Volume 256, number 2 PHYSICS LETTERS B 7 March 1991

Supersymmetry and maximal acceleration

M. T o l l e r Dipartimento di Fisica dell'Universitgt Trento, and lstituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trent, Italy

Received 19 October 1990

We discuss a supersymmetric classical field theory in which the anticommutators of supersymmetry generators generate both the local translations and the local Lorentz transformations. This feature is suggested by the existence of an upper limit to accel- eration due to quantum gravitational effects. The spontaneous breaking of supersymmetry is caused by a spinor field ~ with non- linear transformation properties similar to the one introduced by Nakanishi.

We describe the mot ion of an extended body in Minkowski space by means o f a moving frame 2--, s (2), The inf ini tes imal Poincar6 t ransformat ions acting on it when the pa ramete r 2 varies are given by the following elements of the Poincar6 Lie algebra:

to the velocity r, but no l imi ta t ion to the other quan- tit ies to and a that appear in eq. (2) . The existence of an upper l imit to accelerat ion has been suggested in refs. [4,5] and Brandt [6,7] has shown that the inequal i ty

ds (2 ) = b k A k + ½blrs]At~s I , ( 1 ) d2

where the elements Ak and A [rsl = --A tsr] form a basis of the algebra and represent, respectively, infinitesi- mal spacet ime t ranslat ions and inf ini tes imal Lorentz t ransformat ions . The vectors

f 2 ~ , b O , ,

{bt231 bt3'l bll2l , bo

/bl,Ol br2Ol b.Ol a = k if6 , b o , b o j (2)

Ilall <~c2l - l , l = k l p l = k ( h G c - 3 ) w2 (3)

should appear as a consequence of the ideas of quan- tum theory and general relat ivi ty (k is a numerical factor near to one and lpl is the Planck length). Only after this deve lopment a symmetr ic t rea tment of all the generators of the Poincar6 group becomes possible.

The general form of a condi t ion on the right hand side ofeq . ( 1 ) describing a l imit to accelerat ion and angular velocity has been der ived in ref. [4] under rather reasonable assumptions. It can be writ ten in a s imple form by introducing a Majorana spinor for- mal ism and by replacing the quanti t ies b* and b trs]

by the real symmetr ic 4 × 4 matr ix

are, respectively, the velocity, the angular velocity and the accelerat ion of the moving frame s (2) measured with respect to the frame itself. I f we consider rest frames we have v= 0.

Several authors [ 1-3 ], with various mot ivat ions , have suggested that a physical theory should treat symmetr ica l ly all the coordinates o f the Poincar6 Lie algebra. This symmet ry does not exist in the usual relat ivist ic theories, in which there is an upper bound

bAB=l - lbk (C- l~ )k )AR- -½b t r s l (C- l y rYs )AB . (4)

We adopt the metr ic tensor - goo = gl ] = g22 = g33 = 1 and real 7 matrices. Then we can put C = 7o. The con- d i t ion is that the matr ix b A8 has to be semidefini te posit ive, namely

UA bABuB >/0 ( 5 )

for every real spinor u. This condi t ion implies the

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Volume 256, number 2 PHYSICS LETTERS B 7 March 1991

limitation (3) and defines a cone stable with respect to the transformations

b ~ a V b a , aeGL(4, ~) . (6)

Note that these transformations mix the coordinates b k and b M. The kinematics which follows from these ideas is discussed in ref. [8] and some attempts to build classical and quantum field theories are given in ref. [ 9 ].

In this framework, also the four-momentum Pk and the relativistic angular momentum Ptrs] have to be treated in a symmetric way. In particular, if we intro- duce a supersymmetry assumption with four super- symmetry generators QA, their anticommutators must have the GL (4, N ) covariant form

[QA, QB]+ = - - l ( y k f ) A B e k - - ½ ( y r ~ s f ) A B P [ r s ] . (7)

This formula can be considered as a new formulation of the maximal acceleration principle, if we specify that the generators of "feasible" Poincar6 transfor- mations are just the sums of squares of supersym- merry generators.

Note that in the "macroscopic limit" l-~0 we do not obtain the usual supersymmetry [ I0,I I ], but a different theory, which has been studied by Nakani- shi and Abe [ 12,13 ]. The last term in eq. (7) is usu- ally excluded (see e.g. ref. [ 11 ], p. 204) because it is not compatible with the structure constants of the Poincar6 algebra and the Jacobi identities. It follows that the supersymmetry has to be spontaneously bro- ken, namely the vacuum state cannot be supersym- metric. Note that the vacuum, described by the struc- ture constants of the Poincar6 Lie algebra, is not even invariant with respect to the group GL(4, ~ ). Only a subgroup locally isomorphic to the orthochronous Lorentz group survives the symmetry breaking.

In order to treat a classical model, we consider a graded commutative algebra of fields, in which pairs of fermionic fields anticommute and bosonic fields commute with everything. We concentrate our atten- tion on the algebraic properties of the model [ 14,15 ], leaving the more geometric aspects to a following more detailed paper. Infinitesimal parallel displace- ments and infinitesimal local Lorentz transforma- tions are represented by derivations of the field alge- bra, which we indicate by Ak and At~s~ and the infinitesimal supersymmetry transformations are represented by the anti-derivations AA. The anti-

commutation relation (7) takes the form

[AA,AB]+ =--i l(ykC)ABAk--½i(~rTsC)ABAtrsl (8)

and we write the other commutation relations in the form

[A~,,At~ ]_ _F~pA~ c = - F ~ p A c , (9)

[A~, AB] r c _ = - F ~ s A y - F , B A c , (10)

where the greek indices a, fl, ~ label a basis of the Poincar6 algebra and stand for a four-vector index or a pair of antisymmetric tensor indices. The "struc- ture coefficients" F on the right hand sides are not necessarily constant numbers: for instance in the presence of gravitation the quantities F~[ sj are the components of the curvature tensor. As a conse- quence, the Jacobi identities are not algebraic rela- tions between these coefficients, but give rise to gen- eralized Bianchi identities, which contain derivations of the fields F.

In the following we describe some solutions of the generalized Bianchi identities which are compatible with eq. (8) and describe sourceless gravitational fields. In these solutions the local Lorentz transfor- mations act in the usual way, and the structure coef- ficients with a subscript of the kind [ rs] have the fa- miliar constant values, for instance:

i - - i B F [rs]k -2gk[~s ] , F[rs] A = 1 ( ~ [ r ~ ) s ] ) A B . ( 1 1 )

We assume that torsion is absent and the curvature satisfies the Einstein sourceless field equations:

F~k=0, F}['] = 0 . (12)

The usual Bianchi identities are also assumed:

eJiksF![, "1 = 0 , eJlikAiF![, ~l = 0 . (13)

The other structure parameters are given by the formulas

C FiA - - 0 , (14)

F~A = -- ~ (7~U~)A , (15)

F!ff qj = ll(TiTr~s~)AFtrsPql , ( 16 )

FAk = ¼i(C-'~r~'s~) A V![ s~ , ( 17 )

where ~A is a fermionic spinor field introduced by Nakanishi [ 12 ] (see also ref. [ 16 ] ), which avoids the appearance of the "gravitino" degrees of free- dom. It has the properties

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Volume 256, number 2 PHYSICS LETTERS B 7 March 1991

Ars~= l -~71,7s1~, A k ~ = 0 , (18)

AA~e =iCA, - ½~A~, + ½ (75~)a (75~), • (19)

The fermionic derivations of the curvature are given

by

AAF[[n'q = - A i F ~ 'm] + AkFt,~ "1 + F ~ F ~ m"l

]ff'j l~ [ m n ] ----iA--kj q-l(~2[mTn]~rYs~)AFl~ sl (20)

and the fermionic derivations of all the other struc- ture coefficients can be obtained from the formulas given above.

Under these assumptions, one can show that all the commuta t ion relations and all the Bianchi identities are satisfied. Note that the sourceless Einstein field equations (12) are necessary in order to get these re- suits. The vacuum state is characterized by the addi-

tional condit ions

F [ [ ~j = 0 , ~A=0- (21)

We see from the inhomogeneous nature of eq. (19) that the last condi t ion is not invar iant under infini- tesimal supersymmetry transformations and super- symmetry is spontaneously broken.

The solutions described above are interesting, be- cause they show that eq. (8) is compatible with sev- eral other requirements. However, the extension to gravitational fields interacting with matter is not im- mediate. Moreover, these solutions do not agree

completely with the ideas from which we have started: in fact, eq. (3) cannot be satisfied with l> 0 and h = 0 (classical theory). One can speculate about the pos- sibility that eq. (3) follows from a quan tum treat- ment of eq. (8) or from the stronger requirement of

a broken symmetry of the quan tum theory under the group GL (4, ~ ) [ or at least under its subgroup Sp (4, R) ] . In this case, for instance, eqs. (12) and (14), which have not the required symmetry, cannot hold for the quan tum field operators, but only for their vacuum expectation values.

References

[ 1 ] F. Lurqat, Physics 1 (1964) 95. [2] M. Toller, Nuovo Cimento B 40 (1977) 27. [3] T. Regge, Phys. Rep. 137 (1986) 31. [4] M. Toiler, Nuovo Cimento B 64 ( 1981 ) 471. [5] E.R. Caianiello, Lett. Nuovo Cimento 32 (1981 ) 65. [6] H.E. Brandt, Lett. Nuovo Cimento 38 (1983) 522. [7] H.E. Brandt, in: Proc. Fifth Marcel Grossmann Meeting,

eds. D.G. Blair and M.J. Buckingham (World Scientific, Singapore, 1989) p. 777.

[8] M. Toiler, Intern. J. Theor. Phys. 29 (1990) 963. [9] M. Toiler, Nuovo Cimento B 102 (1988) 261.

[ 10] P. Fayet and S. Ferrara, Phys. Rep. 32 (1977) 250. [ 11 ] P. van Nieuwenhuizen, Phys. Rep. 68 ( 1981 ) 191. [ 12] N. Nakanishi, Prog. Theor. Phys. 77 (1987) 1533. [131 M. Abe, Intern. J. Mod. Phys. A 17 (1990) 3277. [ 14] R. Geroch, Commun. Math. Phys. 26 (1972) 271. [ 15 ] G. Landi and G. Marmo, Phys. Left. B 193 ( 1987 ) 61. [ 16] H. Kanno, Intern. J. Mod. Phys. A 4 (1989) 2765.

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