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

266 P. van Nieuwenhuizen, SupergravitvAll sixteen momentum definitions in (2) are primary constraints, and can be written as—! 0 ~O 0. (10)Thus, as stated above, the field variables are equal to their conjugate momenta.Adding the primary constraints to the canonical Hamiltonian leads to the total HamiltonianHT=Hc+~McM. (11)In terms of HT one has the usual Hamilton equations valid for unconstrained systems= —a’L/a4, = —~, a’HT/3ff = 4,. (12)The Lagrange multipliers CMare anticommuting functions. Consistency, i.e., 4’M =0 for all time, thus{4’M, HT}0, then leads to new, so-called secondary, constraints. From (d/dt)4” =0 one finds~(743r4,s— y,8,4i4) + j~krs7 = 0. (13)This equation fixes c,. but does not lead to secondary constraints. The solution is unique and can befound by multiplication by lcmhIymCk = “iêk4,4’ k~nn(~y~ôm4’n) _~j )l y4(a~4,k— 8k4,~). (14)From (d/dt)~° =0 one finds with {4’~,4’~}= 0 four secondary constraints,~a {~oa,Hc}elmhhiai(~mysyn)a 0. (15)These constraints hold whether or not one has substituted the solution for Ck. Since ,~.and H~do notdepend on momenta, consistency of ~ =0 leads to {~,4’k}ck 0. This seems to lead to yet newconstraints, but one finds{,~a4’kc,(} = _jE~dlmy4y5yk8lCm (16)which is just 8,. times (13). Hence the total set of constraints consists of the sixteen ~ in (10) and thefour ,~ in (15).One now divides these constraints into constraints which commute with all other constraints (firstclass constraints) and the rest (second class constraints). One finds eight first class constraints3k9!IIYSY (17)= IT,,, Xa + 8k4’,, = 3,.ir,, +~ Itmand twelve second class constraints (namely the çb” plus any linear combination of (17)). To check this