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

P. van Nieuwenhuizen, Supergravily 231As before, we choose a second gauge, maintaining the previous gauge y~i/i ~0(9)One still has k . = = =0 on-shell. Thus, in particular(10)where we used that %~%~ =0 and defined °k = ~Jky 1yJ/(2i).Thus u is the spin projection operator, andu~has helicity + 1/2. Since e’~has helicity + 1, we conclude that ~ describes two physical modes withhelicities ±3/2.We now prove that (1) and (2) are unique. We only treat the spin 3/2 case. For the spin 2 case see forexample, Nucl. Phys. B 60 (1973) 478. The most general action with one derivative is given by2’ = i/i,.[ay,.t9~ + f3y~3,.+ y,,Jy~+ ~IO,.~]i/’~. (11)We introduce spin projection operatorsP3/2 — ‘~ ~ — — — ~ 2’fl~— ~ 3Y,.Yv, )‘~~— )1~ 00k, W~— (i~,V Li‘P 1/2\ — 1’ ‘ f 1/2\ — 1 — —I. Ii J,,v — ~ ~ 12 ),.v — ~,r 7,.00,~, !/~p — U~~v 01,.01v,(P~2),.~ = w4c, (P~2),.~ = w,.w~. (12)They have three properties(i) orthonormality: ~ =(ii) decomposition of unity: P312 + P ~ + F ~ = I. Hence the sum of the diagonal projection operatorsequals the unit operator I;(iii) completeness: the operators P~span the space of all field equations of the form (11). This isobvious since the five P~span the four-dimensional space in (11) and they must satisfy one linearrelation to cancel the 8,~8~/El terms. We have chosen this set, but one may use any other set satisfying(i), (ii), and (iii) (for example, with 5,.,, — (k,.~,,+ k,,le,.)/(k . k)).We now consider first massive spin 3/2 fields and then the massless case.Massive fields: On-shell one has y~i/c = 3 . i/i =0, hence on-shell i/i,. = ~ [use(ii)1. However, onecannot define as field equation (P3121 — M)i/i = 0 since P3”2 contains E’ singularities. The choice(P3”202 — M2)./i = 0 is local (i.e., free from El1 singularities) but a higher derivative equation. However,if one writes for the field equation (01— M)i/i = 0 such that 01 is a linear combination of theoperators in (12), which is local, then iteration yields OlOlili = M2i/i = E1P312i/i. Hence 01 must be asquare root of EJP3”2. This method is called the root method and is due to Ogievetski and Sokatchev.The most general square root is0=P312+a(P2+P~+$P~+l3’P~2). (13)