SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

292 P. van Nieuwenhuizen. Supergravitv{~C(x), ~‘n(y)} = [~2,n,f”Jpq+ ~(Dm4’n — D,,4’,,,)] 63(x — y) (18){O”(x), ~‘ 3(x — y) (19)m(y)}= (EYsY,nDa4’b) ””Jcä 6where 12 contains the supercovariantized Riemann curvature. Indeed, in the flat space limit one findsback the global algebra. We stress that these results could have been found by a straightforwardHamiltonian analysis, in which case one would have found that in all these (anti)commutators only p’sand q’s appear.The truly interesting thing is the off-shell closure of the gauge algebra. To establish its relation to thegauge algebra when evaluated on fields F, we note that the Jacobi identity yields{F, {C, C,,}} = (F fm,,P}ç, +f,,,,,~{F,C~}. (20)Whenever (F, fm,,”} is nonzero, one finds extra terms in the gauge algebra proportional to C,,. These arethe equation of motion terms known from the Lagrangian approach (which can only be gotten rid of byintroducing the auxiliary fields into the theory). Apparently, in supergravity the structure constantsdepend on the canonical momenta. (In gravity they depend only on the canonical coordinates in thecostumary parallel-transverse basis, but they depend also on momenta if one uses the same as oneused above.)In the anticommutator (0~Q} ~ no fields appear in the structure functions as a result of defininginstead of general coordinate transformations the covariant translations with parameters f”. There arequite a number of fine points. For example, the brackets are really Dirac brackets, but we refer to theliterature for further study.Let us now discuss why the {Q, Q} - ~‘ relation implies that supergravity is the theory of spinningspace. We are led to consider the Dirac equation as a square root of the Klein—Gordon equation2)4’ = 0. (21)(ytm3 + m)~/i= 0, (111 — mMultiplying by y~and defining y~ym= 0”, 7ç = g5, we can introduce, in addition to the usual canonicalvariables (x, p,,) five anticommuting variables (0, 0~)satisfying{xm, p,,} = 8~’, {o”, o~}= —26~”, {o~,o”’} = 0, {o~,o~}= 2. (22)One now finds a set of first class (i.e. commuting) constraintsS = Otmp,,, + m05, ,9~?= p2 + m2 (23){S,S}=2~r, {s,H}=o, {~r,~r}=0. (24)An action for this system leading to these (anti)commutators can also be given. Hence one can take asquare root of the constraint ~‘.Just as taking the square root of the Klein—Gordon field leads to a spinning particle, taking thesquare root of (the generators Hm of) ordinary space leads to (the generators Q~of) spinningspace.