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

378 P. van Nieuwenhuizen, Supergraviiysame dimension. One might expect that this is the formalism best suited for supersymmetry. Let us takea closer look.The action for a real spin 0 field is given by~E=—~A(p”o,,+M)A, A=AT~. (1)The matrices f3” are supposed to be Hermitian and to satisfyf3~/3~J3A+ 13Ap~p~= 3~~13A+ 0PA13~~. (2)Just as the equation (a,.u” — 1)(b — 1 is solved by the Dirac matrices a,,, = b~= y,,, in thesame way the f3” are a particular 0.u” solution + 1) =of* uTuT(a,~u”u°’ + c~u~— 1)(bAu” + 1) = u~uT— 1. (3)The Lorentz generators are Utm” = ~[13m 13”] and satisfy the Lorentz algebra as a consequence of (2).One can define A = At~by requiring that AA is Lorentz invariant. This leads to[~,Umn]”O and {~,cr~4}=0withm,n=1,3.A Majorana conjugate boson A = ATn is defined by requiring that A transform as A. This leads toflo~,,,vfl_l= ~ A Majorana boson is defined by A = A. In order that this relation is maintained intime under (1), one needs -113~,T =There exists a 5 X 5 representation of (2) for spin 0 fields and a 10 X 10 representation for spin 1fields. A Majorana representation of the former is given by4\ —(~ )5j — (J) )j5 — 01, I,p )54 — ~(‘P )4~— 1.— — ~k io~\ — 1nIn this representation ~ = ~= 2~3~/3” + 1 = diag(1, 1, 1, —1, —1) commutes with 134 and anticommuteswith13k• Also /3~L1 = 3~,T However, ~T = ~ whereas for fermions C’~= —cThe field equations of (1) read in this representation MA,. + OkA5 = 0 and MA4 — iO4A5 = 0 so that Aequals the five-vector (OkA5, —i04A5, —MA5). Inserting these equations into the action, one finds theKlein—Gordon action for ~ = iA5q~iA5. (5)Consider now as a model of supersymmetry a Majorana boson A and a Majorana fermion ~!i.Theaction reads= —~A(8”a,.+ M)A — ~y”0,, + M)~/i. (6)It is invariant under* I thank Dr. Dresden for pointing this out to me.