MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 18 doc

20 GREEN'S FUNCTIONS and PATH INTEGRALS In 1827 Brown investigates the random motions of pollen suspended in wa- ter under a microscope. The irregular movements of the pollen particles are due to their random collisions with the water molecules. Later it becomes clear that many small objects interacting randomly with their environment behave the same way. Today this motion is known as Brownian motion and forms the prototype of many different phenomena in diffusion, colloid chem- istry, polymer physics, quantum mechanics, and finance. During the years 1920- 1930 Wiener approaches Brownian motion in terms of path integrals. This opens up a whole new avenue in the study of many classical systems. In 1948 Feynman gives a new formulation of quantum mechanics in terms of path integrals. In addition to the existing Schrodinger and Heisenberg formu- lations, this new approach not only makes the connectlion between quantum and classical physics clearer, but also leads to many interesting applications in field theory. In this Chapter we introduce the basic features of this technique, which has many interesting existing applications and tremendous potential for future uses. 20.1 BROWNIAN MOTION AND THE DIFFUSION PROBLEM Starting with the principle of conservation of matter, equation as we can write the diffusion (20.1) 633 634 GREEN’S FUNCTIONS AND PATH INTEGRALS where p(T‘,t) is the density of the diffusing material and D is the diffusion constant, which depends on the characteristics of the medium. Because the diffusion process is also many particles undergoing Brownian motion at the same time, division of p(7,t) by the total number of particles gives the probability, w(+,t), of finding a particle at 7 and t as (20.2) 1 N w(7,t) = -p(7,t). Naturally, w(7, t) also satisfies the diffusion equation: (20.3) For a particle starting its motion from 7 = 0, we have to solve Equation (20.3) with the initial condition limw(7,t) + S(?). (20.4) t-0 In one dimension we write Equation (20.3) as (20.5) and by using the Fourier transform technique we can obtain its solution as 1 ZU(X,t) = ~ { -& } . (20.6) Note that, consistent with the probability interpretation, W(X, t) is always positive. Because it is certain that the particle is somewhere in the interval (-co, co), W(X, t) also satisfies the normalization condition = 1. (20.7) For a particle starting its motion from an arbitrary point, (zo,t~), we write the probability distribution as where W(X, t, 20, to) is the solution of (20.9) WIENER PATH INTEGRAL APPROACH TO BROWNIAN MOTION 635 satisfying the initial condition lim W(X, t, zo, to) -+ S(X - ZO) (20.10) t-to and the normalization condition &W(z, t, Xo, to) = 1. (20.11) .la_ From our discussion of Green’s functions in Chapter 19 we recall that W(X, t, XO, to) is also the propagator of the operator (20.12) Thus, given the probability at some initial point and time, w(z0, to), we can find the probability at subsequent times, w(z, t), by using W(z, t, so, to) as 00 w(z,t) = d50W(z,t)X~,to)w(20,tO)) t > to. (20.13) s, L Combination of propagators gives us the Einstein-Smoluchowski-Kolmogorov- Chapman (ESKC) equation: 00 W(X,~,XO,~O) = &’W(X,t,z’,t’)W(z’,t’,ico,to), t > t’ >to. (20.14) The significance of this equation is that it gives the causal connection of events in Brownian motion as in the Huygens-Fresnel equation. 20.2 WIENER PATH INTEGRAL APPROACH TO BROWNIAN MOTION In Equation (20.13) we have seen how to find the probability of finding a particle at (z,t) from the probability at (zo,to) by using the propagator W(z, t, XO, to). We now divide the interval between to and t into N + 1 equal segments: At, = ti - ti-1 t - to N-i-1’ - - (20.15) which is covered by the particle in N steps. The propagator of each step is given as W(Xi, ti, 2i-1, ti-1) = 1 }. (20.16) J47rD(ti -ti-,) 4D(ti - ti-1) 636 GREEN’S FUNCTIONS AND PATH INTEGRALS fig. 20.1 Paths C[zo,o,to;z,t] for the pinned Wiener measure Assuming that each step is taken independently, we combine propagators N times by using the ESKC relation to get the propagator that takes us from (20, to) to (z, t) in a single step as This equation is valid for N > 0. Assuming that it is also valid in the limit as N -+ 00, that, is as At; -+ 0, we write W(2, t, 20, to) = (20.18) Here, T is a time parameter (Fig. 20.1) introduced to parametrize the paths as ~(7). We can also write W(z, t, zo,to) in short as W(z,t,zc),to) = Njexp{-& p(T)dT} i)z(7), (20.20) WIENER PATH INTEGRAL APPROACH TO BROWNIAN MOTION 637 where N is a normalization constant and Dx(T) indicates that the integral should be taken over all paths starting from (z0,to) and end at (z,t). This expression can also be written as W(z, t, zo,to) = 1 &&), (20.2 1) C[zo.to;~,tl where d,z(~) is called the Wiener measure. Because dwz(r) is the measure for all paths starting from (zo, to) and ending at (z, t), it is called the pinned (conditional) Wiener measure (Fig. 20.1). Summary: For a particle starting its motion from (zo,to), the propagator W(z, t, zo, to) is given as This satisfies the differential equation with the initial condition limt4to W(z,t, zo, to) + b(z - zo). In terms of the Wiener path integral the propagator W(z, t, 20, to) is also expressed as W(z, t, 20, to) = .i’ dW47). (20.24) C[~O,tO;Z.Jl The measure of this integral is Because the integral is taken over all continuous paths from (20, to) to (3, t), which are shown as C[zo, to; z, t], this measure is also called the pinned Wiener measure (Fig. 20.1). For a particle starting from (zo,to) the probability of finding it in the interval Ax at time t is given by (20.26) In this integral, because the position of the particle at time t is not fixed, d,z(~) is called the unpinned (or unconditional) Wiener measure. At 638 GREEN’S FUNCTIONS AND PATH INTEGRALS Fig. 20.2 Paths C[zo,lo;t] for the unpinned Wiener measure time t, because it is certain that the particle is somewhere in the interval z E [-oo,oo], we write (Fig. 20.2) The average of a functional, F[z(t)], found over all paths C[zo, to; t] at time t is given by the formula In terms of the Wiener measure we can express the ESKC relation as (20.28) THE FEYNMAN-KAC FORMULA AND THE PERTURBATIVESOLUTION OF THE BLOCH EQUATION 639 20.3 THE FEYNMAN-KAC FORMULA AND THE PERTURBATIVE SOLUTION OF THE BLOCH EQUATION We have seen that the propagator of the diffusion equation, aw(z,t) a2w(z, t) = 0, at 8x2 (20.30) can be expressed as a path integral [Fq. (20.24)]. However, when we have a closed expression as in Equation (20.22), it is not clear what advantage this new representation has. In this section we study the diffusion equation in the presence of interactions, where the advantages of the path integral approach begin to appear. In the presence of a potential V(z), the diffusion equation can be written as aw(x, t) 82w(z, t) = -V(z, t)w(z, t). ax2 at (20.3 1) We now need a Green’s function, WD, that satisfies the inhomogeneous equa- tion - awD(Z, t, Z‘, t’) at z’)6(t - t’), (20.32) so that we can express the general solution of (20.31) as ~(2, t) = WO(X, t) - W~(Z, t, z’, t’)V(z’, ~’)w(z’, t’)&’dt’, (20.33) where wo(z, t) is the solution of the homogeneous part of Equation (20.31), that is, Equation (20.5). We can construct WD(Z, t, z’, t’) by using the prop agator, W(z, t, z’, t’), that satisfies the homogeneous equation (Chapter 19) 11 dW(z,t,x’,t’) d2W(z,t,z’,t’) at ax2 -D = 0, (20.34) as WD(z,t,z’,t’) = W(z,t,Z’,t‘)e(t - t’). (20.35) Because the unknown function also appears under the integral sign, Equation (20.33) is still not the solution, that is, it is just the integral equation version of Equation (20.31). On the other hand, WB(Z, t, x’, t’), which satisfies 640 GREEN'S FUNCTIONS AND PATH INTEGRALS The first term on the right-hand side is the solution of the homogeneous equation [Eq. (20.34)], which is W. However, because t > to we could also write it as W,. A very useful formula called the Feynman-Kac formula (theorem) is given as 1 t WB(Z, t, Zo, 0) = J' ci,z(T) exp { - ciTv[x(T), 7-1 . (20.38) This is a solution of Equation (20.36), which is also known as the Bloch equation, with the initial condition Iim WB(x, t, XI, t') = 6(z - d). (20.39) The Feynman-Kac theorem constitutes a very important step in the develop ment of path integrals. We leave its proof to the next section and continue by writing the path integral in Equation (20.38) as a Riemann sum: G[zo,O;z,t] t+ t' We have taken E = ti - ti-1 t -to Nfl' - (20.41) The first exponential factor in Equation (2.40) is the solution [&. (2.18)] of the homogeneous equation. After expanding the second exponential factor as (20.42) N . NN we integrate over the intermediate x variables and rearrange to obtain WB(Z, t, xo, to) = W(x, t, xo, to) (20.43) j=1 DERIVATION OF THE FEYNMAN-KAC FORMULA 641 In the limit as E -+ 0 we make the replacement EX~ -+ h”,tj. We also suppress the factors of factorials, (l/n!), because they are multiplied by E~, which also goes to zero as E -+ 0. Besides, because times are ordered in Equation (20.43) as we can replace W with WD in the above equation and write WB as (20.44) Now WB(z,t,~o,tO) no longer appears on the right-hand side of this equa- tion. Thus it is the perturbative solution of Equation (20.37) by the itera- tion method. Note that W~(x,t,xo,to) satisfies the initial condition given in Equation (20.39). 20.4 DERIVATION OF THE FEYNMAN-KAC FORMULA We now show that the Feynman-Kac formula, is identical to the iterative solution to all orders of the following integral equation: which is equivalent to the differential equation with the initial condition given in Equation (20.39). We first show that the Feynman-Kac formula satisfies the ESKC [Eq. (20.14)] relation. Note that we write V[Z(T)] instead of V[Z(T),T] when there 642 GREEN'S FUNCTIONS AND PATH INTEGRALS (20.48) In this equation x, denotes the position at t, and x denotes the position at t. Because C[ZO, 0; x,, t,; z, t] denotes all paths starting from (xo,O), passing through (x,,t,) and then ending up at (x,t), we can write the right hand-side of the above equation as dwx(7) exp { - Lts d~v[z(~)]} (20.49) 1: dxs ~xo,ox*,ts;x,tl (20.50) = WB(Z, 4 z0,O). (20.51) From here, we see that the Feynman-Kac formula satisfies the ESKC relation as 00 dx, WB (z, t, zs, tS)WB (xs, ts, 20,O) = WB (274 xo, 0). (20.52) With the help of Equations (20.21) and (20.22), we see that the Feynman- .I_, Kac formula satisfies the initial condition lim WB(x, t, xo, 0) + 6(z - zg) (20.53) t-0 and the functional in the Feynman-Kac formula satisfies the equality (20.54) We can easily show that this is true by taking the derivative of both sides. Because this equality holds for all continuous paths x(s), we take the integral of both sides over the paths C[zo, 0; z, t] via the Wiener measure to get (20.55) [...]... fi/2m Now the path integral in Equation (20.142) can be taken as a Wiener path integral, and then going back to real time, we can obtain the propagator of the Schrijdinger equation as Equation (20.138) 658 GREEN’S FUNCTIONS AND PATH INTEGRALS 20.7.2 Schrodinger Equation in the Presence of Interactions In the presence of interactions the Schrdinger equation is given as a*(x,t) ~- at Making the transformation... converted into an Ndimensional integral [Eq (20.92)] 20.6.2 Evaluating Path Integrals with the ESKC Relation We introduce this method by evaluating the path integral of a functional F [ x ( T ) ] z ( T ) , in the interval [O,t] via the unpinned Wiener measure Let = 7 be any time in the interval [O,t] Using Equation (20.28) and the ESKC relation, we can write the path integral ~ C l x O , O ;~ l, x... restrictions To bring this integral into a form that can be evaluated in practice, we introduce the phase space lattice by dividing the time interval t E [t’, into t”] N 1 slices as + (20.158) Now the propagator becomes K(q”,t”, q’,t’) (20.159) In this expression, except for the points at qN+1 = q” and qo = q’, we have t o integrate over all q and p Because the Heisenberg uncertainty principle forbids... dw4T), (20.82) where C[zo,to; 2 , t] denotes all continuous paths starting from (20,t o ) and ending a t ( 2 ,t) Before we discuss techniques of evaluating path integrals, we METHODS OF CALCULATING PATH INTEGRALS 647 should talk about a technical problem that exists in Equation (20.80) In this expression, even though all the paths in C[ZO, X,t ] are continuous, because to; of the nature of the Brownian... GREEN’S FUNCTIONS AND PATH INTEGRALS From Equation (20.27), t,he value of the last integral is one Finally, using Equations (20.24) and (20.22), we obtain = xo 20.6.3 (20.95) Path Integrals by the Method of Finite Elements We now evaluate the path integral we have found above for the functional F[x(-r)] x ( r )by using the formula [Eq (20.17)]: = (20.96) (20.97) (20.98) = xo (20.99) In this calculation... 4(t)function as (20.114) METHODS OF CALCULATING PATH INTEGRALS 653 Fig 20.4 Path and deviation in the “semiclassical” method Finally the propagator is obtained as W(Z, t ,xo,0) = (x- o)2 (20.115) eexp{4Dt } 1 ~ In this case the ‘‘semiclassical’’ method has given us the exact result For more complicated cases we could use the method of time slices to find the factor 4(t - to) In this example we have also given... ; + ,tl)]} (20.163) Substituting this in Equation (20.159) and taking the momentum integral, we find the propagator as K(q",t", q', t ' ) = lim N+E'O where film [d dql ] m exp{ is}, (20.164) S is given as N 1 =o )] (20.165) In the continuum limit this becomes where is the classical action In other words, the phase space path integral reduces to the standard Feynman path integral 661 ... dz d dL dL (20.154) - In most cases this extremum is a minimum (Morse and Feshbach, p 281) As in the applications of path integrals to neural networks, sometimes a system can have more than one extremum In such cases, a system could find itself in a local maximum or minimum Is it then possible for such systems to reach the desired global minimum? If possible, how is this achieved and how long will it... help For this reason in 1951 Feynman introduced the phase space version of the 660 GREEN’S FUNCTIONS AND PATH INTfGRALS path integral: This integral is to be taken over t , where t E [t’,t’’] Dq means that the integral is taken over the paths q ( t ) , fixed between q”(t”) = q” and q’(t’) = q’ and which make S[z] an extremum The integral over momentum p is taken over the same time interval but without... p(z,t ) is now positive definite and a Gaussian, which can be normalized Can we also write the propagator of the Schrodinger equation as a path 2h integral? Making the D -+ - replacement in Equation (20.132) we get 2m K(x,t,d,t/) = (20.142) where This definition was given first by Feynman, and d F x ( 7 ) is known as the Feynman measure The problem in this definition is again the fact that the argument . also leads to many interesting applications in field theory. In this Chapter we introduce the basic features of this technique, which has many interesting existing applications and tremendous potential. denotes all continuous paths starting from (20, to) and ending at (2, t). Before we discuss techniques of evaluating path integrals, we METHODS OF CALCULATING PATH INTEGRALS 647 should. this method by evaluating the path integral of a functional F[x(T)] = z(T), in the interval [O,t] via the unpinned Wiener measure. Let 7 be any time in the interval [O,t]. Using Equation