A Remarkable New Identity Satisfied by the Dirac Matrices of a Bilocal Field Theory
aa r X i v : . [ g r- q c ] M a y A Remarkable New Identity Satisfied by the Dirac Matrices of aBilocal Field Theory
Patrick L. Nash ∗ Department of Physics and Astronomy, RetiredThe University of Texas at San AntonioSan Antonio, Texas 78249-0697 (Dated: October 29, 2018)
Abstract
In 1925 Elie Cartan described ‘triality’ [4], [5] as a symmetry between SO(8; C ) vectors and thetwo types of Spin(8; C ) spinor. It is known that the reduced generators of the Clifford algebra C defined on the real, eight-dimensional Euclidean space E satisfy an identity that guarantees theexistence of matrix representations (acting on the vector and spinor bundles of E ) of triality.Analogously, let E , denote a real eight-dimensional pseudo-Euclidean vector space that is en-dowed with an indefinite inner product with signature (+ , + , + , − ; − , − , − , +). As a normed vectorspace, E , ∼ = M , × ∗ M , , where M , and ∗ M , denote real four-dimensional Minkowski space-times, with opposite signatures. The reduced generators (i.e., the Dirac matrices) of the pseudoClifford algebra C , defined on E , satisfy an identity [10] , [11] that guarantees the existence ofinvertible linear mappings between each of the two types of S , R ) spinor and the S , R )vector, thereby realizing matrix representations of triality that act on the vector and spinor bundlesof the spacetime E , .In this note we generalize this identity (see Eq.[13]). PACS numbers: 02.10.YnKeywords: ∗ Electronic address: [email protected] . INTRODUCTION AND NOTATION In 1925 Elie Cartan described ‘triality’ [4], [5] as a symmetry between three types ofgeometrical objects that may be defined on real, eight-dimensional R and transform linearlyunder either SO(8; C ) or Spin(8; C ), namely a symmetry between SO(8; C ) vectors and thetwo types of Spin(8; C ) spinor (semi-spinors of the first type and semi-spinors of the secondtype, in the terminology of Cartan).Analogously, let E , denote a real eight-dimensional pseudo-Euclidean vector space thatis endowed with an indefinite inner product with signature (+ , + , + , − ; − , − , − , +) (seeGray [6]). As a normed vector space, E , ∼ = M , × ∗ M , , where M , denotes a realfour-dimensional Minkowski spacetime manifold that is endowed with the pseudo-Euclideanmetric η , = diag(1, 1, 1, -1), and ∗ M , denotes a real four-dimensional Minkowski spacetimethat is endowed with the pseudo-Euclidean metricdiag(-1, -1, -1, 1) = − η , . M , × ∗ M , may be regarded as a classical phase space of asingle relativistic point particle, or a spacetime that carries a bilocal Minkowski field theory(appropriate restrictions on the automorphism groups of E , ∼ = M , × ∗ M , are implied).The reduced generators (i.e., the Dirac matrices) of the pseudo Clifford algebra C , defined on E , satisfy an identity [10] , [11] that guarantees the existence of invertiblelinear mappings between each of the two types of S , R ) spinor and the S , R )vector, thereby realizing matrix representations of triality that act on the vector and spinorbundles of the spacetime E , . In this note we generalize this remarkable identity Eq.[11] toEq.[13]. Simple applications of this formalism are given in Sections [IV] and [V]. E , is an orientable differentiable manifold that, of course, admits a global, right-handedCartesian atlas (as well as many other “curvilinear” and general coordinate systems). Let x ∈ E , and let the 8 scalars x A ∈ R , A, B, ... = 1 , , ...,
8, denote the Cartesian coordinatesof x with respect to a global, right-handed Cartesian atlas. Let T x ( E , ) denote the tangentspace at x . T x ( E , ) is isomorphic to E , . The right-handed frame (cid:8) ∂∂ x A : A = 1 , . . . , (cid:9) that is adapted to these coordinates is orthogonal and pseudo-normal with respect to themetric defined below, and comprises a basis of T x ( E , ). This coordinate system and frameare simply called a “canonical frame”. A vector field V at x, V x = V A ( x ) ∂∂ x A ∈ T x ( E , ),has contravariant components V A ( x ) with respect to a canonical frame. Here the A, B, ...= 1, ... , 8 are to regarded as T x ( E , ) vector indices, and not as indices that enumerate the2 ❡ (cid:0)(cid:0) ❡ ❅❅ ❡ FIG. 1: Dynkin diagram for D scalars x A ; the interpretation of an index should always be clear from context. II. DIRAC MATRICES ON E , A. Representations of SO (8; C ) There is a well known relationship between Clifford algebras C n and the spinor represen-tations of the classical complex orthogonal groups; see, for example, Boerner, The Represen-tations of Groups [2]. In particular, the Clifford algebra C may be defined as the algebragenerated by a set of eight elements e j , j, k = 1 , . . . , , that anticommute with each other andhave unit square e j e k + e k e j = 2 δ j k I × , where I × = 16 ×
16 unit matrix. The scaledcommutators ( e j e k − e k e j ) computed from an irreducible 16-dimensional representationof the e j are the infinitesimal generators of a reducible 16-dimensional representation ofSpin(8; C ), which is the universal double covering of the special orthogonal group SO(8; C ).This 16-dimensional representation of is fully reducible to the direct sum of two inequivalentirreducible 8 × C ) [4], [5],[16], [7], [1]. The fundamental irreducible vector representation of SO(8; C ) is also 8 × D ∼ = SO(8; C ) is symmetrical and pictured in Figure 1. The cen-tral node corresponds to the adjoint representation. The three outer nodes correspond tothe vector representation (left-most node), type 1 spinor and type 2 spinor representationsof Spin(8; C ). The “left-handed” and “right-handed” Spin(8; C ) spinors have S , R )counter parts that are denoted ψ (1) and ψ (2) in this paper, and transform, respectively, un-der two inequivalent real 8 × S , R ) that we havecalled D (1) (type 1) and D (2) (type 2). SO (4 , R ) is a real form of the classical complex orthogonal group SO(8 , C ). O (4 , R )(respectively, SO (4 , R )) may be defined as the group of all real matrices (respectively, withunit determinant) that preserve the norm squared of V x ∈ T x ( E , ), which is the quadratic3orm ( V ) + ( V ) + ( V ) + ( V ) − (cid:2) ( V ) + ( V ) + ( V ) + ( V ) (cid:3) .O (4 , R ) is a pseudo-orthogonal Lie group that possess two connected components [2],[8],with SO (4 , R ) being the identity component (the connected component containing theidentity matrix). Spin(4 , , R ), alternatively denoted SO (4 , R ), is the 2-to-1 coveringgroup of SO (4 , R ). E , may be endowed with both SO (4 , R )-invariant and SO (4 , R )-invariant pseudo-Euclidean metrics that may each be represented in terms of an 8 × E , frame the SO (4 , R )-invariant pseudo-Euclidean metric tensor G (respectively, inverse G − ) has components G A B (respectively, ( G − ) A B = G B A = G A B )that are given by G A B = G A B = η , − η , (1)The indefinite inner product is realized as T x ( E , ) × T x ( E , ) ∋ ( V x , V ′ x ) < V x , V ′ x > = G A B V Ax V ′ Bx ∈ R . B. Spinor representations of S , R ) There exist two inequivalent real S , R ) basic 8-component spinor representations of S , R ). They are defined in Eqs.[47] and simply denoted as D (1) (type 1) and D (2) (type2). The S , R ) invariant metric, denoted σ , is invariant under the action of both D (1) and D (2) . Let S ( j ) x ( E , ), j = 1 ,
2, denote the two distinct basic real 8-component spinorvector spaces at x , endowed with respective automorphism groups D ( j ) . As vector spaceseach is isomorphic to E , . (Thus, as vector spaces, both of the S ( j ) x ( E , ) and T x ( E , )are each isomorphic to E , but with different automorphism groups.) A spinor element ψ ( j ) ∈ S ( j ) x ( E , ) has components ψ a ( j ) ∈ R . In this note, for simplicity, we do not distinguishthe spinor index on ψ (1) from that on ψ (2) [using a convention such as ψ (1) a and ψ (2) ˙ a forspinor components, for example].The disjoint union of tangent spaces T x ( E , ) at all points x ∈ E , gives the SO (4 , R )tangent bundle T ( E , ) over E , . In this case, it is a trivial bundle E , × T x ( E , ) π → E , ,4ith the natural projection π of the first factor in the Cartesian product. Clearly there alsoexist two distinct trivial 16-dimensional real basic 8-component spinor bundles S (1) ( E , ) and S (2) ( E , ), each with base space E , but with fibers S (1) x ( E , ) and S (2) x ( E , ), respectively.For each of the three bundles we denote the natural projection of the first factor in theCartesian product by π . E , may be endowed with a S , R ) invariant metric σ [10] , [11] that we representas σ = (2)where 0 denotes the 4 x 4 zero matrix and 1 denotes the 4 x 4 unit matrix. The matrixelements of σ are denoted σ ab = σ ba , where a, b, . . . = 1 , . . . , S , R ) spinor indices(elaborated in Eqs.[47] through [53] below). Note that σ is equal to the unit matrix, sothat the eigenvalues of σ of are ±
1. Since the trace of σ is zero, these eigenvalues occur withequal multiplicity.The S , R ) invariant (pseudo) norm-squared k ψ ( j ) k of basic real 8-component spinors ψ ( j ) ∈ S ( j ) x ( E , ) is the S , R )-invariant quadratic form ψ a ( j ) σ ab ψ b ( j ) . We define anoriented spinor basis e a of S (1) x ( E , ) normalized according to < e a , e b > = σ ab (3)(the oriented spinor basis of S (2) ( E , ) also satisfies Eq.[3]), so that < ψ (1) , ψ (1) > = < ψ a (1) e a , ψ b (1) e b > = ψ a (1) σ ab ψ b (1) = e ψ (1) σ ψ (1) , where the tilde denotestranspose. For brevity we employ the shorthand u ∈ S (1) ( E , ) and u a ∈ S (1) ( E , ) to denote e a u a ∈ S (1) ( E , ), with similar conventions implied for T ( E , ) and S (2) ( E , ).We also define an oriented vector basis ǫ A of T x ( E , ) normalized according to < ǫ A , ǫ B > = G A B . (4)These two sets of basis vectors are related by Eq.[34] below.The basic spinor representation of the pseudo-orthogonal group SO (4 , R ) may be con-structed from the irreducible generators t A , A = 1, ... , 8 , of the pseudo-Clifford algebra C , [2] , [3] , [9]. Following Brauer and Weyl we call such irreducible C n − generators“reduced Brauer-Weyl generators” [3]. We begin the construction of a representation of5 , R ) by defining eight real 8 × τ A , τ A , A,B,... = 1,...,8 , of the pseudo-Clifford algebra C , that anticommute and have square ± τ A matrices play the role of the Dirac Matrices on E , .) We realize this by requiringthat the tau matrices satisfy (the tilde denotes transpose) στ A = g στ A = e τ A σ (5)and τ A τ B + τ B τ A = 2 I × G A B = τ A τ B + τ B τ A , (6)where I × denotes the 8 × τ A by τ A a b , wemay write Eq.[5] as τ Aab = τ Aba , (7)where we have used σ to lower the spinor indices. In general, σ (respectively, σ − ) willbe employed to lower (respectively, raise) lower case Latin indices (i.e. a S , R ) spinorindex of either type).The following identity is occasionally useful. Let ψ ∈ S (1) ( E , ) be an arbitrary real eightcomponent type-1 spinor field (a section of the type-1 spinor bundle S (1) ( E , )). Consider e ψ σ τ A τ B ψ = (cid:16) e ψ σ τ A τ B ψ (cid:17) T = (cid:16) e ψ σ τ B τ A ψ (cid:17) , by Eq.[5]= 12 e ψ σ (cid:0) τ A τ B + τ B τ A (cid:1) ψ = δ AB e ψ σ ψ , using Eq.[6] . (8)We adopt a real irreducible 8 × X -axis, in which τ = I × = τ . Then, by Eq.[6], τ A = − τ A for A = 1, ... ,7.Hence, again by Eq.[6], ( τ A ) is equal to − I × for A = 1,2,3 and is equal to + I × for A =4,5,6, 7, 8. The Appendix displays one possible representation. III. THE NEW IDENTITY
The tau matrices verify an important identity [10] , [11] that encodes triality: Let Mbe any 8 × g σ M = σ M (9)6i.e., σM is a symmetric matrix) and moreover transforming under S , R ) according to M D (1) M D (1) − (10)(see Eq.[41], below). Then [10] , [11] τ A M τ A = I × tr ( M ) , (11)where, as above, I × denotes the 8 × × σM is proportional to the unit matrix, and there are 36 linearly independent real 8 × M such that σM is a symmetric matrix (these are given below in Eq.[14]).This is a special case of another simple, but also remarkable, general identity that werecord as Theorem III.1
Let M be an arbitrary × matrix that transforms under S , R ) ac-cording to M D (1) M D (1) − . M has matrix elements M ab . Note that M − σ − ^ ( σ M ) istwice σ − times the anti-symmetric part of σM . The generalization of Eq.[11] is τ ( µ ) M τ ( µ ) = − τ ( µ ) tr (cid:0) τ ( µ ) M (cid:1) + 2 (cid:16) I × tr ( M ) + M − σ − ^ ( σ M ) (cid:17) (12) or (cid:0) τ ( µ ) (cid:1) ab (cid:0) τ ( µ ) (cid:1) cd = − (cid:0) τ ( µ ) (cid:1) ad (cid:0) τ ( µ ) (cid:1) cb + 2 ( δ ab δ cd + δ ad δ cb − σ ac σ bd ) (13)The Proof of Theorem[III.1] is straightforward. Firstly, if σM is symmetric then Eq.[12]devolves to Eq.[11]. What if σM has no symmetry? Eqs.[12,13] are linear in M . Expand M in terms of a linear combination of the 64 basis 8 × M s ∈ S × such that σM s is symmetric, plus the 28 = 7 + 21 basismatrices M a ∈ A × such that σM a is anti-symmetric, and verify the theorem componentby component. The set of 35 + 1 matrices S × is given by S × = n τ ( A ) τ ( B ) τ ( C ) (cid:3) { A,B,C }∈{ ,..., } && A>B>C , I × o , (14)and each element of this set clearly verifies Theorem[III.1].The 7 + 21 matrices M a ∈ A × such that σ M a is anti-symmetric are given by A × = n τ ( A ) (cid:3) A ∈{ ,..., } , τ ( A ) τ ( B ) (cid:3) { A,B }∈{ ,..., } && A>B o . (15)7ach of M a ∈ n τ ( A ) (cid:3) A ∈{ ,..., } o satisfies τ ( µ ) M a τ ( µ ) = − M a as well as τ ( µ ) tr (cid:0) τ ( µ ) M a (cid:1) =+8 M a . Each of M a ∈ n τ ( A ) τ ( B ) (cid:3) { A,B }∈{ ,..., } && A>B o satisfies τ ( µ ) M a τ ( µ ) = +4 M a aswell as τ ( µ ) tr (cid:0) τ ( µ ) M a (cid:1) = 0. Therefore each element of A × satisfies Eqs.[12,13] and theTheorem[III.1] is proven. (cid:4) IV. BILOCAL TETRAD
Let u = u ( x α ) ∈ S (1) ( E , ) be a real eight component type-1 spinor field (a section of thetype-1 spinor bundle S (1) ( E , )). u is called the “unit field” for reasons that are explainedin Section [VI]. In a quantum theory the u a satisfy commutation relations rather than anti-commutation relations because of triality. We assume that < u , u > = e u σ u > E , = π (cid:0) S (1) ( E , ) (cid:1) .For brevity a vielbein set of 8 independent vector fields is simply referred to as a tetrad(vierbein). In this Section and the next we replace the indices A, B, . . . = 1 , . . . , µ ) , ( ν ) , . . . , where µ, ν, . . . = 1 , . . . ,
8, in order to display this information is a moreconventional form. Summarizing, α, β, . . . , µ, ν, . . . , a, b, . . . = 1 , . . . ,
8. We also employ α , β , . . . , µ , ν , . . . = 1 , . . . , ψ ∈ S (2) ( E , ) denote a real eight component type-2 spinor field that realizes thebilocal Cartesian coordinates x ≡ ( x α , x α ) of π (cid:0) S (2) ( E , ) (cid:1) ∼ = M , × ∗ M , . The Carte-sian coordinates x α = { x } α are assumed to be C ∞ functions of ψ , x α = x α ( ψ ) such that det ( ∂x α ∂ψ a ) = 0, so that the inverse ψ a = ψ a ( x α ) always exists. We abuse notation and write u = u ( x α ) = u ( x α ( ψ )) = u ( ψ a ). The mass dimension of ψ , [ ψ ], is -1: [ ψ ] = LENGTH =1/MASS = [ G / ] = [Planck length].We define a spacetime tetrad E ( µ ) with components E ( µ ) α as E ( µ ) α = 1 √ e u σ u e u σ τ ( µ ) ∂∂x α ψ. (16) Remark : Let f : E , → R and ∂ f∂x α = f, α . Let r : E , → R + . The tetrad E ( µ ) may bemade to transform covariantly under the local projective transformation u u ′ = r ( x α ) u ψ ψ ′ = r ( x α ) ψ (17)8y replacing the gradient operator ∂∂x α with D α = I × ∂∂x α − e u σ u τ ( µ ) ∂ u ∂x α ⊗ e u σ τ ( µ ) (18)because D ′ α ψ ′ = ( r ψ, α + r, α ψ ) − r e u σ u τ ( µ ) ( r u , α + r, α u ) ⊗ r e u σ τ ( µ ) ( r ψ ) = r D α ψ ,since r e u σ u τ ( µ ) ( r, α u ) ⊗ r e u σ τ ( µ ) ( r ψ ) = r, α ψ (cid:0) e u σ u τ ( µ ) u ⊗ e u σ τ ( µ ) (cid:1) = r, α ψ , usingEq.[11] or Eq.[12].Therefore, if E ( µ ) α = 1 √ e u σ u e u σ τ ( µ ) D α ψ. (19)then E ( µ ) α E ′ ( µ ) α = r ( x ) E ( µ ) α (20)under the local projective transformation Eq.[17].This local projective transformation generates a conformal transformation of the metrictensor. (cid:4) Lemma IV.1
The inverse of the tetrad has components E α ( µ ) E α ( µ ) = 1 √ e u σ u ∂x α ∂ψ τ ( µ ) u . (21)Proof : E α ( µ ) E ( µ ) β = (cid:18) √ e u σ u ∂x α ∂ψ τ ( µ ) u (cid:19) (cid:18) √ e u σ u e u σ τ ( µ ) ∂∂x β ψ (cid:19) = 1 e u σ u ∂x α ∂ψ (cid:0) τ ( µ ) u e u σ τ ( µ ) (cid:1) ∂ ψ∂x β = ∂x α ∂ψ ∂ ψ∂x β by Eq.[11] or Eq.[12] = ∂ x α ∂x β = δ αβ (22)QED (cid:4) Since a matrix commutes with its inverse we also have E ( µ ) α E α ( ν ) = δ ( µ )( ν ) . (23)9et’s look at an example. We make the self-consistent assumption that there existsa constant spacetime tetrad (0) E ( µ ) with constant components (0) E ( µ ) α , which might verify (0) E ( µ ) α = δ ( µ ) α , for example. Pick a constant unit field u that satisfies e u σ u >
0, define ψ = 1 √ e u σ u τ ( ν ) u (0) E ( ν ) β x β (24)(compare with the twistor [14] type) and compute (0) E ( µ ) α = 1 √ e u σ u e u σ τ ( µ ) ∂∂x α ψ = 1 e u σ u e u σ τ ( µ ) ∂∂x α (cid:16) τ ( ν ) u (0) E ( ν ) β x β (cid:17) = 1 e u σ u (cid:0)e u σ τ ( µ ) τ ( ν ) u (cid:1) (0) E ( ν ) α = (0) E ( µ ) α using the identity of Eq.[8] . (25)The pseudo-Riemannian metric associated to this tetrad field is g αβ = E ( µ ) α η ( µ )( ν ) E ( ν ) β = 1 e u σ u e u σ τ ( µ ) ∂∂x α ψ η ( µ )( ν ) e u σ τ ( ν ) ∂∂x β ψ = 1 e u σ u (cid:18)e u σ τ ( µ ) ∂∂x α ψ (cid:19) T η ( µ )( ν ) e u σ τ ( ν ) ∂∂x β ψ = 1 e u σ u ∂∂x α e ψ σ τ ( µ ) u η ( µ )( ν ) e u σ τ ( ν ) ∂∂x β ψ , by Eq.[5]= 1 e u σ u ∂∂x α e ψ σ (cid:0) τ ( ν ) u e u σ τ ( ν ) (cid:1) ∂∂x β ψ = ∂ e ψ∂x α σ ∂ ψ∂x β using Eq.[11] or Eq.[12]= ∂ ψ a ∂x α σ ab ∂ ψ b ∂x β = ∂ ψ a ∂x α ∂ ψ b ∂x β σ ab , (26)which is not an induced metric but, as one may expect, is the coordinate-transform of σ ab .The 4 + 4 = 8 dimensional spacetime π (cid:0) S (1) ( E , ) (cid:1) endowed with this metric has zerocurvature. V. SCHWINGER REAL REPRESENTATION OF QED
Julian Schwinger [15] has given a representation of charged fermion field operators for anelectron in terms of real anti-commuting 8-component spinor fields. Therefore it may be of10nterest to evaluate, using the new identity Eq.[12] and the above tetrad, and with arbitrarySchwinger spinor (bilocal) fields F and H , the operator H γ ( µ ) E α ( µ ) ∂∂x α F = H τ ( µ ) √ e u σ u ∂x α ∂ψ τ ( µ ) u ∂∂x α F = 1 √ e u σ u H τ ( µ ) ∂F∂ψ τ ( µ ) u = h − (cid:0) τ ( µ ) (cid:1) ad (cid:0) τ ( µ ) (cid:1) cb + 2 ( δ ab δ cd + δ ad δ cb − σ ac σ bd ) i H c u b √ e u σ u ∂F d ∂ψ a , (27)which may easily be further reduced. VI. ALGEBRAIC SIGNIFICANCE OF THE SPINOR u, THE UNIT FIELD
Let u ∈ S (1) ( E , ) be a type-1 spinor field (a section of the type-1 spinor bundle S (1) ( E , )),with < u , u > = e u σ u > E , , but being otherwise arbitrary. u , may be called a “unit field”. One may define a special E , frame field F in terms of u and the tau matrices as follows. Let M be the real 8 × M = 1 e u σ u u ⊗ e u σ = 1 e u σ u u e u σM ab = 1 e u σ u u a u c σ cb (28)Then M obeys Eq.[9] and transforms under S , R ) according to Eq.[10]. Using Eq.[11]or Eq.[12] to evaluate τ A M τ A yields I × = 1 e u σ u τ A ( u e u σ ) τ A = (cid:18) √ e u σ u τ A u (cid:19) (cid:18) √ e u σ u e u σ τ A (cid:19) (29)This is a resolution of the identity on E , . Alternatively this relation may be interpreted asa completeness condition verified by the E , orthogonal frame F whose components F aA aregiven by F aA = 1 √ e u σ u τ aA b u b (30)and its inverse is F Aa = 1 √ e u σ u u c σ cb τ A ba (31)Accordingly Eq.[29] may be expressed in index notation as { I × } ab = δ ab = F aA F Ab (32)11ince a matrix commutes with its inverse we also have δ AB = F Aa F aB . (33)We have defined an oriented spinor basis e a of S (1) x ( E , ) in Eq[3] and an oriented vectorbasis ǫ A of T x ( E , ) in Eq[4]. The two are related by ǫ A = e a F aA and e a = ǫ A F Aa (34) A. Split octonion algebra over R , O s ( R ) Let O s ( R ) denote the split octonion algebra over R [17], [16], [7], [11], [12], [13] .A nonassociative alternative multiplication of the oriented spinor basis e a (respectively,oriented vector basis ǫ A ) may be defined [11] that endows the real vector space S (1) x ( E , )(respectively, T x ( E , )) with the structure of a normed nonassociative algebra with multi-plicative unit that is isomorphic to the split octonion algebra over R , O s ( R ). This is accom-plished by specifying the multiplication constants m cab (respectively, m CAB ) of the algebra,which verify e a e b = e c m cab ǫ A ǫ B = ǫ C m CAB (35)The set of multiplication constants m cab (respectively, m CAB ) is defined by [11] m cab = F Aa τ cA b m CAB = F Cc τ cA b F bB . (36)It has been shown that the nonassociative product defined by Eq.[35] (respectively, Eq.[36])of the spinor basis e a (respectively, of the vector basis ǫ A ) endows the respective real vectorspace with the structure of the split octonion algebra over the reals [11]. This is explicit inthe multiplication table below, which employs the representation of the tau matrices givenin Appendix 3 and u a = √ (0 , , , , , , , σ with eigenvalue+1. 12 . Multiplicative identity An element Ψ ∈ O s ( R ) may be realized asΨ = e a ψ a = ǫ A ˆ ψ A ˆ ψ A = F Aa ψ a ⇔ ψ a = F aA ˆ ψ A . (37)The normalized fiducial unit field O s ( R ) ∋ √ < u , u > u = √ e u σ u e a u a = √ e u σ u τ a b u b = e a F a = ǫ = multiplicative identity element of the split octonion algebra O s ( R ) [12]:Multiplicative identity = 1 √ e u σ u e a u a = 1 √ < u , u > u = ǫ . (38)Multiplication Table ǫ A × ǫ B = ❍❍❍❍❍❍❍❍❍ ǫ A = ǫ B = ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ − ǫ ǫ − ǫ − ǫ ǫ − ǫ ǫ ǫ ǫ − ǫ − ǫ ǫ − ǫ ǫ ǫ − ǫ ǫ ǫ ǫ − ǫ − ǫ − ǫ − ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ − ǫ − ǫ ǫ − ǫ ǫ ǫ − ǫ ǫ ǫ ǫ − ǫ − ǫ − ǫ − ǫ ǫ ǫ ǫ ǫ − ǫ ǫ − ǫ − ǫ ǫ − ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ VII. CONCLUDING REMARK
The reduced generators (i.e., the Dirac matrices) of the pseudo Clifford algebra C , defined on E , satisfy a remarkable identity Eq.[13] that defines invertible linear mappingsbetween each of the two types of S , R ) spinor and the S , R ) vector, therebyadmitting matrix representations of triality on this spacetime E , . The trialities are givenbelow in Eqs[55] and [56]. 13 III. APPENDIX 1: TRANSFORMATION UNDER ACTION OF S , R ) The special Lorentz transformation properties of the theory may be determined by con-structing a real reducible 16 ×
16 matrix representation of S , R ) utilizing the irreduciblegenerators t A , A = 1, ... , 8 of the (pseudo-) Clifford algebra C , . Following Lord’s generalprocedure [9] we define the irreducible generators t A as t A = τ A τ A . (39)Let g ∈ S , R ). The 16 ×
16 basic spinor representation of S , R ) is reducibleinto the two real 8 × D (1) ( g ) and D (2) ( g ) of S , R ). The reduced generators of the two real 8 × D (1) ( g ) and D (2) ( g ) of S , R ) follow from the calculation of the infinitesimal generators t A t B − t B t A = τ A τ B − τ B τ A τ A τ B − τ B τ A = 4 D (1) AB D (2) AB , (40)of the 16-component spinor representation of S , R ). We see, as is in fact well knownfrom the general theory, that the 16-component spinor representation of S , R ) is thedirect sum of two (inequivalent) real 8 × D (1) = D (1) ( g )and D (2) = D (2) ( g ) of S , R ) ∋ g that are generated by D (1) AB and D (2) AB respectively,where 4 D (1) AB = τ A τ B − τ B τ A (41)and 4 D (2) AB = τ A τ B − τ B τ A (42)For completeness we remark that the generators of the two spinor types are images of14he projection operators χ ± = 12 (1 ± t ) χ + = I ×
00 0 χ − = I × , (43)where t = t t t t t t t t = τ τ . (44)Here τ = τ τ τ τ τ τ τ = τ τ τ τ τ τ τ (45)and τ = τ τ τ τ τ τ τ = − τ τ τ τ τ τ τ = − τ (46)The representation of the tau matrices is irreducible. τ has square equal to + I × andcommutes with each of the τ A matrices (and therefore with all of their products). Thereforewe conclude that τ = ± I × in any irreducible representation.Let ω AB = − ω BA ∈ R , A, B = 1 , . . . ,
8, enumerate a set of 28 real parameters thatcoordinatize g = g ( ω ) ∈ S , R ). Also, let L = L ( g ) ∈ SO (4 , R ) have matrix elements L AB , ω ♯ denote the real 8 × ω AB = G AC ω CB , ω = ω AB D (1) AB and ω = ω AB D (2) AB . We find that D (1) = D (1) ( g ) = exp (cid:18) ω (cid:19) D (2) = D (2) ( g ) = exp (cid:18) ω (cid:19) L AB = L AB ( g ) = (cid:8) exp (cid:0) ω ♯ (cid:1)(cid:9) AB (47)where, under the action of S , R ), g D (1) AB σ = − σD (1) AB ⇒ e D (1) σ = σD (1) − (48) g D (2) AB σ = − σD (2) AB ⇒ e D (2) σ = σD (2) − (49) L AC G A B L BD = G C D = n e L G L o C D (50)15 AB τ B = D (1) − τ A D (2) (51) L AB τ B = D (2) − τ A D (1) (52)The canonical 2-1 homomorphism S , R ) → S , R ) : g L ( g ) is given by8 L AB = tr (cid:0) D (1) − τ A D (2) τ C (cid:1) G CB , (53)where tr denotes the trace. Note that D (1) ( g ( ω )) = D (2) ( g ( ω )) when ω A = 0, i.e., when onerestricts S , R ) to S , R ) = n g ∈ S , R ) | g = exp (cid:0) ω AB D (1) AB (cid:1)
00 exp (cid:0) ω AB D (2) AB (cid:1) and ω A = 0 (54)This is one of the real forms of Spin(7 , C ). IX. APPENDIX 2: TRIALITY AND S , R ) COVARIANT MULTIPLICA-TIONS
Let V , V , and V be vector spaces over R . A duality is a nondegenerate bilinear map V × V → R . A triality is a nondegenerate trilinear map V × V × V → R . A triality maybe associated with a bilinear map that some authors call a “multiplication” [1] by dualizing, V × V → ∗ V ∼ = V .Let u denote the unit field and let ψ (1) ∈ S (1) x ( E , ) and ψ (2) ∈ S (1) x ( E , ) . Underthe action of S , R ) we assume that u u = D (1) u , ψ (1) ψ = D (1) ψ (1) and ψ (2) ψ = D (2) ψ (2) . Consider the following two multiplications that possess covarianttransformation laws under the action of S , R ) ⇒ S , R ). The first multiplication m A : E , × E , → E , is defined by Q A = 1 √ e u σ u e u σ τ A ψ (2) . (55)For fixed u a , Q A ∈ E , depends on 8 real parameters arranged into the type-2 spinor ψ (2) .The second multiplication m AB : E , × E , → V has an image in V ∼ = E , × E , , anddepends on 8 real parameters (for fixed u a ) arranged into the type-1 spinor ψ (1) : Q AB = 1 √ e u σ u e u σ τ A τ B ψ (1) . (56)16or fixed u , Q AB possesses only 8 degrees of freedom corresponding to the 8 independentdegrees of freedom of ψ (1) , so we also refer to this map as a “multiplication.”Eq.[56] may be easily be solved for the components ψ (1) a = ψ (1) a ( Q AB ). Consider1 √ e u σ u ( τ B ) (cid:0) Q AB τ ( A ) u (cid:1) = 1 e u σ u ( τ B ) (cid:0) τ ( A ) u (cid:1) (cid:0) e u σ τ A τ B ψ (1) (cid:1) = 1 e u σ u ( τ B ) (cid:0) τ ( A ) u e u σ τ ( A ) (cid:1) τ B ψ (1) = ( τ B ) τ B ψ (1) by Eq.[11] or Eq.[12]= (cid:0) τ B τ B (cid:1) ψ (1) = 8 ψ (1) (57)Similarly, ψ (2) = 1 √ e u σ u τ A u Q A . (58)In this paragraph Greek indices run from 1 to 4, α, β, . . . , µ, ν, . . . = 1 , . . . ,
4, while Latincontinue to run from 1 to 8,
A, B, . . . , a, b, . . . = 1 , . . . ,
8. It is convenient to define a SO (3 , R )-invariant symplectic structure Ω on E , (and a complex structure on the splitoctonion algebra) by Ω = − (59)where 0 denotes the 4 x 4 zero matrix and 1 denotes the 4 x 4 unit matrix. The Q AB maybe represented in terms of an arbitrary antisymmetric M , rank 2 tensor F β α = − F α β andtwo S , R ) scalars x and x according to Q AB = F α β ∗ F α β ∗ F βα F α β AB + Q Ω AB + Q G AB , (60)where ∗ F α β is dual to F α β and defined by ∗ F µν = − ǫ αβµν F α β . Note that Q [ A B ] = (cid:0) Q A B − Q B A (cid:1) is independent of Q .Clearly, in order for Eq.[60] to possess physical significance the action of S , R )must be restricted to S , R ) in a manner that links transformations of x , x , x , x to x , x , x , x . 17 . Covariance of maps under S , R ) Let u u = D (1) u , ψ (1) ψ = D (1) ψ (1) and ψ (2) ψ = D (2) ψ (2) under S , R ).Consider the transformation law for the Q A Q A : Q A = e u σ τ A ψ = ^ D (1) u σ τ A D (2) ψ = e u σ D (1) − τ A D (2) ψ = L AB e u σ τ B ψ = L AB Q B ,which follows from Eq.[51], Also Q AB Q AB : Q AB = e u σ τ A τ B ψ = ^ D (1) u σ τ A τ B D (1) ψ (1) = e u g D (1) σ τ A D (2) D (2) − τ B D (1) ψ (1) = e u σ (cid:0) D (1) − τ A D (2) (cid:1) (cid:0) D (2) − τ B D (1) (cid:1) ψ (1) = L AC L BD e u σ τ C τ D ψ (1) = L AC L BD Q CD ,which follows from Eq.[51] and Eq.[52]. In summary, under the action of S , R ), u u = D (1) u ψ (1) ψ = D (1) ψ (1) ψ (2) ψ = D (2) ψ (2) .Q A Q A = L AB Q B Q AB Q AB = L AC L BD Q CD = { LQ e L } AB (61) X. APPENDIX 3: IRREDUCIBLE REPRESENTATION OF THE τ MATRICES
We adopt a real irreducible 8 × τ = I × = τ . Then by Eq.[6] τ A = − τ A for A = 1, ... ,7. Hence,again by Eq.[6], ( τ A ) is equal to − I × for A = 1,2,3 and is equal to + I × for A = 4,5,6,7, 8.A particular irreducible representation of the tau matrices is18 = −
10 0 0 0 0 0 − − − τ = −
10 0 0 0 − − − τ = − −
10 0 0 0 0 0 1 00 − − τ = − − − − τ = − − − − τ = − − − − τ = − − − − τ = (62)19
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