_{2} \end{matrix} ] = \frac{1}{2} [ \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \\ −1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ];

π^{+}(J_{3}) = \frac{1}{2} [ \begin{matrix}0 & 0 1 0 \\ \begin{matrix}0 \\ −1 \\ 0 \end{matrix} & J′_{3} \end{matrix} ] = \frac{1}{2} [ \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ] .

Note that the constant \frac{1}{2} is necessary in order that π^{+} will be a representation of so(3). Since π^{+}(K_{k}) = iπ^{+}(J_{k}), π^{+} is a representation of sl(2, C).

Notice also that π^{+}(J_{1}), π^{+}(J_{2}), π^{+}(J_{3}) each have 2 distinct eigenvalues, and using the fact that this number is 2j + 1 where j is called the spin value, for our representation we have j = 1/2, implying that this is a spin \frac{1}{2} representation. This is confirmed in the proposition that follows, whose proof is an elementary calculation.

Proposition 4.4. π^{+} and π^{−} (π^{−} is defined below) are representations of sl(2, C). The representations π^{+} and π^{−} of so(3) into dinv (S^{4}) induce a spin \frac{1}{2} representations of SO(3) in Dinv (S^{4}). For example, the matrix of the operator R_{3}(t) = exp tπ^{+}(J_{3}) with respect to the TCAR basis {u_{0}, u_{1}, u_{2}, u_{3}} is

R_{3}(t) = exp tπ^{+}(J_{3}) = [ \begin{matrix}cos \frac{t}{2} & 0 & 0 & sin \frac{t}{2} \\ 0 & cos \frac{t}{2} & sin \frac{t}{2} & 0 \\ 0 & − sin \frac{t}{2} & cos \frac{t}{2} & 0 \\ − sin \frac{t}{2} & 0 & 0 & cos \frac{t}{2} \end{matrix} ] .

Similarly for R_{1}(t) = exp tπ^{+}(J_{1}) and R_{2}(t) = exp tπ^{+}(J_{2}).

Let us now define the representation π^{−}, which is going to be the skewadjoint part of π with respect to ♯ on J_{1}, J_{2}, J_{3} and the self-adjoint part of π with respect to ♯ on K_{1}, K_{2}, K_{3}. Thus,

π^{−}(J_{k}) = \frac{1}{2}(π(J_{k}) − π(J_{k})^{♯}), π^{−}(K_{k}) = \frac{1}{2}(π(K_{k}) + π(K_{k})^{♯})

and again π^{−}(K_{k}) = iπ^{−}(J_{k}) for any k = 1, 2, 3.

The matrices of π^{−}(J_{1}), π^{−}(J_{2}) and π^{−}(J_{3}) in the basis u_{0}, u_{1}, u_{2}, u_{3} are extensions of the anti-symmetric matrices J′_{1}, J′_{2}, J′_{3} to a multiple of 4 by 4 orthogonal matrices as follows:

π^{−}(J_{1}) = \frac{1}{2} [ \begin{matrix}0 & −1 0 0 \\ 1 0 0 & J′_{1} \end{matrix} ] = \frac{1}{2} [ \begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ];

π^{−}(J_{2}) = \frac{1}{2} [ \begin{matrix}0 & 0 −1 0 \\ 0 1 0 & J′_{2} \end{matrix} ] = \frac{1}{2} [ \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ];

π^{−}(J_{3}) = \frac{1}{2} [ \begin{matrix}0 & 0 0 −1 \\ 0 1 0 & J′_{3} \end{matrix} ] = \frac{1}{2} [ \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ] .

Alternatively, the above construction can be done by using the Hodge operator, also called the star operator. Borrowing the definition from the theory of differential forms, we set

∗D_{jk} = ϵ_{jklm}g^{jj}g^{kk}D_{lm},

where {j, k, l, m} = {0, 1, 2, 3}, g^{pq} is the Lorentz metric: g^{00} = 1, g^{kk} = −1 for k = 1, 2, 3, and g^{pq} = 0 if p ≠ q, and ϵ_{jklm} is the signature of the permutation (j, k, l, m) ↦ (0, 1, 2, 3). Specifically,

∗D_{01} = D_{23}, ∗D_{02} = D_{31}, ∗D_{03} = D_{12},
∗D_{23} = −D_{01}, ∗D_{31} = −D_{02}, ∗D_{12} = −D_{03}.

It is easy to check that the representation π^{+} is the skew-adjoint part of π with respect to the Hodge operator, and the representation π^{−} is the self-adjoint part of π with respect to the Hodge operator.

Next, we consider invariant subspaces of π^{+} and π^{−}.

Proposition 4.5. The representation π^{+} is a direct sum of two copies of the standard spin \frac{1}{2} representation in terms of the Pauli spin matrices σ_{k}, k = 1, 2, 3. (σ_{k} is defined below)

Proof. Since π^{+}(K_{k}) = iπ^{+}(J_{k}), it suffices to consider the restriction of π^{+} to so(3). A simple calculation shows that a complex matrix which commutes with all π^{+}(J_{k}) has the form

[ \begin{matrix}a & b & c & d \\ −b & a & −d & c \\ −c & d & a & −b \\ −d & −c & b & a \end{matrix} ] = aI_{4} + 2bπ^{−}(J_{1}) + 2cπ^{−}(J_{2}) + 2dπ^{−}(J_{3}).

The eigenspaces of π^{−}(J_{2}) and hence two invariant subspaces of the representation π^{+} are

M = sp_{C} {u_{0} + iu_{2}, u_{1} + iu_{3}}

and its orthogonal complement

M^{⊥} = sp_{C} {u_{0} − iu_{2}, u_{1} − iu_{3}}.

By properties of the TCAR basis, the matrix of π^{+}(J_{1})|_{M} with respect to the basis {u_{0} + iu_{2}, u_{1} + iu_{3}} is the Pauli spin matrix

σ_{2} = \frac{1}{2} [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] . (37)

Similarly, the matrices of π^{+}(J_{2})|_{M} and π^{+}(J_{3})|_{M} with respect to the basis {u_{0} + iu_{2}, u_{1} + iu_{3}} are the Pauli spin matrices

σ_{3} = \frac{1}{2} [ \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ] , (38)

and

σ_{1} = \frac{1}{2} [ \begin{matrix}0 & i \\ i & 0 \end{matrix} ] . (39)

Moreover, the matrices of π^{+}(J_{k})|_{M^{⊥}} with respect to the basis {u_{0} − iu_{2}, u_{1} − iu_{3}} turn out to be identical to (37)-(39).

Hence, for k = 1, 2, 3,

π^{+}(J_{k}) = \frac{1}{2} [ \begin{matrix}σ_{k+1} & 0 \\ 0 & σ_{k+1} \end{matrix} ] ,

where of course by σ_{4} we mean σ_{1}.

One can obtain other invariant subspaces by considering the eigenspaces of π^{−}(J_{1}) and π^{−}(J_{3}). The restriction of π^{+} to these subspaces also leads to the Pauli spin matrices, but possibly in a different order and with possible sign changes. By rotating the basis in these subspaces one can obtain the Pauli matrices in the standard order and with the appropriate sign.

For comparison and the sake of completeness, we also analyze the representation π^{−}. The situation is entirely symmetric in the following sense. As remarked above, the commutant of {π^{+}(J_{k}) : k = 1, 2, 3} is sp_{C}[{π^{−}(J_{k}) : k = 1, 2, 3} ∪ {I}], which, when restricted to real scalars, is a four dimensional associative algebra isomorphic to the quaternions. The commutant of {π^{−}(J_{k}) : k = 1, 2, 3} is therefore sp_{C}[{π^{+}(J_{k}) : k = 1, 2, 3} ∪ {I}], which is also (after restriction to real scalars) isomorphic to the quaternions. Thus we can say that the Lie algebra taut (S^{4}) consists of two copies of the quaternions, “glued together” at (i times) the identity.

The representation π^{−} exhibits all of the characteristics of the representation π^{+}; it is a direct sum of two copies of the spin-\frac{1}{2} two-dimensional representation given by the Pauli spin matrices. Once again, these matrices occur not always in the natural order and with some change of sign, both of which can be corrected by a change of basis. We omit the details of the calculation.

4.4. Action of the spin \frac{1}{2} representation in dinv (S^{4})

Let us now lift the representations π^{+} and π^{−} from acting on S^{4} to an action on dinv (S^{4}) = S^{6} that correspond to the momentum and angular momentum operators on the states represented by S^{4}.

We recall first that, from Example 1.2, a spin grid for dinv (S^{4}) = S^{6} is given by

v_{1} = D_{01}, ṽ_{1} = D_{23}; v_{2} = D_{02}, ṽ_{2} = D_{31}; v_{3} = D_{03}, ṽ_{3} = D_{12}.

So it follows from (16) that a TCAR basis for the spin factor S^{6} is given by

D_{01} + D_{23}, D_{02} + D_{31}, D_{03} + D_{12},
i(D_{01} − D_{23}), i(D_{02} − D_{31}), i(D_{03} − D_{12}).

Note that this has the same complex span as

{π^{+}(J_{k}), π^{−}(K_{k}) : k = 1, 2, 3}.

We shall choose

π^{+}(J_{1}), π^{+}(J_{2}), π^{+}(J_{3}), π^{−}(K_{1}), π^{−}(K_{2}), π^{−}(K_{3}) (40)

as a basis for dinv (S^{4}). For any element T in the group Dinv (S^{4}), define an operator T̃ on Dinv (S^{4}) by T̃A = T AT^{−1} for any A ∈ Dinv (S^{4}). By taking T = R_{k}(t) = exp tπ^{+}(J_{k}) or T = B_{k}(t) = exp tπ^{+}(K_{k}) we obtain the Jordan theoretic connection between the two types of fundamental particles in nature.

Theorem 4.1. With respect to the basis (40) of dinv (S^{4}),

R̃_{1}(t) = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & cos t & sin t & 0 \\ 0 & − sin t & cos t & 0 \\ 0 & 0 & 0 & I_{3} \end{matrix} ] , B̃_{1}(t) = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & cosh t & i sinh t & 0 \\ 0 & −i sinh t & cosh t & 0 \\ 0 & 0 & 0 & I_{3} \end{matrix} ] ,

R̃_{2}(t) = [ \begin{matrix}cos t & 0 & − sin t & 0 \\ 0 & 1 & 0 & 0 \\ sin t & 0 & cos t & 0 \\ 0 & 0 & 0 & I_{3} \end{matrix} ] , B̃_{2}(t) = [ \begin{matrix}cosh t & 0 & −i sinh t & 0 \\ 0 & 1 & 0 & 0 \\ i sinh t & 0 & cosh t & 0 \\ 0 & 0 & 0 & I_{3} \end{matrix} ] ,

R̃_{3}(t) = [ \begin{matrix}cos t & sin t & 0 & 0 \\ − sin t & cos t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & I_{3} \end{matrix} ] , B̃_{3}(t) = [ \begin{matrix}cosh t & i sinh t & 0 & 0 \\ −i sinh t & cosh t & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & I_{3} \end{matrix} ] .

Thus our spin \frac{1}{2} representation π^{+} of SO(3) and the Lorentz group on S^{4} induces a spin 1 representation of SO(3) and the Lorentz group on Dinv (S^{4}).

4.5. Concluding Remarks

We have seen that the action of π^{+} and π^{−} on S^{4} is similar to the action of the Lorentz group on the Dirac bispinors. So it may be that S^{4} is equivalent or is a replacement for the bispinors representing the relativistic state of an electron. Moreover, using the polar decomposition from subsection 3.2 of any element of S^{4}, this state could be decomposed into a maximal tripotent and a positive operator acting on it. The maximal tripotent could be identified with the spin of the particle, while the positive operator can be identified with the representation of the energy and momentum of the particle. Our representations act properly on each part of this decomposition, mainly as a spin \frac{1}{2} on the spin part and as a spin 1 action on the four-vector representing energy-momentum.

Another interesting observation is that adding an additional symmetry with respect to the Hodge operator (or the sharp operation) to the representation π that is an extension to S^{4} of the regular Lorentz group representation on four-vectors leads to the representation on the bispinors. It will be interesting to find the physical meaning of this observation.

Since S^{3} is imbedded into S^{4} and the representations π^{+} and π^{−} on S^{4} preserve the determinant, from results of subsection 4.1, these representations act properly on the intensities of the electromagnetic field.

ACKNOWLEDGMENT

The second author wishes to thank the Jerusalem College of Technology for its hospitality during his stays there while this paper was being developed. He also thanks his own institution, the University of California,Irvine for the sabbatical leave which made these trips possible.

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