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

264 P. van Nieuwenhuizen. SupergravityThe Lagrange multiplier leads to the constrainthikyyTh/i = —2y4o~8j4,k= 0. (2)Thus only 4,,” appears in the action, since the longitudinal part~ô 4, vanishes in the first termin (1) after partial integration and use of (2). If we had worked with field equations, we would at thispoint have needed to introduce a gauge choice. Next we note that since we may replace 4’ by1/IT4’k -~ 8k L]= 0, ‘/‘T = (6/ — a,59’V~ 2)4, 1 (3)again due to the constraint in (2). Thus ‘y = 0 since I 8y’ is elliptic (Latin indices run alwaysfrom (1, 3)). Hence, only 4. appears in the action and is transverse both in ordinary space (a . i/i = 0) andin spinor space ~ 4. = 0). Thusa — (p3/2\a b = (~ a\TF 4‘PM — k’ MV) b’p,. — ~‘PM Iwhere F 312 is the spin 3/2 spin projection operator of subsection 1.13. The fields 41”’ and 41T are gaugeinvariant (i.e. invariant under 64,,,. = ~M~) so that the action has been decomposed into gauge invariantdynamical modes.The action becomes equal to7”’ (5)I = J d~x— ~“(y~84 + y’8~)4,if one uses the expansion of 7~~J7kand the double transversability of 4,,”~.Thus there are two physicalmodes, given already explicitly in subsection 1.13.Using electromagnetic notation, one can also rewrite the action as= —~./iy5(yA E + 70B) (6)whereE, = 4,1~rr, B’ = (7)so that the field equations read E + 75B =0.2.14. Dirac quantization of the free gravitinoAlthough a Majorana field is its own conjugate momentum, there is no problem in determininguniquely the canonical commutation relations, if one uses Dirac brackets { }D rather than Poissonbrackets in order to go from the classical to the6}Dquantum is nonzero theory. and Poisson in agreement brackets are with defined as usual,{4,M2, ~.,.b}D Inbut, second as quantized we will see, formthethere Dirac is no bracket problem {4’M’~’4’ at all, since one finds in ~ absorption operators a,, (k) andcreation operators a ~(k) (and not b~(k)since 4’Ma is a Majorana field) so that {4’M”, 4’V} evaluated at the