15 Functional Derivatives 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 δG[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 δG [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 δG[f] = δG[f][δ ] (15.6) δf(y) y in which the function h(x)isδ (x)=δ(x y). Thus in the preceding y − example δG[f] n 1 n 1 = dx nf − (x)δ(x y)=nf − (y). (15.7) δf(y) − Z Functional derivatives of functionals that involve powers of derivatives also are easily dealt with. Suppose that the functional involves the square of the derivative f 0(x) 2 G[f]= dx f 0(x) . (15.8) Z Then its functional derivative is d δG[f][h]= G[f + ✏h] d✏ ✏=0 d 2 = dx f 0(x)+✏h0(x) d✏ Z ✏=0 = dx 2f 0(x)h0(x)= 2 dx f 00(x)h(x) (15.9) − Z Z in which we have integrated by parts and used suitable boundary conditions on h(x) to drop the surface terms. In physics notation, we have δG[f] = 2 dx f 00(x)δ(x y)= 2f 00(y). (15.10) δf(y) − − − Z 15.2 Functional Derivatives 627 Let’s now compute the functional derivative of the action (15.2), which in- volves the square of the time-derivativeq ˙(t) and the potential energy V (q(t)) d δS[q][h]= S[q + ✏h] d✏ ✏=0 d m 2 = dt q˙(t)+✏h˙ (t) V (q(t)+✏h(t)) d✏ 2 − Z ✏=0 ⇣ ⌘ = dt mq˙(t)h˙ (t) V 0(q(t))h(t) − Z h i = dt mq¨(t) V 0(q(t)) h(t) (15.11) − − Z ⇥ ⇤ where we once again have integrated by parts and used suitable boundary conditions to drop the surface terms. In physics notation, this is δS[q] = dt0 mq¨(t0) V 0(q(t0)) δ(t0 t)= mq¨(t) V 0(q(t)). (15.12) δq(t) − − − − − Z ⇥ ⇤ In these terms, the stationarity of the action S[q] is the vanishing of its functional derivative either in the form δS[q][h] = 0 (15.13) for arbitrary functions h(t) (that vanish at the end points of the interval) or equivalently in the form δS[q] = 0 (15.14) δq(t) which is Lagrange’s equation of motion mq¨(t)= V 0(q(t)). (15.15) − Physicists also use the compact notation δ2Z[j] @2Z[j + ✏ + ✏ δ ] y 0 z (15.16) δj(y)δj(z) ⌘ @✏@✏0 ✏=✏0=0 in which δ (x)=δ(x y) and δ (x)=δ(x z). y − z − Example 15.1 (Shortest Path is a Straight Line) On a plane, the length of the path (x, y(x)) from (x0,y0)to(x1,y1)is x1 x1 2 2 2 L[y]= dx + dy = 1+y0 dx. (15.17) x0 x0 Z p Z p 628 Functional Derivatives The shortest path y(x) minimizes this length L[y],so d d x1 δL[y][h]= L[y + ✏h] = 1+(y + ✏h )2 dx d✏ d✏ 0 0 ✏=0 Zx0 ✏=0 x1 x1 p y0h0 d y0 = dx = h dx =0 (15.18) 2 − dx 2 Zx0 1+y0 Zx0 1+y0 since h(x0)=h(x1) = 0.p This can vanish for arbitraryp h(x) only if d y 0 = 0 (15.19) 2 dx 1+y0 which implies y00 = 0. Thus y(x)p is a straight line, y = mx + b. 15.3 Higher-Order Functional Derivatives The second functional derivative is d2 δ2G[f][h]= G[f + ✏h] . (15.20) d✏2 |✏=0 So if GN [f] is the functional N GN [f]= f (x)dx (15.21) Z then d2 δ2G [f][h]= G [f + ✏h] N d✏2 N |✏=0 d2 = (f(x)+✏h(x))N dx d✏2 Z ✏=0 2 d N 2 2 N 2 = ✏ h (x)f − (x) dx d✏2 2 Z ✓ ◆ ✏=0 N 2 2 = N(N 1) f − (x)h (x)dx. (15.22) − Z 2 Example 15.2 (δ S0) The second functional derivative of the action S0[q] (15.1) is 2 t2 2 2 d m dq(t) dh(t) δ S0[q][h]= 2 dt + ✏ d✏ t1 2 dt dt Z ✓ ◆ ✏=0 t 2 2 dh(t) = dt m 0 (15.23) dt ≥ Zt1 ✓ ◆ 15.4 Functional Taylor Series 629 and is positive for all functions h(t). The stationary classical trajectory t t t t q(t)= − 1 q(t )+ 2 − q(t ) (15.24) t t 2 t t 1 2 − 1 2 − 1 is a minimum of the action S0[q]. The second functional derivative of the action S[q] (15.2) is 2 t2 2 2 d m dq(t) dh(t) δ S[q][h]= 2 dt + ✏ V (q(t)+✏h(t)) d✏ t1 " 2 dt dt − # Z ✓ ◆ ✏=0 t2 dh(t) 2 @2V (q(t)) = dt m h2(t) (15.25) dt − @q2(t) Zt1 " ✓ ◆ # and it can be positive, zero, or negative. Chaos sometimes arises in systems of several particles when the second variation of S[q] about a stationary path is negative, δ2S[q][h] < 0whileδS[q][h] = 0. The nth functional derivative is defined as dn δnG[f][h]= G[f + ✏h] . (15.26) d✏n |✏=0 The nth functional derivative of the functional (15.21) is n N! N n n δ G [f][h]= f − (x)h (x)dx. (15.27) N (N n)! − Z 15.4 Functional Taylor Series It follows from the Taylor-series theorem (section 4.6) that 1 δn 1 1 dn eδG[f][h]= G[f][h]= G[f + ✏h] = G[f + h] (15.28) n! n! d✏n n=0 n=0 ✏=0 X X which illustrates an advantage of the present mathematical notation. The functional S0[q] of Eq.(15.1) provides a simple example of the func- 630 Functional Derivatives tional Taylor series (15.28): d 1 d2 eδS [q][h]= 1+ + S [q + ✏h] 0 d✏ 2 d✏2 0 ✓ ◆ ✏=0 t2 2 m d 1 d 2 = 1+ + q˙(t)+✏h˙ (t) dt 2 d✏ 2 d✏2 Zt1 ✓ ◆ ✏=0 t2 ⇣ ⌘ m 2 2 = q˙ (t)+2˙q(t)h˙ (t)+h˙ (t) dt 2 t Z 1 ⇣ ⌘ m t2 2 = q˙(t)+h˙ (t) dt = S0[q + h]. (15.29) 2 t Z 1 ⇣ ⌘ If the function q(t) makes the action S0[q] stationary, and if h(t) is smooth and vanishes at the endpoints of the time interval, then S0[q + h]=S0[q]+S0[h]. (15.30) More generally, if q(t) makes the action S[q] stationary, and h(t) is any loop from and to the origin, then 1 1 dn S[q + h]=eδS[q][h]=S[q]+ S[q + ✏h] . (15.31) n! d✏n |✏=0 n=2 X If further S2[q] is purely quadratic in q andq ˙, like the harmonic oscillator, then S2[q + h]=S2[q]+S2[h]. (15.32) 15.5 Functional Di↵erential Equations In inner products like q f , we represent the momentum operator as h 0| i ~ d p = (15.33) i dq0 because then ~ d ~ d ~ ~ d q0 pq f = q0 q f = q0 q0 f = + q0 q0 f (15.34) h | | i i dq h | | i i dq h | i i i dq h | i 0 0 ✓ 0 ◆ which respects the commutation relation [q, p]=i~. So too in inner products φ f of eigenstates φ of φ(x,t) h 0| i | 0i φ(x,t) φ0 = φ0(x) φ0 (15.35) | i | i 15.5 Functional Di↵erential Equations 631 we can represent the momentum ⇡(x,t) canonically conjugate to the field φ(x,t) as the functional derivative ~ δ ⇡(x,t)= (15.36) i δφ0(x) because then ~ δ φ0 ⇡(x0,t)φ(x,t) f = φ0 φ(x,t) f h | | i i δφ0(x0)h | | i ~ δ = φ0(x) φ0 f (15.37) i δφ0(x0) h | i ~ δ 3 = δ(x x0) φ0(x0) d x0 φ0 f i δφ (x ) − h | i 0 0 ✓Z ◆ ~ δ = δ(x x0)+φ0(x) φ0 f i − δφ (x ) h | i ✓ 0 0 ◆ = φ0 i~δ(x x0)+φ(x,t) ⇡(x0,t) f h |− − | i which respects the equal-time commutation relation [φ(x,t),⇡(x0,t)] = i ~ δ(x x0).