15 Functional Derivatives

Transcription

1 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 imension maps the coorinate q(t), which is a function of the time t, into a number the action of the process. If the particle of mass m is moving slowly an freely, then for the interval (,t ) its action is S 0 [q] = t m q(t). (15.1) If the particle is moving in a potential V (q(t)), then its action is S[q] = t " m q(t) V (q(t))#. (15.) 15. Functional Derivatives A functional erivative is a functional G[f][h] = G[f + h] (15.3) of a functional. For instance, if G n [f] is the functional G n [f] = x f n (x) (15.4)

2 15. Functional Derivatives 619 then its functional erivative is the functional that maps the pair of functions f,h to the number G n [f][h] = G n[f + h] = x (f(x)+ h(x)) n = x nf n 1 (x)h(x). (15.5) Physicists often use the less elaborate notation G[f] = G[f][ (x y)] (15.6) f(y) in which the secon function h(x)is (x y). Thus, in the preceing example G[f] f(y) = x nf n 1 (x) (x y) =nf n 1 (y). (15.7) Functional erivatives of functionals that involve powers of erivatives also are easily ealt with. Suppose that the functional involves the square of the erivative f 0 (x) G[f] = x f 0 (x). (15.8) Then its functional erivative is G[f][h] = G[f + h] = x f 0 (x)+ h 0 (x) = x f 0 (x)h 0 (x) = x f 00 (x)h(x) (15.9) in which we have integrate by parts an use suitable bounary conitions to rop the surface terms. In physics notation, we have G[f] f(y) = x f 00 (x) (x y) = f 00 (y). (15.10)

3 60 Functional Derivatives Let s now compute the functional erivative of the action (15.), which involves the square of the time-erivative q(t) an the potential energy V (q(t)) S[q][h] = S[q + h] = apple m q(t)+ ḣ(t) V (q(t)+ h(t)) h i = m q(t)ḣ(t) V 0 (q(t))h(t) = m q(t) V 0 (q(t)) h(t) (15.11) where we once again have integrate by parts an use suitable bounary conitions to rop the surface terms. In physics notation, this is S[q] q(t 0 ) = m q(t) V 0 (q(t)) (t t 0 )= m q(t 0 ) V 0 (q(t 0 )). (15.1) In these terms, the stationarity of the action S[q] is the vanishing of its functional erivative either in the form S[q][h] = 0 (15.13) for arbitrary functions h(t) (that satisfy the bounary conitions) or equivalently in the form which is Lagrange s equation of motion S[q] q(t) = 0 (15.14) m q = V 0 (q). (15.15) Here s a shortcut to the functional erivative in the notation of physics: G[f] f(y) = G[f + y] in which the function h(x) is replace by y (x) = (x y). (15.16) Example 15.1 (Shortest Path is a Straight Line) of the path (x, y(x)) from (x 0,y 0 )to(x 1,y 1 )is On a plane, the length L[y] = x1 x1 p p x + y = 1+y 0 x. (15.17) x 0 x 0

4 15.3 Higher-Orer Functional Derivatives 61 The shortest path y(x) minimizes this length L[y] L[y] y = L[y + h] = x1 y 0 h 0 = p x = x 0 1+y 0 x1 x 0 x1 x 0 p 1+(y 0 + h 0 ) x y 0 h p x (15.18) x 1+y 0 since h(x) satisfies h(x 0 )=h(x 1 ) = 0. Di erentiating, we fin L[y] y = x1 x 0 h y 00 x (15.19) 1+y0 which can vanish for arbitrary functions h(x) only if y 00 = 0, which is to say only if y(x) is a straight line, y = mx + b Higher-Orer Functional Derivatives The secon functional erivative is G[f][h] = G[f + h]. (15.0) So if G N [f] is the functional G N [f] = f N (x)x (15.1) then G N [f][h] = G N[f + h] = (f(x)+ h(x)) N x = N h (x)f N (x) x = N(N 1) f N (x)h (x)x. (15.) Example 15. ( S 0 ) (15.1) is The secon functional erivative of the action S 0 [q] S 0 [q][h] = t m q(t) + h(t) t h(t) = m 0 (15.3)

5 6 Functional Derivatives an is positive for all functions h(t). The stationary classical trajectory is a minimum of the action S 0 [q]. q(t) = t t q(t )+ t t t q( ) (15.4) The secon functional erivative of the action S[q] (15.) is S[q][h] = t " = t m " m q(t) h(t) + h(t) V (q(t)+ h(t))# V h (t) (t) (15.5) an it can be positive, zero, or negative. Chaos sometimes arises in systems of several particles when the secon variation of S[q] about a stationary path is negative, S[q][h] < 0while S[q][h] = 0. The nth functional erivative is efine as n G[f][h] = n n G[f + h]. (15.6) The nth functional erivative of the functional (15.1) is n G N [f][h] = N! (N n)! f N n (x)h n (x)x. (15.7) 15.4 Functional Taylor Series It follows from the Taylor-series theorem (section 4.6) that e G[f][h] = 1X n=0 n 1 n! G[f][h] = X n=0 1 n! n n G[f + h] = G[f + h] (15.8) which illustrates an avantage of the present mathematical notation. The functional S 0 [q] of Eq.(15.1) provies a simple example of the func-

6 15.5 Functional Di erential Equations 63 tional Taylor series (15.8): e S 0 [q][h] = S 0 [q + h] = m t q(t)+ ḣ(t) = m t q (t)+ q(t)ḣ(t)+ḣ (t) = m t q(t)+ḣ(t) = S0 [q + h]. (15.9) If the function q(t) makes the action S 0 [q] stationary, an if h(t) is smooth an vanishes at the enpoints of the time interval, then S 0 [q + h] =S 0 [q]+s 0 [h] (15.30) in which the functions q(t) an h(t) respectively satisfy the bounary conitions q(t i )=q i an h( )=h(t ) = 0. More generally, if q(t) makes the action S[q] stationary, an h(t) is any loop from an to the origin, then S[q + h] = e S[q][h] = S[q]+ 1X n= 1 n n! n S[q + h]. (15.31) If further S [q] is purely quaratic in q an q, like the harmonic oscillator, then S [q + h] =S [q]+s [h]. (15.3) 15.5 Functional Di erential Equations In inner proucts like hq 0 fi, we represent the momentum operator as because then hq 0 pq fi = ~ i q 0 hq0 q fi = ~ i p = ~ i q 0 (15.33) ~ q 0 q 0 hq 0 fi = i + ~ q0 i which respects the commutation relation [q, p] = i~. q 0 hq 0 fi (15.34)

7 64 Functional Derivatives So too in inner proucts h 0 fi of eigenstates 0 i of (x,t) (x,t) 0 i = 0 (x) 0 i (15.35) we can represent the momentum (x,t) canonically conjugate to the fiel (x,t) as the functional erivative (x,t)= ~ i 0 (x) (15.36) because then h 0 (x 0,t) (x,t) fi = ~ i 0 (x 0 ) h 0 (x,t) fi = ~ 0 i 0 (x 0 (x)h 0 fi (15.37) ) = ~ i 0 (x 0 (x x 0 ) 0 (x 0 ) 3 x 0 h 0 fi ) = ~ (x x 0 )+ 0 (x) i 0 (x 0 h 0 fi ) = h 0 i~ (x x 0 )+ (x,t) (x 0,t) fi which respects the equal-time commutation relation [ (x,t), (x 0,t)] = i ~ (x x 0 ). (15.38) We can use the representation (15.36) for (x) to fin the wave function of the groun state 0i of the hamiltonian H = 1 +(r ) + m 3 x (15.39) where we set ~ = c = 1. We will use the trick we use in section.11 to fin the groun state 0i of the harmonic-oscillator hamiltonian In that trick, one writes H 0 = p m + m! q. (15.40) H 0 = 1 m (m!q ip)(m!q + ip)+i! [p, q] = 1 m (m!q ip)(m!q + ip)+1 ~! (15.41) an seeks a state 0i that is annihilate by m!q + ip hq 0 m!q + ip 0i = m!q 0 + ~ q 0 hq 0 0i =0. (15.4)

8 15.5 Functional Di erential Equations 65 The solution to this i erential equation is hq 0 0i = q 0 hq0 0i = m! ~ m!q0 ~ hq0 0i (15.43) 1/4 m!q 0 exp ~ (15.44) in which the prefactor is a constant of normalization. So extening that trick to the hamiltonian (15.39), we factor H H = 1 h p i r + m i ihp r + m + i 3 x + C (15.45) in which C =(i/) R [, p 4 + m ] 3 x is an (infinite) constant. The groun state 0i of H therefore must satisfy the functional i erential equation h 0 p r + m + i 0i = 0 or h 0 0i 0 (x) = p r + m 0 (x) h 0 0i. (15.46) The solution to this functional i erential equation is 1 h 0 0i = N exp 0 (x) p r + m 0 (x) 3 x (15.47) in which N is a normalization constant. To see that this functional oes satisfy equation (15.46), we compute the erivative h 0 + h 0i = N apple exp h p 4 + m 0 + h 3 x (15.48) which at =0is h 0 + h 0i = 1 apple h(x) p 4 + m 0 (x) 3 x + 0 (x) p 4 + m h(x) 3 x h 0 0i. (15.49) We integrate the secon term by parts an rop the surface terms because the smooth function h goes to zero quickly as its arguments go to infinity. We then have h 0 + h 0i = Letting h(x 0 )= (3) (x 0 x), we arrive at (15.46). h(x 0 ) p 4 + m 0 (x 0 ) 3 x 0 h 0 0i. (15.50)

9 66 Functional Derivatives The spatial Fourier transform 0 (p) 0 (x) = e ip x 0 (p) 3 p ( ) 3 (15.51) satisfies 0 ( p) = 0 (p) since 0 is real. In terms of it, the groun-state wave-function is 1 h 0 0i = N exp 0 (p) p p + m 3 p. (15.5) Example 15.3 (Other Theories, Other Vacua) We can fin exact groun states for interacting theories with hamiltonians like H = 1 h p i r + m n + c n i ihp r + m + c n n + i 3 x. (15.53) The state i will be an eigenstate of H with eigenvalue zero if h 0 i 0 (x) = hp r + m 0 (x)+c n 0n i h 0 i. (15.54) By extening the argument of equations ( ), one may show (exercise 15.3) that the wave functional of the vacuum is apple h 0 1 i = N exp p 0 r + m 0 + c n 0n+1 3 x (15.55) n +1 which is normalizable only when n is o. Exercises 15.1 Compute the action S 0 [q] (15.1) for the classical path (15.4). 15. Use (15.5) to fin a formula for the secon functional erivative of the action (15.) of the harmonic oscillator for which V (q) =m! q / Show that (15.55) satisfies (15.54) Derive (15.5) from equations (15.47 & 15.51).