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Contractions yielding new supersymmetric extensions of the poincaré algebra
Vol. 30 (1991)
REPORTS
CONTRACTIONS EXTENSIONS
ON MATHEMATICAL
YIELDING
PHYSICS
No. 1
NEW SUPERSYMMETRIC
OF THE POINCARI? ALGEBRA
J. A. DE AZCARRAGA Department of Theoretical Physics, University of Valencia and IFIC (CSIC), 46100 Burjasot (Valencia), Spain J. LUKIERSKI Institute for Theoretical Physics, University of Wrodaw, 50 205 Wrodaw, Poland and J. NIEDERLE Institute of Physics, Czechoslovak Academy of Sciences, 180 40 Prague 8, Czechoslovakia (Received April 12, 1990)
Two new PoincarC superalgebras are analysed. They are obtained by the Wigner-In6nl contraction from two real forms of the superalgebra OSp(2; 4; C) - one describing the N = 2 anti-de-Sitter superalgebra with a non-compact internal symmetry SO(l, 1) and the other corresponding to the de-Sitter superalgebra with internal symmetry SO(2). Both are 19-dimensional self-conjugate extensions of the Konopel’chenko superalgebra. They contain 10 PoincarC generators and one generator of internal symmetry in addition to 8 odd generators half of which, however, do not commute with translations.
1. Introduction
The construction of various extensions of the Poincart algebra - the symmetry algebra of relativistic physics - is closely related with two developments in particle physics. The first is an attempt to unify the Poincari algebra with the internal symmetry algebra within a large Lie algebra or superalgebra; the second development is the gauge or supergauge descriptions of existing as well as hypothetical fundamental interactions in nature. As is well known, the difficulties in realizing the first programme led to a series of no-go theorems both in the Lie algebra [l] and superalgebra frameworks [2] (for details see [3]). The aim of this paper is to consider new real supcrsymmetric extensions of the PoincarC algebra P. These extensions of P go beyond the standard axiomatic framework of QFT (in particular the axiom of positive-definite metric in the space of states) and were not treated by Haag, Lopuszafiski and Sohnius in [2]. The first alternative to the supersymmetric extensions classified in [2] was found by Konopel’chenko [4]. However, as shown in [3], [5], Konopel’chenko’s superalgebra can hardly be used in particle physics since it is a 1331
34
J. A. DE AZCARRAGA,
J. LUKIERSKI
and J. NIEDERLE
complex Lie superalgebra (due to the fact that complex conjugation is not an involution) which does not admit hermitian representations (except trivial ones) for odd generators. But this unphysical feature of Konopel’chenko superalgebra might be overcome as the extension recently proposed by Khan [6] and the one discussed in this paper indicate. In fact, we shall show that both extensions can be derived by contracting the appropriate real forms of the orthosymplectic superalgebra OSp(2; 4; C).i Thus, in Section 2, all real forms of OSp(2; 4; C) which can be connected with the symmetries of the Minkowski space-time are introduced. They are three - OSp(2; 4; R), OSp(l, 1; 4; R) and UU,(l, 1; 1; H) (Appendix I) - in accordance with the classification of real simple Lie superalgebras given in [7]. Section 3 is devoted to their contractions. It is shown that a contraction of the N = 2, D = 4 ant-de-Sitter superalgebra OSp(1, 1;4; R) whose internal symmetry sector consists of the non-compact algebra SO(l) 1) yields Khan’s proposal. Another supersymmetric enlargement of P is derived by contracting the N = 1, D = 4 de-Sitter superalgebra UU,(l, 1; 1; H) - the real form of OSp(2; 4; C) whose odd part is subjected to a quaternionic or symplectic Majorana reality condition. Finally, the third real supersymmetric enlargement of P is obtained by contracting the superalgebra OSp(2;4; R), whose internal symmetry consists of SO(2) and whose odd part is given by two-four-component Majorana supercharges. It is just the conventional real N = 2 Poincare superalgebra treated in [2] and, therefore, it will be only mentioned here. In Section 4 some final remarks concerning possible applications of the derived superalgebras are presented. 2. Three real forms of OSp(2; 4; C) The complex orthosymplectic superalgebra OSp(2; 4; C) is determined by the commutation relations of the algebra SO(5, C) @ SO(2, C)
Wp~,Mp~l = i(g,&L, + gvpMILo - g&&o - gvc#pp) , [M,,, RI = Gvx~p - g,xPv)>
(2.la) (2. lb) (2.lc)
[P,, P,,] = -im2MpLv,
[h&w1 = 0, PTP,l = 0,
(2.ld) (2.le)
plus the relations
Oil= -l/W,v)ABQ;, [P,,QAl= -1/2m(y,)ABQ&,
[Mclv,
[t3,Q\]
= -i@Q;
,
(24 (2.W (2.lh)
and {Q>, Q”,} = -@j[(y,C)~~p~ ’ Our discussion
can, however,
+ 1/2m(C,,C)ABMP”]
be extended
+ i@rnCABB,
to any OSp(2; N; CT) with even N.
(2.li)
CONTRACTIONS
YIELDING
EXTENSIONS
OF POINCARi
ALGEBRA
35
where p,u,(~,X = 0.1.2,3: i.,j = 1.2; A,B = 1.2.3.4 and gr,,,, -,1v, y 3’V,-~~3 and C are defined in Appendix II. Thus, the complex superalgebra OSp(2; 4; C) supersymmetrizes the algebra SO(5, C) 9 SO(2, C); it includes, besides 11 even generators (&Lb,,, Pk, and B), 8 odd generators written as two independent , non-self-conjugate, four-component spinors &>, i = 1.2. The 19-dimensional complex superalgebra OSp(2; 4; C) may also be considered as the 38-dimensional real superalgebra OSp(2; 4; C)n of generators (MV,,,, iM p’/, ‘PIL,iPLl,B, il3. Qi. i&i). Any real subalgebra of OSp(2; 4; C)n whose complexification is equal to OSp(2; 4; C) is a real form of OSp(2;4;C). It turns out that any real form of OSp(2; 4; C) is the subalgebra of fixed points of an involutive semimorphism of OSp(2; 4; C) [7]. Consequently, the real forms of OSp(2; 4; C) can be obtained by finding involutive semimorphisms of OSp(2; 4; C) in the group of automorphisms of OSp(2; 4; C). With our metric, the y matrices (and C = rO) are purely imaginary in the Majorana realization, and (y”C) and (C,,,C) are real. Nevertheless, since the defining relations C = -CT are invariant under the mapping Y” = YO+,Ti = -?‘I+, Cyp”c-’ = +T, Qi
= [lQli,
.-/‘l
= u+-f”u,
C’ = UTCU
(2.2)
with U+U = 14, U E U(4), equations (2.1 f-i) are, of course, valid for any realization of the 0(1 13) Dirac matrices. In particular, in the Weyl realization we may write Q;
=
($f).
Q",
= (z)
(2.3)
in terms of four independent SL(2. C) spinors. Different pairs of self-conjugate spinors may be extracted from the two general QA by imposing different reality conditions. By requiring them to be involutions of the superalgebra (2.1), we can halve the number of the generators of OSp(2; 4; C)R to obtain the associated real forms of OSp(2; 4; C). 2.1 N = 2 anti-de-Sitter superalgebra with compact internal symmetry: OSp(2; 4; R) The simplest reality condition spinors,
is to require that the Qi
themselves are Majorana
Q\ = CyoTQ$ I (QTJ)C. In the Weyl realization (2.3), eq. (2.4) leads to R, = Qa, T, = S,. that (2.4) is an involution of (2.1) gives P@ = P;
= P,,.
M,,
= Mzy = A&,
B=U+-B
(2.4) The requirement
(2.5)
so that the even part (P,, n/r,,,, B) generates the algebra SO(3.2) $ SO(2) with spacetime metric (+, -, -, -; +). Thus, for Majorana Q;a and Hermitian P,,, iIf{,, and B. (2.1 j describes the real superalgebra OSp(2: 4; R), well-known in supersymmetric theories of a world with anti-de-Sitter geometry (see, e.g. [8] and also [9]).
2.2 N = 2 anti-de-Sitter superalgebra with now-compact internal symmetry: OSp( 1, 1; 1: R) This corresponds
to choosing Q;
=
(#(Q;)c
WI::
36
J. A. DE AZCARRAGA,
J. LUKIERSKI
and J. NIEDERLE
which in (2.3) leads to the conditions S, = R,, T” = QQ. The linear combinations (2.7) allow us to write the Qf4 which satisfy (2.6) in terms of two Majorana spinors Q> = (Q>)C. The condition that the new reality condition (2.4) is an involution of (2.1) leads to Q,, = P; =P@.
M,,
= M;”
a = -B+
= kf,,,,
E iB’,
w3)
Comparing with (2.5) we see that generator B of the compact SO(2) has been replaced by the generator B’ of the non-compact SO(l, 1). The commutators of A.JWu,P@ are as in (2.1 a-c) and the rest of the superalgebra is determined by [B’, PJ = o>
(2.9a)
[B’, ,%&,I = 0.
(2.9b)
[h/l,,, Q,] = -1/2(C,,)AB& [PkL,&I = -m/2(-r,,JAB& LB’, Q;] = -i(rlE)i.jQi , {ai,
9j,}
= -(q)zJ[(ypc)Pp
(2.9~)
:
(2.9d)
,
+ ‘m/2(~pl,C)hfpv],JB
(2.9e) + i’mc”c~BB’,
(2.9f)
where the inner metric is now 7 = g3 and thus (7~) = 0 I. Consequently. B’ generates the SO(l, 1) rotation (2.9e) SQL N (~l)~j&, and the real superalgebra (2.la-c), (2.9a-f) is the anti-de-Sitter superalgebra OSp( 1; 1; 4; R). Supersymmetric theories based on antide-Sitter superalgebras OSp(p, 4; 4; 12) have already been considered by some authors (see e.g. [lo]). However, due to the appearance of the semidefinite metric n E g3 in (2.9f), the superalgebra OSp(1, 1;4; R) leads e.g. to supergravities with one physical and one “ghost” gravitinos. 2.3 de-Sitter superalgebra with compact internal symmetry: UU,(l,1; 1; H) This is obtained by imposing the so-called symplectic [ll-131 or quaternionic Majorana reality condition ?sQ\
= EijcyaTQjAI E
&J(QJA)C
[14] (2.10)
which is fulfilled if we put S, = -R, and T” = Q” in the expression of Q’, [(2.3)]. As before, one can obtain from the QA fulfilling (2.10) two independent self-conjugate (Majorana) spinors Qi now given by 0; = 5’“;
- r5Q;),
i?‘A = -$+?Qi
+ Q6>.
(2.11)
Imposing that (2.10) is an involution of (2.1), we get the conditions PW = -P;L’ = iK,,
M,,
= M$
= ikf,,,,
t?=l?+=B.
(2.12)
Introducing an SO(5, C) covariant notation with j&b, a, b = 0, 1,2,3; 5, we see (Appendix II) that the replacement Pp = iK, changes the metric to the de-Sitter one (+, --, -, -; -).
CONTRACTTONS
YIELDING
EXTENSIONS
OF POINCARk
ALGEBRA
37
Thus, the real form obtained from (2.10) is the superalgebra UU,(l: 1; 1;H) with inner symmetry U(H) - SO(2). The commutators of the pure even part are (2.1 a,b,d,e) {with M/&v = iwp”, PP = iK,, 23 = B) and [KIL: K,]
(2.13)
= im2hfp,,
The mixed and odd part, expressed in terms of Q-4 E Qa and Q, (Q,C determined by [~~pv,Q~l
=
-1/2(cFuQ)~,
[~lJpu,O,tl
[KWL)QAI= i/2m(T,Q)~; [B, QA] {QA:
QB}
[K,,
= ~($CCI))A, =
Q.,]
[B,QA]
-~(Y~C)ARK’
-
(2.14a)
= ~/~(Q~w)A,
(2.14b)
= ~P~~(QY,)A.
(2.14c)
= ~($CQ)A.
(‘.14d)
112n~(CpvC),~~~““,
= 2m&B
{QA. Q,}
j
(2.14e)
{Q;1. QB} = Z(C~J.~BK~‘ - l/2 t?6(~~p,)A&‘f’li’. Another expression for UU,(l, Majorana spinors (2.11). Then {@,.&} (&,
Q;}
1; 1; H), which is explicitly (2.14 d-f) take the form f0:
= -l/2rn(CC,,V)ABi\lPV = i&iJ{($“iilC)A&? {B,QA}
= Q’,T~), is
i = 1
+ mC.413B}
(2.14f)
real, is obtained
by using the
and i = 2,
(2.15aj
for i #
(2.15b)
.A.
= --1m&3QJA.
j ,
(2.15c)
It can be seen (e.g. from (2.14e)) that Tr(Q.4, Qp} = 0 and thus the superalgebra UU,( I. 1: 1; H), which is associated with supersymmetric theories in a world with de-Sitter geometry. can be realized in a quantized theory only in a space of states with indefinite metric [12, 151. Recent discussions of models with local de-Sitter symmetry [12, 161 show that the ghost problem is also present in de-Sitter supergravity.
3. Contractions
of real forms of OSp(2; 4; C)
3.1 OSp(2; 4; R) It is well known that the contraction of OSp(2;4; R) leads to the N = 2 Poincare superalgebra with or without a central charge [17]. Let us then discuss the contractions of
the two other real forms; to this aim we write first the odd generators of OSp(l, 1; 4: R) and UU,(l. 1; 1;H) in terms of the Weyl charges R,, Q, and their complex conjugate ones. which determined the different versions of self-conjugate spinors. 3.2 OSp(l,1;4;
R)
This algebra can be read from (2.1) by using (2.6) and = (Ra, Q”). ‘The even part is given by (2.1 a-c) with M,, (2.9a,b). The rest of OSp(1 ~I; 4; R) is given by
2.8) which requires (3: = M{,,,, F’,, = F’,&plus
38
J. A. DE AZCARRAGA,
J. LUKTERSKI
and J. NIEDERLE
{R”,Rj} = o>
{Qa.Q,g} = 0.
{Q,, Rj}
{&Cl,.R&j} = -mE,;,R’.
(3.ltJ)
= (o&P
(3.lc)
and [‘ld/Lv. QN] = -l/2
[A$,.
(c+,)o~Q~:
&a]= -l/2 m(a,&R’,
R
[P,l. R,]
[R/L>
[B’, Qa] = -R,
~
(c~,,)Q~R,J
= -1/2~z(a,),~Q’~,
[B’> R”] = -Q’
.
(3.ld) (3.le) (3.lf)
plus their complex conjugate relations. The generators (Al,,,, R,, R’) determine a subalgebra of OSp( 1,l; 4; R) with respect to which we perform the Wigner-Inonti contraction (we could equally choose (M,,,, Q,,, Using the same symbols for the generators of the contracted algebra, we obtain that its defining commutation/anticommutation relations are given by (2.1 a, b, d, e) for A[~,, Pp, B’ together with
Q").
[R,,RVl {Qa, QJ} = 0,
= 0,’
(3.2)
{R% R”} = 1/2m(a,,)“%kY,
(3.3a)
{QcuQg} =O> {R,,Rfi} =O, {QAa} = (oJnOPp {Qa,Ro} = -~7wd3’,
(3.3b) (3.3c)
and [fief,>
RN] Qal= -W(qw)~aQs,[M/w. [B’, Rd] = -Q”,
[I’,, R”] = -l/2
= -1/2(0,&‘R~.
(3.3d)
~(cJ”~Q,,
(3.3e)
,
=0 [B',Qal = 0, [J',,Qal
(3.3f)
to which one has to add the complex conjugate expressions. The above superalgebra the one described by Khan [6] (Q and R correspond to T and 5’ in Khan’s notation). 3.3 U&(1,1;
is
l;H)
Similarly to the previous case, the superalgebra UU,(l, 1; l;H) is read from (2.1) by using (2.10) which requires Q 54 = (-RCY, Q”) and (2.12). In this way we get for the purely even part (2.1 a, b, d, e) (with M,, = iVlp,,, Pp = iK,, B = B) and (2.13). The rest of UU,(l, 1; l;H) is given by (3.1. a, b, d) and
{Qa,Ra] = -imE&, {Q,,Ra] = i(n),,+', [K@, Qa] = i/2m(o,),bRR, [R,
Qal= i&,
(3.4a)
[K,, R”] = i/2m(c,)@Qo,
(3.4b)
[B, R’]
(3.4c)
= -iQ”
plus their complex conjugate relations. Now, contracting again with respect to the superUU,( 1,l; 1; H) tends to a superalgebra generated by (MP,, R,, R”) the superalgebra algebra which differs from that of Khan in relations (3.3~) and (3.3e), which should be replaced now by (3.4a) above and [B, R”] = A&“,
[K,,, R”] = i/2 m(a,)“IPQp
(3.5)
CONTRACTIONS
and their complex conjugate isomorphic to that of Khan.
YIELDING
EXTENSIONS
OF POINCARi
ALGEBRA
It can be shown that this superalgebra
expressions.
39 is not
4. Discussion In this paper we have looked for new superalgebras which include the Poincare algebra and which turned out to be self-conjugate extensions of the Konopel’chenko superalgebra. To obtain them, we performed Wigner-InGnu contractions of real simple superalgebras containing SO(3,2) or SO(4,l) which, after contractions abelianizing the fifth momenta, lead to the physical Poincare algebra. We have shown that the anti-de Sitter (with internal symmetry SO(N, N)) or de-Sitter superalgebra can be used for that purpose. The next step would be to construct field models which are invariant under the rigid supersymmetries generated by the superalgebras treated in the paper and, subsequently, N = 2 supergravities invariant under the local supersymmetries. It should be mentioned that such theories can be obtained by suitable contractions of N = 2 models with SO(l. 1) anti-de-Sitter [lo] or SO(2) de-Sitter supersymmetry 112, 1.51. As follows from the relations (3.16) all these models will contain states with indefinite metric. Another direction for study is to consider real forms of OSp(2;4; C) in which the Lorentz subalgebra SO(3.1) is replaced by the Euclidean one SO(4) [18]. Since the Euclidean metric describes a theory with analytically continued time (t + r = it), the ghosts appearing in a Euclidean quantum field theory do not describe physical states.
Appendix
I
The quaternionic algebra U(l, quaternionic matrices A satisfying
1; H) is the subalgebra of algebra gl(2! H) of 2 x 2 real the condition Ag = -gA+, where A+ denotes quater-
nionic hermitian conjugation (A+ = XT under which q = qo + qzei + ?j = qo - qie,) and the metric g is given by g = diag( 1, -1). The quaternionic algebra U,(l; H) is the algebra of quaternions fulfilling the condition qe2 = -ezq. One can show that U(l, and U,(l;H) - SO(2). The quaternionic superalgebra UU,(l, 1; 1; H) infinitesimal transformations preserving the scalar product ij1q2 - ij2q2 where n = 770+ ~7% describes a Grassmann-valued quaternion. It is quaternionic supermatrices a satisfying the relation ag = where
a# denotes
graded-quaternionic
a= and the supermetric
-pz#,
conjugation
(a :) --+a#=(_%+$)
g has the form
( > 9
0
0
e2
1; H) - SOE4) is determined by + qezq (see [14]) defined by 3 x 3
40
J. A. DE AZCARRAGA,
Appendix
J. LUKIERSKI
and J. NIEDERLE
II
Conventions: gPL”= (+: _., _, _); “,o = ?o+, “is = -yi+, (7’)’ = 1 and C,, = i/2[ yP, r,,]. In the Weyl realization
?s = iy0,iiy2y3, ?S+ = y”,
a,,!? = 1,2, A = 1,2;3,4, where (ap&
= (o”, a’), (a~)“0 = (9. -ai), (q&3 = i/2[(aq,,(o”)+‘7 + (/L t-t v)], (cT,,)/ii = i/2 [(flqq&)y& - (p tf v)]. The dot on the index of a Weyl spinor means complex conjugation. Charge conjugation is defined by Qc = CyoTQ*. In four dimensions C)Y{‘C-i = -y@r, CT = -C. In the Weyl realization
with
Ed
= _+,
@b = -$,
El2 =
_E12
=
$2 = -&ii = 1. Weyl spinor indices are
raised and lowered from the left: E&~QB = QLc,PflQ,
= Q”.
In 5-dimensional space-time C’s satisfies CsyaC;’ = yaT, C’s = --CT, a = 0, 1: 2,3; 5 (see e.g. [ll]). C’s is required to write the de-Sitter algebras in 5-dimensional covariant notation, in which mMPs = Pti (anti-de-Sitter) or K, (de Sitter). In this last case, the i accompanying Kp (see (2.12)) transforms g55 = 1 (anti-de-Sitter) into g[ss = -1 (de Sitter) and, in the odd part, introduces a new 7s’ E iy”, (Y’~)’ = -1. REFERENCES
[‘I
PI [31
1101 I”1 [t21
[l’,]
Rev. Leff. 14 (1965), 575; Whys. Rev. B139 (1965). 1052; Flato, M., Sternhcimer, D.: Phys. Rev. Left. 15 (1965) 934: 16 (1966) 1185; Jost, R.: Helv. P/~_y.s.Acm 39 (1966) 369; Coleman, S.. Mandula. J.: Phys. Rev. 159 (1967), 1251; Garber, W. D., Reeh. H.: J. ~!&th. Phys. 19 (1978) 59; Commun. Math. Phys. 70 (1979), 169. Haag, R., Lopuszanski, J. T., Sohnius, M.: ‘%‘uc~.P/zys. BSS (1975) 257. Niederle, J.: in Proc. V. Int. Sem. Problems of High Energy Physics and Quantum Field Theoty, Protvino 1982, Vol. II, p. 369. Konopel’chenko, B. Cr.: JEPTLett. 20 (1974), 314; 21 (1975), 287. Hlavat)i, L., Niederlc, .i.: Lett. Math. Phys. 4 (l%O), 301: Hlavat$, L.: J. Phys. A. (Ma/h. Gen) 15 (1982). 2973. Khan. I.: Lett. Math. P\zy~. 9 (1985). 265. Kac. V. G.: Ad!,. in Math. 26 (1985). 8; Parker, M.: .I. Math. Phys. 21 (1980) 689. lbwnscnd, P.. van Nieuwenhuizen, P.: Phys. Left. 67B (1977) 43Y. Flato. M., Fronsdal. C.: in Ex?a)s on Supersymmetry (C. Fronsdal, ed.), Reidel 1986, p, 123. Hull, C. M.: Phyx Letf. 142B (1984), 39: 148B (1984),297; Volkov, D. V., Soroka, V. A.,Tkach. V. I.: Soy. J. NLII L F’h):c. 44 (19X6), 522. Nieuwenhuizcn P.: in Relatil?n; Groups and Topolo~ II, I.cs Hoxheh 1983, Elsevicr (19X4). p, 843. Pilch, I;., van Yicuwcnhuizen, P., Sohnius, M. F.: Cornrnwr. ~lf~/l:. Plgr. 98 (1985), 105. Kugo, ‘I’., Townsend, P.: &i/cl. Phys. B221 (1983), 357. Lukierski, J., Nowicki. A.: Fortschr. der Phyx 30 (19S2), 75: Am. Phvs. 166 (1986) 164. Lukterski, J., Nowicki, A.: Phys. hf. 151Bb (19S.i), 382. de Witt. B.. Zwartkruis. A.: Cluss. Qutzn~um G’mv. 4 (1987), LSY. Ferrara. S.: P,$s. Lett. 69R (1977), 48 I. Chaichian, M . de Azcarraga, J. A.. Lukierski, J.: Phys. bft. B222 (lY89), 72.
REPORTS
CONTRACTIONS EXTENSIONS
ON MATHEMATICAL
YIELDING
PHYSICS
No. 1
NEW SUPERSYMMETRIC
OF THE POINCARI? ALGEBRA
J. A. DE AZCARRAGA Department of Theoretical Physics, University of Valencia and IFIC (CSIC), 46100 Burjasot (Valencia), Spain J. LUKIERSKI Institute for Theoretical Physics, University of Wrodaw, 50 205 Wrodaw, Poland and J. NIEDERLE Institute of Physics, Czechoslovak Academy of Sciences, 180 40 Prague 8, Czechoslovakia (Received April 12, 1990)
Two new PoincarC superalgebras are analysed. They are obtained by the Wigner-In6nl contraction from two real forms of the superalgebra OSp(2; 4; C) - one describing the N = 2 anti-de-Sitter superalgebra with a non-compact internal symmetry SO(l, 1) and the other corresponding to the de-Sitter superalgebra with internal symmetry SO(2). Both are 19-dimensional self-conjugate extensions of the Konopel’chenko superalgebra. They contain 10 PoincarC generators and one generator of internal symmetry in addition to 8 odd generators half of which, however, do not commute with translations.
1. Introduction
The construction of various extensions of the Poincart algebra - the symmetry algebra of relativistic physics - is closely related with two developments in particle physics. The first is an attempt to unify the Poincari algebra with the internal symmetry algebra within a large Lie algebra or superalgebra; the second development is the gauge or supergauge descriptions of existing as well as hypothetical fundamental interactions in nature. As is well known, the difficulties in realizing the first programme led to a series of no-go theorems both in the Lie algebra [l] and superalgebra frameworks [2] (for details see [3]). The aim of this paper is to consider new real supcrsymmetric extensions of the PoincarC algebra P. These extensions of P go beyond the standard axiomatic framework of QFT (in particular the axiom of positive-definite metric in the space of states) and were not treated by Haag, Lopuszafiski and Sohnius in [2]. The first alternative to the supersymmetric extensions classified in [2] was found by Konopel’chenko [4]. However, as shown in [3], [5], Konopel’chenko’s superalgebra can hardly be used in particle physics since it is a 1331
34
J. A. DE AZCARRAGA,
J. LUKIERSKI
and J. NIEDERLE
complex Lie superalgebra (due to the fact that complex conjugation is not an involution) which does not admit hermitian representations (except trivial ones) for odd generators. But this unphysical feature of Konopel’chenko superalgebra might be overcome as the extension recently proposed by Khan [6] and the one discussed in this paper indicate. In fact, we shall show that both extensions can be derived by contracting the appropriate real forms of the orthosymplectic superalgebra OSp(2; 4; C).i Thus, in Section 2, all real forms of OSp(2; 4; C) which can be connected with the symmetries of the Minkowski space-time are introduced. They are three - OSp(2; 4; R), OSp(l, 1; 4; R) and UU,(l, 1; 1; H) (Appendix I) - in accordance with the classification of real simple Lie superalgebras given in [7]. Section 3 is devoted to their contractions. It is shown that a contraction of the N = 2, D = 4 ant-de-Sitter superalgebra OSp(1, 1;4; R) whose internal symmetry sector consists of the non-compact algebra SO(l) 1) yields Khan’s proposal. Another supersymmetric enlargement of P is derived by contracting the N = 1, D = 4 de-Sitter superalgebra UU,(l, 1; 1; H) - the real form of OSp(2; 4; C) whose odd part is subjected to a quaternionic or symplectic Majorana reality condition. Finally, the third real supersymmetric enlargement of P is obtained by contracting the superalgebra OSp(2;4; R), whose internal symmetry consists of SO(2) and whose odd part is given by two-four-component Majorana supercharges. It is just the conventional real N = 2 Poincare superalgebra treated in [2] and, therefore, it will be only mentioned here. In Section 4 some final remarks concerning possible applications of the derived superalgebras are presented. 2. Three real forms of OSp(2; 4; C) The complex orthosymplectic superalgebra OSp(2; 4; C) is determined by the commutation relations of the algebra SO(5, C) @ SO(2, C)
Wp~,Mp~l = i(g,&L, + gvpMILo - g&&o - gvc#pp) , [M,,, RI = Gvx~p - g,xPv)>
(2.la) (2. lb) (2.lc)
[P,, P,,] = -im2MpLv,
[h&w1 = 0, PTP,l = 0,
(2.ld) (2.le)
plus the relations
Oil= -l/W,v)ABQ;, [P,,QAl= -1/2m(y,)ABQ&,
[Mclv,
[t3,Q\]
= -i@Q;
,
(24 (2.W (2.lh)
and {Q>, Q”,} = -@j[(y,C)~~p~ ’ Our discussion
can, however,
+ 1/2m(C,,C)ABMP”]
be extended
+ i@rnCABB,
to any OSp(2; N; CT) with even N.
(2.li)
CONTRACTIONS
YIELDING
EXTENSIONS
OF POINCARi
ALGEBRA
35
where p,u,(~,X = 0.1.2,3: i.,j = 1.2; A,B = 1.2.3.4 and gr,,,, -,1v, y 3’V,-~~3 and C are defined in Appendix II. Thus, the complex superalgebra OSp(2; 4; C) supersymmetrizes the algebra SO(5, C) 9 SO(2, C); it includes, besides 11 even generators (&Lb,,, Pk, and B), 8 odd generators written as two independent , non-self-conjugate, four-component spinors &>, i = 1.2. The 19-dimensional complex superalgebra OSp(2; 4; C) may also be considered as the 38-dimensional real superalgebra OSp(2; 4; C)n of generators (MV,,,, iM p’/, ‘PIL,iPLl,B, il3. Qi. i&i). Any real subalgebra of OSp(2; 4; C)n whose complexification is equal to OSp(2; 4; C) is a real form of OSp(2;4;C). It turns out that any real form of OSp(2; 4; C) is the subalgebra of fixed points of an involutive semimorphism of OSp(2; 4; C) [7]. Consequently, the real forms of OSp(2; 4; C) can be obtained by finding involutive semimorphisms of OSp(2; 4; C) in the group of automorphisms of OSp(2; 4; C). With our metric, the y matrices (and C = rO) are purely imaginary in the Majorana realization, and (y”C) and (C,,,C) are real. Nevertheless, since the defining relations C = -CT are invariant under the mapping Y” = YO+,Ti = -?‘I+, Cyp”c-’ = +T, Qi
= [lQli,
.-/‘l
= u+-f”u,
C’ = UTCU
(2.2)
with U+U = 14, U E U(4), equations (2.1 f-i) are, of course, valid for any realization of the 0(1 13) Dirac matrices. In particular, in the Weyl realization we may write Q;
=
($f).
Q",
= (z)
(2.3)
in terms of four independent SL(2. C) spinors. Different pairs of self-conjugate spinors may be extracted from the two general QA by imposing different reality conditions. By requiring them to be involutions of the superalgebra (2.1), we can halve the number of the generators of OSp(2; 4; C)R to obtain the associated real forms of OSp(2; 4; C). 2.1 N = 2 anti-de-Sitter superalgebra with compact internal symmetry: OSp(2; 4; R) The simplest reality condition spinors,
is to require that the Qi
themselves are Majorana
Q\ = CyoTQ$ I (QTJ)C. In the Weyl realization (2.3), eq. (2.4) leads to R, = Qa, T, = S,. that (2.4) is an involution of (2.1) gives P@ = P;
= P,,.
M,,
= Mzy = A&,
B=U+-B
(2.4) The requirement
(2.5)
so that the even part (P,, n/r,,,, B) generates the algebra SO(3.2) $ SO(2) with spacetime metric (+, -, -, -; +). Thus, for Majorana Q;a and Hermitian P,,, iIf{,, and B. (2.1 j describes the real superalgebra OSp(2: 4; R), well-known in supersymmetric theories of a world with anti-de-Sitter geometry (see, e.g. [8] and also [9]).
2.2 N = 2 anti-de-Sitter superalgebra with now-compact internal symmetry: OSp( 1, 1; 1: R) This corresponds
to choosing Q;
=
(#(Q;)c
WI::
36
J. A. DE AZCARRAGA,
J. LUKIERSKI
and J. NIEDERLE
which in (2.3) leads to the conditions S, = R,, T” = QQ. The linear combinations (2.7) allow us to write the Qf4 which satisfy (2.6) in terms of two Majorana spinors Q> = (Q>)C. The condition that the new reality condition (2.4) is an involution of (2.1) leads to Q,, = P; =P@.
M,,
= M;”
a = -B+
= kf,,,,
E iB’,
w3)
Comparing with (2.5) we see that generator B of the compact SO(2) has been replaced by the generator B’ of the non-compact SO(l, 1). The commutators of A.JWu,P@ are as in (2.1 a-c) and the rest of the superalgebra is determined by [B’, PJ = o>
(2.9a)
[B’, ,%&,I = 0.
(2.9b)
[h/l,,, Q,] = -1/2(C,,)AB& [PkL,&I = -m/2(-r,,JAB& LB’, Q;] = -i(rlE)i.jQi , {ai,
9j,}
= -(q)zJ[(ypc)Pp
(2.9~)
:
(2.9d)
,
+ ‘m/2(~pl,C)hfpv],JB
(2.9e) + i’mc”c~BB’,
(2.9f)
where the inner metric is now 7 = g3 and thus (7~) = 0 I. Consequently. B’ generates the SO(l, 1) rotation (2.9e) SQL N (~l)~j&, and the real superalgebra (2.la-c), (2.9a-f) is the anti-de-Sitter superalgebra OSp( 1; 1; 4; R). Supersymmetric theories based on antide-Sitter superalgebras OSp(p, 4; 4; 12) have already been considered by some authors (see e.g. [lo]). However, due to the appearance of the semidefinite metric n E g3 in (2.9f), the superalgebra OSp(1, 1;4; R) leads e.g. to supergravities with one physical and one “ghost” gravitinos. 2.3 de-Sitter superalgebra with compact internal symmetry: UU,(l,1; 1; H) This is obtained by imposing the so-called symplectic [ll-131 or quaternionic Majorana reality condition ?sQ\
= EijcyaTQjAI E
&J(QJA)C
[14] (2.10)
which is fulfilled if we put S, = -R, and T” = Q” in the expression of Q’, [(2.3)]. As before, one can obtain from the QA fulfilling (2.10) two independent self-conjugate (Majorana) spinors Qi now given by 0; = 5’“;
- r5Q;),
i?‘A = -$+?Qi
+ Q6>.
(2.11)
Imposing that (2.10) is an involution of (2.1), we get the conditions PW = -P;L’ = iK,,
M,,
= M$
= ikf,,,,
t?=l?+=B.
(2.12)
Introducing an SO(5, C) covariant notation with j&b, a, b = 0, 1,2,3; 5, we see (Appendix II) that the replacement Pp = iK, changes the metric to the de-Sitter one (+, --, -, -; -).
CONTRACTTONS
YIELDING
EXTENSIONS
OF POINCARk
ALGEBRA
37
Thus, the real form obtained from (2.10) is the superalgebra UU,(l: 1; 1;H) with inner symmetry U(H) - SO(2). The commutators of the pure even part are (2.1 a,b,d,e) {with M/&v = iwp”, PP = iK,, 23 = B) and [KIL: K,]
(2.13)
= im2hfp,,
The mixed and odd part, expressed in terms of Q-4 E Qa and Q, (Q,C determined by [~~pv,Q~l
=
-1/2(cFuQ)~,
[~lJpu,O,tl
[KWL)QAI= i/2m(T,Q)~; [B, QA] {QA:
QB}
[K,,
= ~($CCI))A, =
Q.,]
[B,QA]
-~(Y~C)ARK’
-
(2.14a)
= ~/~(Q~w)A,
(2.14b)
= ~P~~(QY,)A.
(2.14c)
= ~($CQ)A.
(‘.14d)
112n~(CpvC),~~~““,
= 2m&B
{QA. Q,}
j
(2.14e)
{Q;1. QB} = Z(C~J.~BK~‘ - l/2 t?6(~~p,)A&‘f’li’. Another expression for UU,(l, Majorana spinors (2.11). Then {@,.&} (&,
Q;}
1; 1; H), which is explicitly (2.14 d-f) take the form f0:
= -l/2rn(CC,,V)ABi\lPV = i&iJ{($“iilC)A&? {B,QA}
= Q’,T~), is
i = 1
+ mC.413B}
(2.14f)
real, is obtained
by using the
and i = 2,
(2.15aj
for i #
(2.15b)
.A.
= --1m&3QJA.
j ,
(2.15c)
It can be seen (e.g. from (2.14e)) that Tr(Q.4, Qp} = 0 and thus the superalgebra UU,( I. 1: 1; H), which is associated with supersymmetric theories in a world with de-Sitter geometry. can be realized in a quantized theory only in a space of states with indefinite metric [12, 151. Recent discussions of models with local de-Sitter symmetry [12, 161 show that the ghost problem is also present in de-Sitter supergravity.
3. Contractions
of real forms of OSp(2; 4; C)
3.1 OSp(2; 4; R) It is well known that the contraction of OSp(2;4; R) leads to the N = 2 Poincare superalgebra with or without a central charge [17]. Let us then discuss the contractions of
the two other real forms; to this aim we write first the odd generators of OSp(l, 1; 4: R) and UU,(l. 1; 1;H) in terms of the Weyl charges R,, Q, and their complex conjugate ones. which determined the different versions of self-conjugate spinors. 3.2 OSp(l,1;4;
R)
This algebra can be read from (2.1) by using (2.6) and = (Ra, Q”). ‘The even part is given by (2.1 a-c) with M,, (2.9a,b). The rest of OSp(1 ~I; 4; R) is given by
2.8) which requires (3: = M{,,,, F’,, = F’,&plus
38
J. A. DE AZCARRAGA,
J. LUKTERSKI
and J. NIEDERLE
{R”,Rj} = o>
{Qa.Q,g} = 0.
{Q,, Rj}
{&Cl,.R&j} = -mE,;,R’.
(3.ltJ)
= (o&P
(3.lc)
and [‘ld/Lv. QN] = -l/2
[A$,.
(c+,)o~Q~:
&a]= -l/2 m(a,&R’,
R
[P,l. R,]
[R/L>
[B’, Qa] = -R,
~
(c~,,)Q~R,J
= -1/2~z(a,),~Q’~,
[B’> R”] = -Q’
.
(3.ld) (3.le) (3.lf)
plus their complex conjugate relations. The generators (Al,,,, R,, R’) determine a subalgebra of OSp( 1,l; 4; R) with respect to which we perform the Wigner-Inonti contraction (we could equally choose (M,,,, Q,,, Using the same symbols for the generators of the contracted algebra, we obtain that its defining commutation/anticommutation relations are given by (2.1 a, b, d, e) for A[~,, Pp, B’ together with
Q").
[R,,RVl {Qa, QJ} = 0,
= 0,’
(3.2)
{R% R”} = 1/2m(a,,)“%kY,
(3.3a)
{QcuQg} =O> {R,,Rfi} =O, {QAa} = (oJnOPp {Qa,Ro} = -~7wd3’,
(3.3b) (3.3c)
and [fief,>
RN] Qal= -W(qw)~aQs,[M/w. [B’, Rd] = -Q”,
[I’,, R”] = -l/2
= -1/2(0,&‘R~.
(3.3d)
~(cJ”~Q,,
(3.3e)
,
=0 [B',Qal = 0, [J',,Qal
(3.3f)
to which one has to add the complex conjugate expressions. The above superalgebra the one described by Khan [6] (Q and R correspond to T and 5’ in Khan’s notation). 3.3 U&(1,1;
is
l;H)
Similarly to the previous case, the superalgebra UU,(l, 1; l;H) is read from (2.1) by using (2.10) which requires Q 54 = (-RCY, Q”) and (2.12). In this way we get for the purely even part (2.1 a, b, d, e) (with M,, = iVlp,,, Pp = iK,, B = B) and (2.13). The rest of UU,(l, 1; l;H) is given by (3.1. a, b, d) and
{Qa,Ra] = -imE&, {Q,,Ra] = i(n),,+', [K@, Qa] = i/2m(o,),bRR, [R,
Qal= i&,
(3.4a)
[K,, R”] = i/2m(c,)@Qo,
(3.4b)
[B, R’]
(3.4c)
= -iQ”
plus their complex conjugate relations. Now, contracting again with respect to the superUU,( 1,l; 1; H) tends to a superalgebra generated by (MP,, R,, R”) the superalgebra algebra which differs from that of Khan in relations (3.3~) and (3.3e), which should be replaced now by (3.4a) above and [B, R”] = A&“,
[K,,, R”] = i/2 m(a,)“IPQp
(3.5)
CONTRACTIONS
and their complex conjugate isomorphic to that of Khan.
YIELDING
EXTENSIONS
OF POINCARi
ALGEBRA
It can be shown that this superalgebra
expressions.
39 is not
4. Discussion In this paper we have looked for new superalgebras which include the Poincare algebra and which turned out to be self-conjugate extensions of the Konopel’chenko superalgebra. To obtain them, we performed Wigner-InGnu contractions of real simple superalgebras containing SO(3,2) or SO(4,l) which, after contractions abelianizing the fifth momenta, lead to the physical Poincare algebra. We have shown that the anti-de Sitter (with internal symmetry SO(N, N)) or de-Sitter superalgebra can be used for that purpose. The next step would be to construct field models which are invariant under the rigid supersymmetries generated by the superalgebras treated in the paper and, subsequently, N = 2 supergravities invariant under the local supersymmetries. It should be mentioned that such theories can be obtained by suitable contractions of N = 2 models with SO(l. 1) anti-de-Sitter [lo] or SO(2) de-Sitter supersymmetry 112, 1.51. As follows from the relations (3.16) all these models will contain states with indefinite metric. Another direction for study is to consider real forms of OSp(2;4; C) in which the Lorentz subalgebra SO(3.1) is replaced by the Euclidean one SO(4) [18]. Since the Euclidean metric describes a theory with analytically continued time (t + r = it), the ghosts appearing in a Euclidean quantum field theory do not describe physical states.
Appendix
I
The quaternionic algebra U(l, quaternionic matrices A satisfying
1; H) is the subalgebra of algebra gl(2! H) of 2 x 2 real the condition Ag = -gA+, where A+ denotes quater-
nionic hermitian conjugation (A+ = XT under which q = qo + qzei + ?j = qo - qie,) and the metric g is given by g = diag( 1, -1). The quaternionic algebra U,(l; H) is the algebra of quaternions fulfilling the condition qe2 = -ezq. One can show that U(l, and U,(l;H) - SO(2). The quaternionic superalgebra UU,(l, 1; 1; H) infinitesimal transformations preserving the scalar product ij1q2 - ij2q2 where n = 770+ ~7% describes a Grassmann-valued quaternion. It is quaternionic supermatrices a satisfying the relation ag = where
a# denotes
graded-quaternionic
a= and the supermetric
-pz#,
conjugation
(a :) --+a#=(_%+$)
g has the form
( > 9
0
0
e2
1; H) - SOE4) is determined by + qezq (see [14]) defined by 3 x 3
40
J. A. DE AZCARRAGA,
Appendix
J. LUKIERSKI
and J. NIEDERLE
II
Conventions: gPL”= (+: _., _, _); “,o = ?o+, “is = -yi+, (7’)’ = 1 and C,, = i/2[ yP, r,,]. In the Weyl realization
?s = iy0,iiy2y3, ?S+ = y”,
a,,!? = 1,2, A = 1,2;3,4, where (ap&
= (o”, a’), (a~)“0 = (9. -ai), (q&3 = i/2[(aq,,(o”)+‘7 + (/L t-t v)], (cT,,)/ii = i/2 [(flqq&)y& - (p tf v)]. The dot on the index of a Weyl spinor means complex conjugation. Charge conjugation is defined by Qc = CyoTQ*. In four dimensions C)Y{‘C-i = -y@r, CT = -C. In the Weyl realization
with
Ed
= _+,
@b = -$,
El2 =
_E12
=
$2 = -&ii = 1. Weyl spinor indices are
raised and lowered from the left: E&~QB = QLc,PflQ,
= Q”.
In 5-dimensional space-time C’s satisfies CsyaC;’ = yaT, C’s = --CT, a = 0, 1: 2,3; 5 (see e.g. [ll]). C’s is required to write the de-Sitter algebras in 5-dimensional covariant notation, in which mMPs = Pti (anti-de-Sitter) or K, (de Sitter). In this last case, the i accompanying Kp (see (2.12)) transforms g55 = 1 (anti-de-Sitter) into g[ss = -1 (de Sitter) and, in the odd part, introduces a new 7s’ E iy”, (Y’~)’ = -1. REFERENCES
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PI [31
1101 I”1 [t21
[l’,]
Rev. Leff. 14 (1965), 575; Whys. Rev. B139 (1965). 1052; Flato, M., Sternhcimer, D.: Phys. Rev. Left. 15 (1965) 934: 16 (1966) 1185; Jost, R.: Helv. P/~_y.s.Acm 39 (1966) 369; Coleman, S.. Mandula. J.: Phys. Rev. 159 (1967), 1251; Garber, W. D., Reeh. H.: J. ~!&th. Phys. 19 (1978) 59; Commun. Math. Phys. 70 (1979), 169. Haag, R., Lopuszanski, J. T., Sohnius, M.: ‘%‘uc~.P/zys. BSS (1975) 257. Niederle, J.: in Proc. V. Int. Sem. Problems of High Energy Physics and Quantum Field Theoty, Protvino 1982, Vol. II, p. 369. Konopel’chenko, B. Cr.: JEPTLett. 20 (1974), 314; 21 (1975), 287. Hlavat)i, L., Niederlc, .i.: Lett. Math. Phys. 4 (l%O), 301: Hlavat$, L.: J. Phys. A. (Ma/h. Gen) 15 (1982). 2973. Khan. I.: Lett. Math. P\zy~. 9 (1985). 265. Kac. V. G.: Ad!,. in Math. 26 (1985). 8; Parker, M.: .I. Math. Phys. 21 (1980) 689. lbwnscnd, P.. van Nieuwenhuizen, P.: Phys. Left. 67B (1977) 43Y. Flato. M., Fronsdal. C.: in Ex?a)s on Supersymmetry (C. Fronsdal, ed.), Reidel 1986, p, 123. Hull, C. M.: Phyx Letf. 142B (1984), 39: 148B (1984),297; Volkov, D. V., Soroka, V. A.,Tkach. V. I.: Soy. J. NLII L F’h):c. 44 (19X6), 522. Nieuwenhuizcn P.: in Relatil?n; Groups and Topolo~ II, I.cs Hoxheh 1983, Elsevicr (19X4). p, 843. Pilch, I;., van Yicuwcnhuizen, P., Sohnius, M. F.: Cornrnwr. ~lf~/l:. Plgr. 98 (1985), 105. Kugo, ‘I’., Townsend, P.: &i/cl. Phys. B221 (1983), 357. Lukierski, J., Nowicki. A.: Fortschr. der Phyx 30 (19S2), 75: Am. Phvs. 166 (1986) 164. Lukterski, J., Nowicki, A.: Phys. hf. 151Bb (19S.i), 382. de Witt. B.. Zwartkruis. A.: Cluss. Qutzn~um G’mv. 4 (1987), LSY. Ferrara. S.: P,$s. Lett. 69R (1977), 48 I. Chaichian, M . de Azcarraga, J. A.. Lukierski, J.: Phys. bft. B222 (lY89), 72.