Jordan algebra dynamics

Volume 210, number 1,2

PHYSICS LETTERS B

l 8 August 1988

JORDAN ALGEBRA DYNAMICS Martin C E D E R W A L L Institute of Theoretical Physics, S-412 69, GOteborg,Sweden Received 22 April 1988; revised manuscript received 4 June 1988

Classical particle dynamics in spaces described by Jordan algebras is discussed, with concentration on invariances, light-likeness and twistor transforms.

The present paper investigates the classical motion of particles in spaces described by Jordan algebras and the corresponding super-extensions, the motivation is to generalise the work of ref. [ 1 ] and to examine the consequences of alternative space-time structures and symmetries. The arising "space-times" do in general not have a norm, but I will show how the concept of light-like lines and twistors are carried over. A certain knowledge of the division algebras is assumed. Mathematical discussions o f Jordan algebras and their symmetries etc. can be found e.g. in refs. [2,3 ]. Standard papers on twistors are found in ref. [4]. The division algebras are denoted K , = ~ , C, H, O for v = 1, 2, 4, 8, respectively. The elements of H,, ( ~ . ) , (n × n )-hermitean matrices with entries in N., form a Jordan algebra with respect to the commutative, but in general nonassociative, product x o y = ½( x y + y x ) ,

(1)

i.e. the product satisfies Xo(X2oy)=x2o(xoy) ,

(2)

for any n when v ~<4 and for n ~<3 when v = 8. Such a matrix contains ½n ( n - 1 ) u + n independent real components. It is known that the case n = 2 relates to particle dynamics in d = v + 2 Minkowski space and its twistor formulation [ i ]. When n = 2, Postal address: Haga 0stergata 4b, S-413 01 G6teborg, Sweden.

satisfies the generic equation X 2 -- T ( X ) X " ] - O ( x )

= O,

(3)

where T ( x ) =~, +~2

(4)

is the trace, and

O(x) = ½ [ T ( x ) 2 - T ( x 2) ] = ~, ~2 - aa

(5)

is the Minkowski norm. Obviously, the time variable is T ( x ) . It is unchanged under rotations o f the spacelike coordinates, i.e. under g = s o ( v + 1 ), which is the algebra of derivations of the Jordan algebra [3], D e r H 2 (N,). Q ( x ) is invariant under the full Lorentz algebra which is the reduced structure algebra [3], ~ = S t r ' H2(Kv)=sl(2; ~ ) = s o ( 1, v + 1 ). A full analysis is given in ref. [ 3 ]. Taking the step to particle mechanics, the on-shell statement Q(P) =0,

(6)

following from an action S = I d r [ T(J(oP) - V Q ( P ) ] = f dzlO(j(),

(7)

is Lorentz invariant. Using the generic equation (3) it is reformulated as 169

Volume 210, number 1,2

p2 = T ( P ) P ,

PHYSICS LETTERS B (8)

which I will refer to as P being scale-invariantly idempotent. It is this property that makes a twistor formulation possible. Any P fulfilling eq. (8) can be written P = 7-'g-a

(9)

with ~P a column 2-spinor with entries in ~,. The twistor variable 7/is (eomponentwise) canonically conjugated to

o~= ~ x

(lO)

and the full twistor Z = [S] satisfies the ("spin-shell") constraint

ZgZ* =0

( 11 )

( v ¢ 8 ) , where g is the symplectic metric [_o ~]. A thorough discussion oftwistor space, including its relation to ordinary space-time and quantisation, is found in ref. [ 1 ], except for the case v = 8, which will be the subject o f a separate publication. For what follows, I would like to stess two points. Firstly, eq. (9) for a null-vector (i.e. a scale-invariantly idempotent matrix) gives a clear characteristic of the light-cone. Identifying P's differing by real rescaling, eq. (9) has an invariance amounting to right rescaling o f ~ with an element in ~ , (to be exact, when v = 8 the components are multiplied by different elements in ~3; a field-dependent S7-transforma tion). ~ubecomes a set of homogeneous coordinates for the projective space 0%PL= S", clearly displaying the (quite trivial) topological structure of the proj ecrive light-cone. Secondly, the constraint ( 11 ) manifests the entire conformal invariance of the theory under e ~ = C o n H 2 ( ~ , ) = s p ( 4 ; ~ , ) = s o ( 2 , v + 2 ) [31. Turning to (3 × 3)-matrices and the algebras H3(~v), among which is the exceptional Jordan algebra H 3(O), things become more complicated, but by following the same lines of thought as for n = 2, I will draw some conclusions that generalise the ordinary space-time picture. The generic equation for

X=

170

~2

a

a

~3J

eH3(~.)

18 August 1988

is now third order:

x 3- T(x)x2+ Q(x)x-N(x)

=0,

(12)

with T as the trace, Q still being a Minkowski norm,

Q ( x ) = ½[ T ( x ) ~-- T ( x 2) ]

=~2 +~2~3+~3~ -aa-6b-¢c,

(13)

and N a determinant function

N ( x ) = -~[2 T ( x 3) - 3 T ( x 2) T ( x ) + T(x) 3 ] =~l~2~3-~,da-~2bb-~3¢(.'-}-2 Sc (abc)

(14)

(Sc z = ½( z + g) ). It is easy to convince oneself that the signature of the space (with respect to Q) is -t ... and that T ( x ) plays the role of a time coordinate. T ( x ) and Q(x) are invariant under ~ = Der H3 (K~) that generalises the algebra of rotations among the space-like coordinates. N ( x ) has the full invariance under the "Lorentz algebra" ~ = Str' H3 (K~) = sl (3; ~ ) . These algebras are listed in table 1. The coordinate X transforms under a representation r = 3( v + 1 ) of 5 °. There is also a representation r, contained in the tensorial product r × r. The Jordan algebra does not distinguish between r and r (this is quite analogous to the principle of triality for so (8) in the octonionic algebra), and in order to manifest the ~-invariance the product must be covariantised. This is achieved by the "Freudenthal cross product"

x×y=xoy-

½[ T ( x ) y + T ( y ) x ]

+½ [ T ( x ) T ( y ) - T(xoy) ] .

(15)

Then, if x, ye r, x × ye r, and vice versa. One can form a scalar function r × r × r ~ 1:

S(x, y, z) = T ( ( x X y ) oz),

(16)

Table 1 Symmetry groups for n = 3. For simplicity in notation the compact versions are given. Group

v 1

2

4

8

so(3) so(3) sp(6)

su(3) su(3) 2 su(6)

sp(6) su(6) so(12)

F4 E6 E7

V o l u m e 210, n u m b e r 1,2

PHYSICS LETTERS B

completely symmetric in the three arguments. One also finds the relations

(x×X)ox=N(x)~,

(17)

T(x×x) = Q(x),

(18)

Table 2 Effective d i m e n s i o n a l i t i e s . p

l

4N(a)

/1 2

and the inversion formula

y=aXx~,x= - -

18 A u g u s t 1988

[ ( a X a ) X y - T(aoy)a].

3

4

5

6

[

3

4

5

6

7

2 4 8

4 6 l0

6 10 18

8 14 ...

10 18

12 22

... ... ...

(19) The situation is now that the "space-time" is equipped with a certain invariance group 50but it is not a normed space. In ordinary Minkowski space the norm is (classically) used only to select the null-vectors (as long as no massive particles ase involved), defining a light-cone. If there are alternative ways of doing this, there is no reason to insist on the existence of a norm. It might also happen that a (lowerdimensional) orthogonal structure arises via symmetry breaking from a space without a metric. The scale-invariant idempotency condition of eq. (8) provides with a fruitful generalisation of the onshell condition. To see this, consider the possible covariant constraints. N ( P ) = 0 is ruled out, being cubic in momenta. One is left with PXP=0,

(20)

superficially containing 3 ( v + 1 ) equations, but in fact being degenerate so to give only v + 2 real independent conditions. Using eqs. (17) and (18) one gets

P X P = beN(P) =p2

T ( P ) P = Q(P) ,

(21)

stating that a P fulfilling eq. (20) extremises N ( P ) and that eq. (20) is equivalent to the scale-invariant idempotency of P (The same two equations give the on-shell vanishing of N ( P ) and Q(P)). The 50lightcone lies entirely in the Minkowski light-cone, and is exactly the part of it invariant under 5 °. Eq. (20) is obtained from an action S= f dr[T(J(oP)-½S(V,P,P)],

(22)

or, using eq. (19) and ~'= V,P,

The projective light-cone is the space of idempotents with trace 1 (the real scale has been factored out), a known realisation o f ~ P 2. In the twistor formulation, still formally given by eqs. (9) and (10), appears as a set of homogeneous coordinates for this space, except when v = 8 , in which case there seems to be no twistor transform (a result connected to the non-existence of homogeneous coordinates for Qp2).

The twistor zero-norm relation ( 11 ) manifests the conformal invariance ~ = C o n H 3 ( G . ) = s p ( 6 ; G,,) [3] (of course present also for p=8). It is tempting to introduce an "effective dimensionality" d¢rf defined as two plus the dimensionality of the projective light-cone (see table 2). Of course, this latter dimensionality is significant as measuring the physical degrees of freedom, and d, frjust provides a comparison in a Minkowski space terminology. It is striking that the two models with n=2, u=8 and n = 3, v = 4 both share the same effective dimensionality and the same conformal algebra, though the topologies of the light-cones differ. For v ¢ 8, the discussion can be continued to higher n. The on-shell constraint (8) always contains a number of equations equal to the number of components in an element ofHn_ j (0~), and there is always a twistor transform. The topology of the projective light-cone is G.pn-~, and the effective dimensionality d, fr= ( n - 1 ) u + 2 (table 2). Any value of n gives a magic square of algebras, given in table 3. Supersyrnmetrisation is easily come about by everywhere substituting X with J ( + i ( 0 0 * - 0 0 * ) , 0 being a column spinor of sl(n; ~ , ) [5,6,1J. The super-twistor transforms [ 7,6,1 ] take the form p = ~t.,t,

S = f drs-~-~1

{T[ ( VX V)o (J(XJ() ]

o9= 7u*( X - i O 0 ~) , - [ T(VoX)

] 2}.

(23)

~=x~Frt0,

(24)

171

PHYSICS LETTERS B

Volume 210, number 1,2

References

Table 3 Symmetry groups for n arbitrary. Group

v

~'

l

2

4

so(n) su(n) sp(2n)

su(n) su(n) 2 su(2n)

sp(2n) su(2n) so(4n)

a n d t h e c a n o n i c a l s t r u c t u r e is t a k e n o v e r to t h e su-

iol

persymplectic metric

g~

-1

o

I

I

172

i

•

18 August 1988

[1] 1. Bengtsson and M. Cederwall, Nucl. Phys. B 302 (1988) 81. [2] R.D. Schafer, An introduction to nonassociative algebras (Academic Press, New York, 1966). [3] A. Sudbery, J. Phys. A 17 (1984) 939, [41 R. Penrose, J. Math. Phys. 8 (1967) 345; Intern. J. Theor. Phys. 1 (1968) 6t; R. Penrose and M.A.H. McCallum, Phys. Rep. 6 (1972) 241; L.P. Hughston, Twistors and particles, Lecture Notes in Physics (Springer, Berlin, 1979 ). [5] R. Casalbuoni, Nuo,,'o Cimemo 33A (1976) 389. [6] A.K.H. Bengtsson, I, Bengtsson, M. Cederwall and N. Linden, Phys. Rev. D 36 ( 1987 ) 1766. [7] A. Ferber, Nucl. Phys. B 132 (1978) 55; T. Shirafuji, Progr. Theor. Phys. 70 (1983) 18.