15.1 Functionals A functional G[f] is a map from a space of functions to a set of numbers. For instance, the action functional S[q] for a particle in one dimension maps the coordinate q(t), which is a function of the time t, into a number—the action of the process. If the particle has mass m and is moving slowly and freely, then for the interval (t1,t2) its action is
t2 m dq(t) 2 S [q]= dt . (15.1) 0 2 dt Zt1 ✓ ◆ If the particle is moving in a potential V (q(t)), then its action is
t2 m dq(t) 2 S[q]= dt V (q(t)) . (15.2) 2 dt Zt1 " ✓ ◆ #
15.2 Functional Derivatives A functional derivative is a functional
d G[f][h]= G[f + ✏h] (15.3) d✏ ✏=0 of a functional. For instance, if Gn[f] is the functional
n Gn[f]= dx f (x) (15.4) Z 626 Functional Derivatives then its functional derivative is the functional that maps the pair of functions f,h to the number
d G [f][h]= G [f + ✏h] n d✏ n ✏=0 d = dx (f(x)+ ✏h(x))n d✏ Z ✏=0 n 1 = dx nf (x)h(x). (15.5) Z
Physicists often use the less elaborate notation