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

quantum level is nonzero. For example, in the gauge y• 4, =0 one hasP. van Nieuwenhuizen, Supergravity 265= ~ ~ [a,,(k) u~(k)eu1~x+ a ~(k) u~(k)*e~~] (1)where u~= ~u,,and A = + or —.Consider a theory with fermionic variables. We define canonical momenta as left derivatives of theLagrangianp = a~L/a4. (2)Then the Hamiltonian is independent of 4 ifone defines it asH~=4p—L, 5H~=4~p_5q(3eL/~q). (3)The equations of motion read (r = right)a’~H~/ap=4, a’H~/aq=—atLI3q=—ji (4)where in the second equation we have used the Euler—Lagrange equationsatL/aq = ~j (a~L/t94)= j5. (5)Afunctional F(q, p) evolves then in time asdF/dt = (3rHJ3p~~eF/~q) — (a’F/ap)(a’H~/aq)= {F, H~}. (6)This defines the Poisson bracket for two variables f and g if at least one of them is bosonic.In the general case, the Poisson bracket is defined as{f, g} = —(a’f/apXa~gIaq)+ (— )fa(arg/ap)(aef/aq) (7)where f = 0(1) if f is bosonic (fermionic). The extra sign (~)ft is needed in order that for two fermionic fand g one has {f, g} = ~g, f}. The overall sign is fixed by requiring that {f, gh} = {f, g}h + (— )~{g,fh}. Wenow consider the free spin ~field, following P. Senjanovië [608].Thus, for example, for the gravitino field {4’M”,4,b} = {*Ma lTb} =0 where we denote the conjugatemomentum by *~a rather than ITMa in order to obtain manifestly Lorentz covariant results. In particular3(x—y). (8){4,Ma(X, t), lTb(y, t)} = —6g6~oThe Hamiltonian is (as usual for fermionic systems) independent of time derivatives= Jd 3x iEM~rc~4’yya4, (r= 1,2,3). (9)