The Dirac equation in a non-Riemannian manifold: III. An analysis using the algebra of quaternions...

The Dirac equation in a non-Riemannian manifold III: An analysis using the algebra of quaternions and octonions

Sirley Marques-Bonham@ NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, MS 209, P. 0. Box 500, Batavia, IlIinois 60510 and Departamento de Campos e Partkulas, Centro Brasileiro de Pesquisas Fkicas, Rua Xavier Sigaud, 150, 22.290, Rio de Janeiro, RJ, Brazil

(Received 26 March 1990; accepted for publication 14 November 1990)

The geometrical properties of a flat tangent space-time local to the manifold of the Einstein- Schrodinger nonsymmetric theory to which an internal octonionic space is attached, is developed here. As an application of the theory, an octonionic Dirac equation for a spin-l/2 particle is also obtained, where is now used an octonionic-like gauge field. It is shown that the (quaternionic) nonsymmetric Yang-Mills theory can be easily recovered and from there, the usual gauge theory on a curved space.

1. INTRODUCTION

Einstein’s general relativity (GR) is a successful theory where the gravitational field is included in the geometry of a curved Riemannian space-time manifold. The Einstein- Schrbdinger (ES) nonsymmetric theory’ is an extended geometrical theory, where it is supposed the electromagnetic (EM) field is also contained in the geometry. This was the first effort to obtain a geometrical unified theory of fields. Others followed2,” in the analysis of this theory, in an attempt to overcome the problems related to the interpretation of the (geometrical) field equations so that it could be trans- lated in what was known in a common field theory on the curved space-time. The theory developed by Borchsenius4 is another step in the direction of an unified geometrical theory, which includes the Yang-Mills fields.

GR uses the algebra of the real numbers on a Rieman- nian space-time manifold, but the ES theory instead uses the algebra of the complex numbers in a non-Riemannian space- time manifold. The Borchsenius theory uses a matrix algebra for the SU(2) symmetry group, with realization via Pauli matrices, and can be easily reinterpreted as a quaternionic algebra. In an attempt to extend the Borchsenius geometrical theory, a nonsymmetric theory was developed using the algebra of the octonions,5 which is permitted by the Hurwitz theorem. Nevertheless, this was achieved only in the branch of split octonions using the Zorn (vectorial) matrices and a duplication-like of Yang-Mills fields included in the geometry was obtained there.

The above geometrical theories were developed on the curved space-time, and can also be performed on the local tangent space-time.6 This is achieved through a generalization of the equivalence principle. As an application of the theory, in the last two parts of this work, Refs. 7 and 8, we also obtained a Dirac equation for a spin-l/2 particle placed in a non-Riemannian manifold of the ES nonsymmetric theory (the complex case), which includes the EM fields, and the matrix-Borchsenius case, which includes the n-dimen- sional [for SU( n) ] Yang-Mills fields. We are going to present here the same kind of theory now using the quaternions

“’ Present address: Dept. of Physics, Univ. of Pretoria, Pretoria 0002, Re- public of South Africa.

and the split octonions in the way developed in Refs. 5 and 6. Meanwhile, it might be of interest to present a useful math- ematical review.’

A composition algebra over the real numbers is defined as an algebra A with an identity element and with a nondegenerate quadratic form Q defined over A such that Q per- mits composition, i.e., for each element x, y of A,

Q(v) = Q(x)Q(y), Q<ax) = a2Q(x), (1.1) where a is a real number. [A quadratic form is nondegenerate if the scalar product associated to it is nondegenerate, i.e., if x # 0, there exists y such that ( XJ) # 0. ] We can associate with this quadratic form, a scalar product:

(x,Y) = ;[Q(x +Y) - Q(x) - Q(Y) I, (1.2) which is a symmetrical nondegenerated bilinear form. The norm of a vector x is defined as

N(x) = Q(x), i.e., N(x) = (x,x). (1.3)

The Hurwitz theorem states that: “the only composition algebras on the real numbers, except for isomorphisms, are the algebra of real numbers R (dim n = 1 ), the algebra of complex numbers C (dim n = 2), the algebra of quaternions Q (dimn=4), and the algebra of octonions 0 (dim n = 8). Of these algebras the quaternions are noncommutative, and the octonions are noncommutative and nonassociative.” The composition algebra is said to be a division algebra if the quadratic form Q is anisotropic, i.e., if Q(x) = 0, it implies that x = 0. If not, the algebra is called “split.”

The quaternionic algebra Q has four generators {e,,ei,i = 1,2,3}, where e, is the identity element of the algebra. They satisfy the relation,

eiej = eiike, -Sue,. (1.4) Every element x of Q can be written in terms of the generators as

x = xcec + xiei, E = xcec - x,e,, (1.5) where z is the quaternionic conjugate of x. Therefore, the squared norm of x is given by

xX=xX=(x; +xf)l, (1.6) where 1~ e, .

1383 J. Math. Phys. 32 (5), May 1991 0022-2488/91/051383-l 2$03.00 @ 1991 American Institute of Physics 1383

A possible realization of the quaternionic algebra used in this work, is through the Pauli matrices:

w, = i- ‘CT,, i = 1,2,3, where gi are the Pauli matrices and oi are the (matrix) generators of a quaternionic algebra, and where w0 = a, = 1. The numbers {wO,oi,i= 1,2,3), satisfy a product of the type ( 1.4) above:

op, = EyXW1. - Ijljoo. (1.7) The octonionic algebra has eight dimensions and its

base vectors, e,, e,, i = 1, . . . ,7, satisfy the product law: e,e, = e,e, = e,, eiej = eUhe, - See,,, (1.8)

etih is now a completely skew-symmetric object with seven nonzero matrix elements: E 123rE516,E624~E435,E471,~5721E673.

This algebra is also called the Cayley algebra.” It is neither associative, nor commutative, but it pertains to the class of alternative algebras. For any three octonions x, y, z, their associator is given by

Lv,z~ = (XY)Z - x(y.2). (1.9) In an alternative algebra the associator of three of its elements, as defined in ( 1.9), changes sign when any two of its arguments change position. Also, x, y, z are called “Cayley numbers.”

In terms ofbase-vectors any octonion x can be written as

x=x,, +x,e,, F=x,e, -xjei, i= 1 , . . . ,7, (1.10)

where X is the octonionic conjugate of x. Taking ( 1.8) and ( 1. lo), the squared norm of x can be written as

Q(x) = XE = xx = (x:, + xf)l, (1.11) where 1 = e, is the identity element of the algebra.

The Cayley algebra with the base defined by ( 1.8), pertains to the class of division algebras (real base), but it also can be presented as an split algebra if we use a new base in the complex domain. This is given as

u, = ;(eo + ie, 1, u$ = +(eo - ie, ),

u/ =t(e, +ie,+,), UT = $(e, - ie,, 3 ), I= 1,2,3, (1.12)

where i is the complex unit: i = sqvt( - 1). From this definition follows the multiplication table:

u:u*y = Eijh U&, utuj = Edl,UX, u u* = - s..‘.J

1 J I/ 0, up.8, = - s,u,*,

u”uo* = 0, uiu() = 0,

**- * uou, - u, f u,u, = ui,

u *2 - U,*, 2- (1.13) 0 - uo - uo,

uplo = u:, u,u,* = u ,f u,,u: = 0 , [email protected] = 0, l&4, = l4,t.f; = 0. It is of our interest to take a realization for the elements

of the base ( uO, u,, u,*, UT), through the Pauli matrices. This is possible by means of the following identifications:

uo=(; io), a:=(; ;), ui=((Ji ;), +(; -;i>, i=1p2y3;

(1.14) w. and wi are defined in ( 1.7). Therefore, an split-octonion A can be written as

A=au,*+bu,+x;u*+yiu,= ; -y . ( > (1.15)

The conjugate of A is now defined as

~=bt@+aet, -x,u:-yyiu, = . (1.16)

The norm of A is then given by Ax=zA= (ab+x*y)l, (1.17)

where 1 is the identity element of the algebra: 1 = 1 *u$ + 1.~1,. The “matrices” written above are called “Zorn matrices” or “vectorial matrices”.” The product of any two of those matrices is defined by:

AB = (; ,“) (f -d”)

( ac - x*v - (au + dx + yxv) = cy+bv+xxu bd-y-u > , (1.18)

which guarantees the nonassociativity of the product. The multiplication table for the t(‘s is reproduced in this Zorn matrix notation. Some other properties for the octonions:

Al=lA=A, AB=B’;I; A + 2 = Tr(A)l, Tr(AB) = Tr(BA), Tr[ (ABIC] = Tr[A(BC)] = Tr(ABC), (1.19) where the Tr is realized on the Zorn matrices. [We are using here A=Z(A), the Zom matrix of the octonion A, since the Zorn algebra is isomorphic to the algebra of the split octonions. ]

In general, the complex (split) Cayley algebra contain seven Euclidean vectorial subalgebras, as well as seven quaternions subalgebras. This property follows from the multiplication rules for the complex (split) base given in ( 1.13) and from the definition ( 1.12).

Ii. THE QUATERNlONlC TANGENT SPACE

In the second part of this series, Ref. 8, we developed a treatment of a tangent space local to the non-Riemannian space of Einstein-Schrodinger theory connected to an internal (matricial) space of SU(n) type. This includes the Yang-Mills fields in the geometrical theory developed in the first part ( Ref. 7). If we restrict the discussion to the SU (2) symmetry group, this internal vectorial space can be realized via Pauli matrices (a,, i = 1,2, 3). This, on the other hand, can be immediately reinterpreted in terms of quaternions when, as it was mentioned in the last section, we take 1 ‘-57, =wi,wg =a, = 1,. As an example, the quatemionic Vierbein E; (x) can be written in terms of components as

E; (xl = k;,,oc, -I- k;p,, E?“(X) = k *o‘w P No 0 - k **a. P, 1’ (2.1)

1384 J. Math. Phys., Vol. 32, No. 5, May 1991 Sirley Marques-Bonham 1384

ti ol

L i T z > 2 .K $ 0-l r d 8

g d 5 ii s r om 3 5

it

TABLE I. Principal geometrical properties for tangent spaces local to the space-time manifold, using R, C, Q, and split 0 algebras

The Riemannian theory The non-Riemannian theory (R-algebra) (C-algebra)

The qartenionic theory (Q-algebra)

The octonionic theory (split-0 algebra)

ds’ = dx*” dx” quh dx” = e“ dx” ,I ’ dx** = e*” dx’ I’

ds’ = & Tr(dx’” dx” 11 ) dx” = E; -dY: dx+” = ii’ dx’

ds’ = j Tr ( dxt” dxL vu,, ) dx” = H; dx”, dxt” = H,:” dx” Line element

Metric relation

Orthogonality relations

Tangent

d.6’ = dx” dxhq,, dx”= h; dx”

G”“= H;“H;;q,, G,,,. = H:?H$ vu,.

G;,. (~9 = G,,, (s,r) G,,,.(x) = G,,,.(i*)

= s,,,& + S,,v’u* + *,,,. 4 + ‘;,,.-u

S ,zto = *,,,o

g’” = e*“e;quh ‘I *u I? g,,,. = e,. e,, sh

E,. = g,,, && = g,w + k,,,.

G’“‘= E:“E;;Q,,

G,,r = E?::v.,, G;,. = G,,,

G,,,. = q,,>oe + q,,,..~

g- = h :‘h ;p g,,,.=h,“,h?L*

g,,v = g.,,

e*fi’@ = 6; u ,a e*J’e”. = 8: Y, I

E b E :l’ = E ‘YE +b = #‘q, ,’ c < ,r

E TruE: = E :‘.E :: = S:;o,, H”H?‘= H’IHth=6”(u,, + u,:) L C,’ ‘ H$H; = H:HH:: =S;;(u,,+ u$)

h” hh =& u,r h

h”h: = 81 “I %

A ,,Ub = (A,,“,, h,

1 ,‘“6 = - A,,!”

A ,d? = &d? )(% + m Q p”h = - q,dw

A ,‘“b = - A ,dw

(a real number) A ,,Yh = - A,,,w

Internal connection

Tangent

curvature

Internal curvature

I-,, = B,;w r,, = L,;u* + K,;u c,, = A,, (a pure imaginary number) (none)

s,,,. = (A,,.>. - A I..,, - [A,,A.] )(G + UC,) s,,,. = A,,.v -A,:,, - [A,,A,.l s,,,. = A,,.>. - A,:,, - [“,okl (a real number)

s,,,. = (A ,,.,. -A ,..,, - [A,,,A,.I)~u

P ,,,. = r,,.r - r ,..,, - [r,,w P,,. = r ,,.,. - rr.,, - [r,x.i pp. = c,,.,. - cr.,, (an imaginary number) (none)

h a 3 iF 12

B ;;i V

5 I! I

n *: 1 II

‘QJ< * I 3 II : g*

a0

d II

&G t: + Gwk -* r’ + &< ;a m I

au: II >:- UJ

0 II

*i \c 34 -e= + :; cc

-3 23

I ba e It

2f rc

2 .s 4 B 1w i .L *

1386 J. Math. Phys., Vol. 32, No. 5. May 1991

0 0 II II

d $. LE II $ ;$; x t Q- I

:d & 0”

i “;‘3 :’ t 2 caws , E t “Q 8

0 * h II :i

m 2 f ;2

L 1 i: I- :- YS 5

m b v) II II

0 II _^ s;

d a> If <=

;; t

7 & 0 II 4

+

i 2 II

2 $ .o .9 s ‘;;I z

g : .; .$ 9 2 p

b

:: . c

ti

d II

‘Y-. : ; s II

li trl 1

d “1 2 5 II ‘: s II :,

g ;’ ,+

ii =; ” f < 7 ”

II :. c i i

3

^ II &= P 2 k* : 5 ay Jg

t i It - :

t : i 2 tsl 1 2 :: : hc 4: + -2 u

II 3’ ‘! 2 3:’ 4 Is- 4 & -2

It $

L

i :> m

d 6 II L tr ,, grq .

+ r; m rr u- :f *3

5 Fl : I ‘: t-j

II -c 2 :: : I L.2 c-i 12 L- I $: $&d 5 Lq 1%

0 +- 1 II *g j 24

i ’ *‘r_ -2

z. + 75:

c5 f ;&$>a

% H +LG

-L d d z II ” II 2

*n &I gm &; rS a !!I!

-:g = -q2

II I f&v; Y-: 1:9 i;;’ 41 L’ t ;,i I 1 0 “+ g-3; I c II ;$-

k) $ II ,, &?,a

u iw: I! g p 2 i

2 % 2 .s z ‘i;l s s : : .E .- 2 2 p

Sirley Marques-Bonham 1386

because, w: = - wi, k ;, G it$, and where tip, is not a pure imaginary number. This shows that every formal relation obtained in the theory developed in the second part of this series, Ref. 8, restricted to the SU( 2) case, is the same for the quaternionic theory, differing only by a factor of i = m, when we take the components on the internal vectorial space. Therefore, as we have placed the geometrical properties of the generalized space-time (connected to the internal spaces that follow the Hurwitz theorem) in Table I, one can just complete them with some specifications about the internal components of the objects, as well as new definitions necessary to this quaternionic theory. Also, it is not intended to repeat here the real and the complex cases that should be referred to in Refs. 7 and 8.

The transformation law for vectors on the tangent space-time is as usual defined through the Lorentzian rotation matrices L z such that L T~L = 7. However, it is now necessary to define a more generalized transformation law so that it takes into account for the “internal’‘-quaternionic degrees of freedom. That will be included in what we call “quaternionic transformations,” first used in Ref. 5 and thor- oughly developed in Appendix A.

Therefore, the general transformation law for the vector E; (x) on the tangent space is defined as

E;“(x) = L ;(xwwE; (xl u+(x)), (2.2) where now V(x) is a 2 X 2 unitary matrix, or a quaternion with real coefficients:

U(x) = mow0 + miwi, U+(x) = U - ‘(x) = mow0 - miwi, (2.3) rni +mf= 1,

U(x) being a unimodular matrix, U - ’ =u, the quaternionic conjugate of U. This implies that Ut = U - ’ = i? which makes the coefficients real. (See Appendix A).

The internal covariant derivative for the SU(2) situation, is now called quaternionic covariant derivative or Q derivative. Then, the Q derivative of a quaternion K(x) is defined as

KII, =K.I1 + [r,Jl, where

(2.4)

rp = - (dfi)b;(iu)- + (dji)bp-o. (2.5) The total covariant derivative of Ez (x) can be written

as

E ;,v = E;,,. - “;,I; + ArAE; + [L,E;]t (2.6) +

where, C$,., A,.Ob, and I,,, are the curved space-time, local tangent space-time, and quaternionic-internal space affini- ties, respectively. This expression is formally the same as that for a matrix theory. In (2.6), as well as in Table II, it is used for the Einstein convention for a “ + ” (or “ - “) covariant derivative of a vector (or a tensor) in a non-Rieman- nian manifold (see Ref. 1).

Table II lists the field equations for the quaternionic theory, which are formally the same as the ones obtained in Ref. 8.

III. THE OCTONIONIC TANGENT SPACE

Let us consider now, a space-time manifold to which is attached an internal space constituted of octonions. Again, we will be using here the split-octonion algebra with realization via Pauli matrices, as in Ref. 5, because of the conven- ience related to a possible physical interpretation and of ob- taining a limit to the Riemannian theory (the Real algebra theory). Taking the definitions used in Ref. 5, we can write the O-metric as

G,,(x) = s,wp,* + s,,,u? + rpy,uo + r,v,ui

s P% wo - Sp,,‘W = (3.1)

rpv** ‘PV” WO > = G,, (~1,

where r P% = s I, v, = gpv + ipF,,,

- V

where g,,,. is the symmetric metric, and F,; is the Maxwell tensor, and s,, and r,, are such that in the limit s,, --+rp,, =p2fp,., where f,,, are the Yang-Mills pelds for the SU(2)

group. The parameter p is defined such that ?“,d,=ib i,“,“‘,” I = 3.8 X 10 - 32 cm, in the normalization

. . The symmetry property for the 0 metric is defined as

CL,. (s,r) = Gw (~1, (3.2)

G;,.(s,r) = ‘rjL.(,@o s;v*o - r&*0 s* >

= G,, (s,r). (3.3) PO@0

This symmetry property will then require that: s* =s P% TO’ (3.4) . I s* =s PV p’v = - s,, r$ = rpy = - r,,

The quaternionic Vierbein (2.1) is generalized to the octonionic Vierbein, or 0 Vierbein, as

H;(x) = k;@ + k;,u: + I&, + Z;,ui

> = H; (k,Z). (3.5)

Define:

G,, = H t”H; q&a, where vclb is the Minkowski tensor, and such that (3.2) is valid. There exists an inverse of G,,,:

WV = Ht”H;T7,,, (3.7) such that,

G,, (s,r)W”(s,r)‘= GYs,r)G,, (s,r) = 6: (u, + u,*,, (3.8)

in this order. Then, from (3.6), (3.7), and (3.8), the orthogonality relations for 0 Vierbeins are

H;Hp = H;HjP = &(u, + u;), HFH; = H;I”H; = 6; (u, + u,*). (3.9)

These relations are obtained in such way that the following relation is true:

Tr(G,,W”) = Tr[ (H~H~)(H~H;)qob?fd], (3.10)


c E 7 0 TABLE III. The Dirac theory on a non-Riemannian manifold. The Dirac equation. -2 I” -<

The Riemannian theory The non-Riemannian theory The quatemionic theory e R algebra

The octonionic theory C algebra Q algebra split 0 algebra

x Generalized 5

y”=hj;f y=e’:y y = E;;f y-matrices jc’=Ey’r’

r”=H:‘f’ */’ = es”y

VI j.4 = H?‘fk~:l’~

8 {f,y’3 = 2g’“1, Cr,oYJ = zg,,,. 1,

Ci”‘.y’-1 = 2g’“l, fY,o?J = %,,J.l

Tr(Y’,y”l = 2 Tr( G”‘)l, Tr{y”,y”} = 2 Tr(Q”‘)l, = 2~4g’“l, =2X2&q

Tr{y,,,i,.I =2Tr(G,,,.)l, Trb,,,yJ = 2 TrfG,,,.)l, = 2X4g,,,.l, = 2x2g,,,.x4

Relations for the y-matrices

g,,, is the GR metric

Y,‘ I = Y ,‘.’ - bhy,, I- [ A,.J,, 1 = 0

g,,, is the ES metric

Yr: , = Y,t.x - q:,.Y,, -I- G.Y,,

+ [LY,, 1 = 0

Tr on the internal (SU (2) matrix g,,, is the ES metric

YP % = Y,&V - e:;,.y,, + GY,,

+ [LY,,] + [f-.-.Y,, ] =O

Tr on the internal Zorn matrix g,,V is the ES metric

Y, 8 = YJ,.v - @::,.Y, + CY,,

f 1 A,.J,, I + P-,.,Y,, I = 0

PP. = i; L,. - q,i;, - c,.t;, FJ. t = *r.r - e::,, i;, - c, jl,. . . Yn . = Y,,... - q,j;, - c,.?;,

+ [A,.$,, ] = 0 f [A,-$,, ] + [r,.,?,, ] = 0 -i- [LP,,] + [J-‘,.$,,] =O

cul;ii,, -pO$=O Generalized - i@,y,, --PI = 0

cyi;, rpr,$= 0 - im;v,, --PI = 0

w%, y-,0= 0 - ?m;y,, -y) =o

(?q, -J!J)~=o

Dirac-equation V,, = 4 + A,, -kv:,i;, -IO=0

V,, = a,, f A,, -i- C,, V,, = a, (a,, + A,, + C,, ) -i- r,, v,, = (h, I- 63 ca,,

V: = a,, + A,, - C,, q = w, ca,, + A,, - C,,) - r,, + A,, + C,, 1 I- r,,

v; =y = (us + u:) ca,,

-2 f A,, - c,, ) - r,,

8 5

Gauge fields a present in

A,c (gravitation) A,, (gravitation) A,, (gravitation)

8 A,, (gravitation) C,, (electromagnetism)

C,, (electromagnetism) C,, (electromagnetism)

v) the theory I,, [Yang-Mills tields(SU(2))) I,, (“double- f like” Yang-Mills fields)

i9

where the position of the parenthesis is no longer important if the objects are fixed in their positions.

The equivalence principle states that locally, the line element on the curved space-time is equal to the line element on the tangent (flat) space-time, with Minkowski metric. This principle generalized to this split-octonionic theory, can be written as

ds2 = 4 Tr(dti dx”G,,.) = 4 Tr(H~H~~,,, dxp dx”), (3.11)

due to the definition (3.6). Again we are taking the metric on the tangent space as being that of Minkowski. (The Tr is taken on the Zorn matrix.)

The covariant derivative, or 0 derivative, of an octonion K is here defined by

K,,, = K, + FVKI, (3.12) where F, is the “octonionic affinity,” such that it makes K,, transform as an octonion under 0 transformations:

K’=UKU-‘, KL = UK,,U - ‘, (3.13) and

r; =ur,u-‘-u&U-‘, (3.14) [In the case of Eq. (3.13), for the octonionic transformation, it is true that: K’ = U(KU-‘) = (UK)U-‘. See Ap- pendix A. ] The objects U(X) are split octonions written as Zorn matrices, which define here local-octonionic transformations. They also are homomorphic to the rotation group 0,. The Zom matrix U - ’ is defined as being identical to is, the octonionic conjugate of U.

We impose here the condition that FCC be a trace-free Zorn matrix, and of the Yang-Mills type:”

r,, = - L,#u? - Kp,ui, i= 1,2,3,

E 0, L,*w

- K,*o 0, ’ (3.15)

where 0, is the 2x2 null matrix. In the limit K, -+L,, this split-octonionic theory would turn into an “exact-doubling” of the quaternionic theory with realization via Pauli matrices.

The total covariant derivative for the octonionic Vier- bein Hf: , for example, can be written as

H ,“I,,. = Hz,,. - W,,,H; + A,.“,H; + [I?,.,H;]. +

(3.16) In this expression a$,, is the affinity for the nonsymmetric manifold of the Einstein-Schrodinger theory, and A,, is the corresponding tangent space affinity.

The octonionic internal curvature is obtained through the relation:

K,,,,. - K,,,, = Pp,.K - KPp,. + {r,,r,.,K), (3.17)

where {F, ,r,.,K} is the associator of the fields F, ,lY,,,K. The object P,,,. is called an “internal octonionic curvature,” or “0 curvature,‘* and it is written as

p,,,. = r,.,. - r,.,r - [r,,r,.]. (3.18) We define now, the objects:

RApYY = r%,, - r$,, + r$yrplly - r$rppy, (3.19)

r;v=~~yw+~o) +s;r,, (3.20) and

s o vyb = A.,(Ib,Y -AyabP - [LAy];lb, (3.21)

A,Ob =Avab (u: + UO) + wk? (3.22) such that the following expression is true:

Tr[H?$, - S&H; ] = 0, (3.23) which is obtained from the trace of the curl: (a,,dv - 6’,,~9,)H; = 0. Therefore, we can call them the general-

ized “0 curvatures” for the 0 space-time and the 0 tangent space, respectively. They can be rewritten:

R&r,, =RP,,h, + ~$1 +S;tP,, (3.24)

S I7 =s v c voc(~o + u,*) + WV,9 (3.25)

where R &, is the curvature of the space-time written in terms of the R’s, and S,, is the tangent space curvature written in terms of the A’s.

The field equations for the metric, in the split-octonionic theory, are placed on Table II. From them, we obtain the following relations for the octonionic “affinity” A,:

A.” vb= -Ha HP + (Harp P.V P P” )H~J - {A,~,,H;,H~‘-‘) (3.26)

and A ,tab = H;Hi$ + H;WPp,H~“) + (~,“b,H;,H~p),

(3.27)

where now A %,*b = (bob -h%c,,)(U,*$~~) -t&r,.,

and

rzv = ep,,,(~; + 24,) + 6p-,. Therefore, the expression for the tangent affinity A,, in

the case of a tangent (flat) space-time local to a non-Rie- mannian space-time manifold, and in the presence of the split-octonionic internal space defined here, can be written as

A” vb = (aRe{Tr[ -H;,,,Hp

+ f$,H;H~ + H;r,H?]h, (3.28) or

A” vb = (d Re{ Tr[H;H&

+ e&H;HP - H;l?,.H~]}). (3.29)

In (3.28) and (3.29), was used the property that Tr[A (BC)] =Tr[(AB)C] =Tr[ABC], for the trace of the product of three Zorn matrices, A, B, and C, which ac- counts for the nonexistence of parenthesis separating the product of the objects on the right side of the equations [see the properties for octonions in (1.19)]. Also, it should be remembered that here A = Z(A ), the Zorn matrix of the octonion A, due to the isomorphism between the octonion algebra and the Zom matrices algebra.


IV. THE OCTONIONIC DIRAC EQUATION

The generalized set of Dirac equations in the extended manifold, for the case of SU(2), is also directly reinterpreted as a quaternionic set of Dirac equations through the method exposed above. The same is true for the definition of quaternionic y matrices. They are displayed in Table III. We will go forward, then, in developing new definitions for the case of octonions.

Starting from the anticommutation relations for the Dirac y matrices ( Ref. 12)) we will obtain the corresponding generalization for the octonionic (split) case. Therefore, we define

Tr{y,,i,l= 2 Tr(G,,)l, = 2X4gti,14, (4.1) where the Tr is taken on the Zorn matrix (3.1)) and where

H;Y, = rp; H:“Y, = i, (4.2) are the octonionic y matrices. In (4.1)) g,, is the metric of the Einstein-Schrodinger nonsymmetric theory.

The inverse of the relation (4.1) is obtained multiplying the inverse of the commutation relations {f,?} = 2@ 1, by HF and H l, and then taking the trace of the product of the Zorn matrices. This results in

Tr{@,y”) = 2 Tr(w”)l, = 2x4&?,, (4.3) where g+“’ is the inverse of g,,, and

H’64=y; H;y=j,“. (4.4) The total covariant derivative ofthe octonionic y matrix

should in general be

Y@lV = Y&V - %Y, + [A”,Y/L 1 + ILr, I, (4.5) +

where it is important to note that the electromagnetic potential is displayed only when the nonsymmetric space-time affinity is rewritten is terms of the Schriidinger affinity,

fl& = B;,. + (2/3)ss1;,. , B”p,, = 0. (4.6) V V

in (4.5), A,. corresponds to the internal connection of the y matrices (also internal) space. With (4.2) above, we can write the corresponding field equations for the generalized Yp :

or

~plt, = (HffJ, yu 1 = (Hzi,. )yo = 0, (4.7) + + 1%’ +

ip,,. = W+; yc,) = (Hz,. )‘y, = 0, (4.8) - - jV -

Yplv = Y&V - @&Yp -I- C,.Yp +

+ [LY, ] + [Ly, ] = 0, (4.9)

r 1’1” = i,,.v - et$f, - c,*Y,~

where + [A,.& ] + [L,;, ] = 0, (4.10)

c,, FE g a:,.. (4.11) V

Also, we can take C, = ieiA, (n), as in Ref. 7, where e is the electric charge for the electron, A,,(X) is the electromagnetic

potential, and I is a constant such that it balances units. It is again easy to show that the expression for the affin-

ity A,, continues to be similar to that of general relativity:*3p7

A, = ( l/4i)h,“bu,,, (4.12) where now A,, is given in (3.27) or (3.28).

The next step is to write a minimal action principle to obtain field equations for a spin-l/2 particle of mass m, placed in a non-Riemannian manifold, and in presence of the (octonionic) field I’,,, Yang-Mills-like, as deflned in ( 3.15). Therefore, the action,

A= 2?‘d4x, s (4*13)

has the Lagrangian density defined as

2 =J-w@w+phq + [~(c$q]q+&}, (4.14)

where to= [ -$Tr( det GF,,)]t’*, (4.15)

3V11=(UOfUO*)(;j~+Al,+CC,)fr~, (4.16)

~~=(un+ub)(~~+h,-C,)-rr,, (4.17) and wJere p is the-mass term. [The quantities @, ? [ (FV, ) $1, and [ $( VL @) ] @are invariants under octonionic transformations. ] Again, the trace (Tr) is taken on the Zorn matrix and the det is taken on the space-time indices. The function $(x) is a complex object that locally transforms under the representation of Lorentz group. Also, r,&x) is a vector in an internal space that transforms under the octonionic transformation represented by U, defined in Appendix A:

P=W, (4.18) where U can be written as a Zorn matrix:

u=y; ,“). The “octonionic” wave function 3 is defined here as

$4 > -4

x ’ (4.19)

and the functions (b and x are “partial” quaternionic-like wave functions. [The properties of r,&x) are stated in Ap- pendix B.] As we are also using the Pauli matrices representation, we write d, and x as

#I +=L) , ,y= x’ .

2 0 x2 (4.20)

The conjugate wave function is written as usual: 4 =@ro = $T*yO, where y. is the constant y0 matrix. The octonionic transformation of ;3 is written as

ijk@-‘, (4.21) where

Similarly to (4.19), 3 is written as

iJ=(3 21, (4.22)


where

3= (3, 3,), ,p=(,p, X2). (4.23) In (4.23)) $i E ~FY,, Xi =x,r*re, i = 1,2, are now related only to the “internal” Dirac ymatrices degrees of freedom of *.

The operations for the octonionic transformation of $ and $ stated in (4.18) and (4.2 I), respectively, are defined such as the product t+@ is invariant under octonionic transformations (see Appendix B). One therefore obtains for 11, and $‘:

fl=(y;;), (4.24)

iP= Gio -/pl ,ppo -$P). (4.25) The field equations obtained from the above Lagrangian

keep the same form as those we have obtained for the real, complex, and quaternionic theories, that is

(yQ%4=0, (4.26)

- ij(&q -pij=o, (4.27) whze now thegositions of the parenthesis are important, as cr”v, M#u”CV/A).

The Dirac equation of the “charge conjugate” wave function t,P is given by

(~?~)tcf-p~=O (4.28)

where @= C$‘, and C is the charge conjugate matrix, as usual.

V. AN ANALYSIS OF THE OCTONIONIC DIRAC EQUATION

In the former parts I and II of this work, Refs. 7 and 8, it was supposed that the mass term in the generalized Dirac equation also contains internal degrees of freedom, i.e., the components of,u on the internal space are non-null. Here, we will again follow this idea, for the octonionic Dirac equation. However, we will also introduce the idea developed in Ref. 9, which is related to the expansion of the wave function t&x) in terms of the parameterp, now extended to the case of the split octonions.

Take the octonionic Dirac equation (4.26) :

WY’$J$-P~=Q (5.1) where the octonionic Vierbein can be expanded in terms of components as in (3.5):

H:(X) = k”,u,* + k:& + I$,u, + lziui. (5.2) Suppose now, the mass term ,u is also “octonionic-like,” i.e., it can be expanded as

p=aa,u,*+a,uF+&uO +P;ui. As in Refs. 7 and 8, we also assume that

k 6~ = k ir dR + ikl$,,,,, 1% = I& + and that

(5.3)

iI%, (5.4)

a0 = %R + iaO19 PO = fiOR + ifiO1. (5.5) In the interests of simplicity, we suppose further, that:

k$R = l%R =A% aOR =POR =pOt (5.6)

1391 J. Math. Phys., Vol. 32, No. 5, May 1991

where h z are the GR-Vierbeins, and ,u,, the corresponding mass-term.

Define

k%, = l%,, =pG, aoI =& =pm,, (5.7)

and kti =p%$, I$ =p2efi, ai = p2m], #Si = p2mr. (5.8)

This means that the EM-components on the Vierbeins should be as weak as ofp-order, and the YM-components be as weak as of p2-order, when compared to the gravitational components (see Ref. 14.)

When the above quantities are placed in (5.1), the octonionic Dirac equation is expanded as

[(uo*+%Hw~p --po)]q +ip[(u,*+u,)(n~ybV, -m,)]$ +P2[uf(dy+,, -ml)]+ +p*[u,(e:&,, -m]‘)]$=O. (5.9)

Furthermore, we are going to introduce the approxima- tion of expanding the g(x) in a Taylor series in terms of the parameterp, i.e.,

?e) = $0 (xl + Plct, (xl + fP’h (x) + * * *. (5.10) When this expansion is placed in (5.9)) and then comparing terms of equal powers ofp, it is obtained by the following system of equations:

1: cu,* + uom~4ijp -pFL)rco =o, p: cu,* + &),(h’6Y~, -Pu,fA

(5.11)

+i(uo*+u,)(nb’y,?,, -mo)$o =O, p2: cu,* + udky$ -pu,*2

+i(u,*+u,)(nty,ij, -m,)Ilf,

+ [ut(dr”v, - mr)]&

+ [u,(e$y”T,, -ml’)]& =O,

p3: . ..and so on... .

If we do not consider internal mass terms, one obtains a set of equations similar to the one of Ref. 9, for the split 0 case considered here.

In the limit ofp-+0, one obtains the Dirac equation as in general relativity, for a spin-l/2 particle in presence of the gravitational, the electromagnetic, and of an octonionic-like Yang-Mills fields. As in Ref. 9, the other equations in (5.11), should be interpreted as being the result of a more exact Dirac equation, that is corrections to the Dirac equation obtained in an octonionic-gauge theory local to a curved Riemannian space-time.

In the limit of 1: --+ k: in the expression for the octonionic-Vierbein, and K, -. L, in the expression for I?,, , and 4-x in the expression for $, the set (5.11) will correspond to the quaternionic limit:


1: w,(hyT, -/A)&) = 0,

p: a,(h’U’@p -,uU,lcIt +iwQ(nZyo?p -m,)t,& =0, p2: wo(hZf?p -p)h + iwQ(n$y,?p - mo)+bl

+OjCejY”o, -mi)]$O =O7 p3: . ..and so on... , (5.12) which is equivalent to the SU( 2) case in Ref. 8. In (5.12) the wavefunctions $ are now of the form:

VI. CONCLUSION Taking the ES nonsymmetrical theory for a local tan-

gent space on an extended manifold, for the SU(n) internal symmetry group (developed in the second part of this ser- ial), we were able to obtain a quartenionic theory when the theory was restricted to the case of n = 2. The last extension of the algebra permitted by the Hurwitz theorem was then obtained for the case of the split octonions also using the realization of the algebra via Pauli matrices. This was done, as in Refs. 5 and 6, in a tentative way to keep the physical content of the theory. As an application, we again developed an octonionic Dirac theory, and obtained the corresponding Dirac field equation. As a possible interpretation of the Dirac equation, the wave function $ was then series expanded in terms of the parameter p. In the limit of p-+0, one recovers the Dirac equation as in general relativity, for a spin-l/2 particle in presence of the gravitational, electromagnetic, and of an octonionic-like Yang-Mills fields.

A Dirac theory like the one developed in this work is only meaningful in regions of the curved space-time with high intensity fields, and at distances of the order of the Planck length, where it is reasonable to use a nonzero parameterp. Also, it is noticeable that the presence of an octonionic Yang-Mills field, obtained via extension of the algebra on the internal space, suggests a “double-type” SU (2) -gauge theory, not predicted via normal SU( n ) theory. This is possible due to the realization of the split-octonions via the Zorn (vectorial) -matrices, and the realization of the quatemions via Pauli matrices.

It is noteworthy to include here, a motivation for the use in this work of split-o&onion algebra instead of the real octonions. We could very well develop a similar theory using the real octonions, which consists of a unitary element and seven base vectors. It obviously would still be of Yang-Mills type, as it is always possible to define covariant derivative of a vector, where it would be obtained the octonionic-vector potential, and the octonionic curvature. The homomor- phism group in such a theory is the G2 group. However, this real-octonionic theory cannot be interpreted in such a direct way as the split-octonionic theory developed in this work. Therefore, any attempt related to the use of real-octonions was discarded in the present analysis.

ACKNOWLEDGMENTS

This work was supported by the DOE and by the NASA, Grant Number NAGW-1340, at Fermilab. Part of

this work was developed during a visit to the Centro Brasi- leiro de Pesquisas Fisicas (CBPF), under the financing of FAPERJ, Rio de Janeiro, Brazil.

APPENDIX A: INTERNAL TRANSFORMATlONS FOR QUATERNIONS AND OCTONIONS

When we have a complex unimodular matrix A,

A= a ( 9 Y 6 then its inverse is defined as

The conjugation operation of quaternions and o&onions is equivalent to the inversion of a unimodular matrix.

1. Quaternionic transformations

Let us take U as the quatemionic transformation matrix, or “Q transformation,”

u = m&J, + m&o,, (AlI which is also a transformation matrix of the SU(2) group, i.e., it satisfies the condition

U-‘= UT

where Ut = U’* (A21

is the Hermitian conjugate of U. It. is straightforward to show that

U+zn* = mo*wo - mrwi, (A31 where ?? is the quaternionic conjugate of U. Therefore, U, being a unimodular matrix, must satisfy the property of the inverse unimodular matrix shown above, i.e., U -‘E??, which, with the condition (A2) for matrices of the SU(2) group, implies that the coefficients m,, mi, i = 1,2,3, are real. Thus we have

ZW= UV= U-‘U= UU-’ = (rni + mf)wo = 1, (A41

where m$+mf=detU=l. (As)

Then, we must have U=ei~‘u=e-A’~, (A61

where R = (1 1 ,;1, ,/2, ) are real parameters. We define a Q transformation of a quaternion K by K’= MU-‘. (A71

If K = QQWO + cY$o,, (A81

K ’ is given in terms of components, using (A7 ), as K’= (m,a,m, + m,a,m, + mkaOmt - m,a,m,)o,

+[ moapmo - moaomp -k mpaQmQ f mkakmp

-I- Ekrp(mkalmo - mQakml) - Eijb~kipmiaJmi]~,.

(4491

The symmetry group of this transformation is SU(2), which is homomorphic to the rotation group 0, , This can be ex- pressed through the relation:


Uwi U - ’ = R,w,, i, j = 1,2,3, (AlO) where R, is the transformation matrix for the 0, group. In terms of components this can be written as

m$, + mimp - 2etipm,mj + l tikeklpm,mj = R,. (All)

For the local Q transformations, which are used in the space-time manifold connected to an internal quaternionic space, we must have

V= U(x) = mo(x)wo + m,(x)w; (A121 and

U(x) = e - h(x)-, (A13) where now, h(x) = (2, (x),2, (x),/z,(x)) are real functions. Also the coefficients of K in (A8) are functions of space-time coordinates.

2. Octonionic transformations

When we consider octonions, we may define an octonionic transformation, or 0 transformation, by means of the octonion U, in our case, split

U=pou,* +piuT +qouo +qiui, i= 1,2,3, (A14) wherep, 9, ,qo ,qi, i = 1,2,3, are real coefficients. The conjugate of U is then given by

0 = qous -P,UF +pou, - q,u,. (.415)

Comparing with the situation for the quaternions, we can define:

U-l-6 (A16) Actually,

17) which will be equal to 1 if

PO40 +PiQi = l* In this case, we have for U,

(A 18)

u = e - 6.u’ - y-u f (A19)

where 6 = (S, ,S, ,S, ) and y = ( yi ,y2 ,y3 ) are real parameters.

For an extension of the case valid for quaternions, the 0 transformation of an octonion K is defined by

K’=i[(UK)U-‘+U(KU-‘)I. (A201 We can show easily that

(UK)U-’ = U(KU-‘), (4421) which simplifies the relation (A20) above to

K’=UKU-‘. (A221 If K is written in terms of components by

K=pou,*+piu?+KoUo +KiUi, (A231 we have, after (A22),

I<’ = (pop& +popkqk +pkKoqk --pkKk%)Uo* + (qoKo% + qoKkPk + qkPOPk - qkPkPO)UO

+ [PoppPo - PopoPp f P~KoPo + PpKkPk + Ek/p (hK/qk - qoKkq/ ) - EijkEklpPiPjql I ‘,*

+ [ 4oKpqo - 4oKo4p f qkPOPk + qkpkqp f Ekip (Pkpl% -POPkPI) - E!,kEklpqtKjPI I up- (A241

We are going to define a similar relation to (A lo), for 0 transformations, which is

U(u: + u,)U-’ = ktiu: +Zguj, i,j= 1,2,3. (A25) In terms of components it is

PiSip + PtPp - E;/pqj (PO + % ) f ErjkEk/pP,ql = k;p,

dstp + q,qp - EilpP, (PO + 40 ) f Eijli EklpqjPi = lip,

(A261 with the additional condition

Po4, -PI40 = 0. (A271

From the relations (A26), we can see clearly that in the limit qo,, -+po., , the 0 transformations, through U, will be equivalent to the Q transformations, through U. Therefore, the 0 transformations are homomorphic to the rotation group 0, and so, they must be SU(2)-like (in our case). Also, we observe a certain symmetry with regard to the positioning of the components terms, in expressions (A26).

In the case of local 0 transformations, which we use in the octonionic space-time manifold, we must have:

K = K(x) =po(x)uo* +p,(x)u: +Ko(x)uo +K,(X)u,, (A281

1393 J. Math. Phys., Vol. 32, No. 5, May 1991

I u=“(x) ~~O(X)Uo*+~i(X)U~+qQ(X)UQ +4,(x)ut9

(A291 and

U(x) = e- S(x).“* -y(x).” , (A30) where 6(x) and y(x) are now real functions of the space- time coordinates. APPENDIX B: THE OCTONIONIC WAVEFUNCTION I)

The transformation law for the octonionic wave function $ in the Dirac-equation (4.18)) is defined with relation to the octonionic transformation, as

Y=W, (Bl) where U is given in (A 14). The relation (B 1) can be rewritten in terms of components and/or Zorn matrix as

~=U~=(PQu,*+piu~+qQuQ +QjU,)$,

=ti ,“> t$ where

4,

+=L > , x= x1

2 0 x2

(I321

(J33)


The functions, I$ and x, are “partial” quaternionic-like wave-functions.

The octonionic transformation for $, defined in (4.2 1 ), is given by

+&J-1, WI where U-’ is given in (AlS) and (A16). Again, we can rewrite it in terms of components and/or Zorn matrix as

~=Slu-‘~~~=~(q*u,*--iut+p,u, -qrUi),

=t3 Q’q ;), (I=)

where,

$=(A a; X=(X1 &I, are again, “partial” quaternionic-like wave functions.

The products in (B2 ) and (B5 ) , of the octonions U and u, by the wavefunctions 11, and & respectively, have to be defined in such way that the product $$ remains invariant, i.e.,

&I = (;iu-‘,(uq, = i&b, We define these products in the following way:

(B6)

r; ,“> t$= tf+--q:)’ (B7)

tS n($ ;)= 670 -2s JP +jpcJ). (J38)

Therefore,

iv= (&Jo -xs $Pf,@o) tfi-ip9 0

= hoqo + P-q)4 +,moqo + p’4)x = 54 +,g&$, (B9)

where it was used (A 18). It is important to note that, despite of the “vectorial” form of the results in (B7) and (B8), it would still be obtained as a result, a scalar “column” and “line,” respectively. This is due to the use of the Pauli matrices representation for the quaternion basis (u~,w~,~ = 1,2,3). Also note that this definition works because, in this special case, we use a scalar product when multiplying the vectorial parts of the Zorn matrix. The ad- vantage in doing so, is that we obtain the correct signs in the product $‘$‘.

We can then conclude from the above results that

p+ (q&J--‘,(U$, = $<u-‘U,$= &b, (BlN and that the product &/I is invariant under octonionic transformations.

In the quaternionic limit defined in Sec. V, the partial wave functions # and x, will correspond exactly to the quaternionic (SU (2)) wave function.

Other properties from the above definitions are

A(B?b) # km?‘4 (Bll) where A and B are split octonions (or Zorn matrices), and,

i@W) = (@W, 0312) where K is again an split octonion (or Zorn matrix), or an octonionic operator. It is also easy to show that (B 12) is an invariant under octonionic transformations.

’ A. Einstein, Rev. Mod. Phys. IQ, 35 (1048); A. Einstein, TheMeaning@ Relatiuity (Princeton U.P., Princeton, NJ, 195,5), Appendix 2.

2 M. A. Tonnelat, Les Theories Unitaires de ~‘ElectromagnCtisme e de la Grauitation (Gauthier-Villars, Paris, 1965).

’ W. 3. Bonnor, Proc. R. Sot. London Ser. A 226,336 ( 1954). J. W. Mof- fat, D. L. Boal, Phys. Rev. D 11, 1375 (1975);

4K. Borchsenius, Phys. Rev. D 13,3554 ( 1976). Also, see J. W. Malufand S. Okubo in Phys. Rev. D31, 1327 ( 1985).

‘S. Marques and C. G. Oliveira, J. Math. Phys. 26, 3 13 1 ( 1985). “S. Marques, C. G. Gliveira, Phys. Rev. D 36, 1716 ( 1987). ‘S. Marques, J. Math. Phys. 29,2127 (1988). “S. Marques-Bonham, J. Math. Phys 31, 1478 (1990). “S. Marques-Bonham, “A new way to interpret the Dirac equation in ma non-Riemannian manifold,” Fermilab preprint pub-89/147-A.

“‘See K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, RingsthatareNearly Associative (Academic, New York, 1982), Chap. II. See also, M. Gunaydin, F. Gursey, J. Math. Phys. 14, 1651 ( 1973), and references therein.

” M. Zorn, Abh. Math. Sem. Univ. Hamburg 8, 123 ( 1931). ‘*P. A. M. Dirac, Proc. R. Sot. London Ser. A 117,610 (1928). “J. L. Anderson, PrinciplesofRelativity (Academic, New York, 1967), pp.

358,359. “In the present work wechoosenot tointroduce thefactordz 1 I/pi, which

actsas a comparative factor for the internal parts of the Dirac equation, as in Refs. 7 and 8. In the limit of (now a parameter) p- l//1, and in the special case when the components of the (internal) Vierbeins are equal to the gravitational Vie&ins (i.e., the ones on the unitary axis), the Dirac equation would have equal parts on the internal axis. However, as cam- mented on in Ref. 9, this situation results in canceling of the EM and YM parts of the generalized metric.

‘“SeeC. N. Yang and R. L. Mills, Phys. Rev. 96, 191 ( 1954). As it is used here the Pauli-matrices representation, each of the partial-potentials, L,, and K,, are, independently, of Yang-Mills-potentials type, in the sense that they do not havew,, =o,, components in the Zorn-matrix representation.


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