THE MYSTERY OF STOCHASTIC MECHANICS Edward Nelson Deparmen of Mahemaics Princeon Universiy 1
Classical Hamilon-Jacobi heory N paricles of various masses on a Euclidean space. Incorporae he masses in he fla Riemannian meric m ij, he mass ensor. Then if v i is a velociy, v i = m ij v j is a momenum. Kineic energy: 1 2 vi v i. Poenial energy: V. Lagrangian: L = 1 2 vi v i V. Posiion a ime of he configuraion: ξ(). Iniial ime:. Final ime: 1. Hamilon s principal funcion: S(x, ) = 1 L ( ξ(s) ) ds. Subsanial derivaive (derivaive along rajecories): Then DS = L. D = + vi i. 2
Vecor field: v wih dξ d = v. Principle of leas acion in Hamilon-Jacobi heory: v is a criical poin for S, for unconsrained variaions. Tha is, le v be anoher vecor field, le δv = v v, and denoe by a prime quaniies wih v replaced by v. Then D(S S) = D S DS + (D D )S = L L δv i i S = L L δv i i S + o(δv). Now L L = v i δv i + o(δv), S S = 1 (v i i S)δv i ds + o(δv). Since his is rue for all variaions, we have he Hamilon-Jacobi condiion: v i = i S. 3
Togeher wih DS = L; i.e., s + vi i S = 1 2 vi v i V, his gives he Hamilon-Jacobi equaion S + 1 2 i i S + V = 0. If we ake he gradien we obain Newon s equaion F = ma. 4
Sochasic Hamilon-Jacobi heory Following Guerra and Morao, consruc a conservaive Markovian dynamics for a Markov process ξ. Wiener process (Brownian moion) w wih Edw i ()dw i () = hd. Kinemaics: dξ() = b ( ξ(), ) d + dw(). Dynamics: δe L d = 0 (heurisically). The rajecories are non-differeniable, so wha is he meaning of 1 2 dξ i d dξ i d in he Lagrangian L? 5
Le d > 0 and dξ() = ξ( + d) ξ(), so dξ d is a quoien, no a derivaive. Compue E 1 dξ i dξ i up o o(1). 2 d d dξ i = +d Noe: dw is of order d 1/2. b i( ξ(r), r ) dr + dw i. dξ i = where +d + dw i b i( ξ() + r = b i d + k b i W k + dw i + O(d 2 ) b ( ) ξ(s), s)ds + w(r) w(), r dr W k = +d [w k (r) w k ()] dr. Therefore 6
1 2 dξi dξ i = 1 2 bi b i d + b i dw i d + k b i W k dw i + 1 2 dwi dw i + o(d 2 ). Now EW k dw i = hδ k i +d (r )d = h 2 δk i, so k b i W k = h 2 ib i. The erm b i dw i d is singular, of order d 3/2, bu is expecaion is 0. Finally, Edw i dw i = h 2 nd. Hence we have he sough-for resul: E 1 2 dξ i d dξ i d = 1 2 bi b i + h 2 ib i + h 2 n d + o(1). 7
The singular erm h n is a consan, no depending on he rajecory, and i drops ou of he 2 d variaion. Le and L + = 1 2 bi b i + h 2 ib i V I = E 1 L + ( ξ(s) ) ds. Le δb be a vecor field, b = b + δb, and as before denoe by primes quaniies wih b replaced by b. Le ξ saisfy dξ () = b ( ξ (), ) d + dw() wih ξ () = ξ() for he iniial ime. Definiion: ξ is criical for L in case I I = o(δb). 8
Sochasic Hamilon s principal funcion: Then S(x, ) = E x, 1 DS = L + L + ( ξ(s), s ) ds. where now D is he mean forward derivaive: ( ) df() Df = lim E. d 0+ d and As before, D(S S) = D S DS + (D D )S Now = L + L + δb i i S = L + L + δb i i S + o(δb). L + L + = b i δb i + h 2 iδb i + o(δb) ( ) I I = E 1 ( b i i S + h ) 2 i δb i ds + o(δb). 9
Digression on Markovian kinemaics: Markov process: given he presen, pas and fuure are independen. The wo direcions of ime are on an equal fooing. In addiion o he forward drif b here is he backward drif b. v = b + b 2 u = b b 2 curren velociy, osmoic velociy. Le ρ be he probabiliy densiy of ξ. Then v = (vρ) u = h ρ 2 ρ curren equaion, osmoic equaion. End of digression. 10
The equaion (*), namely I I = E 1 ( b i i S + h ) 2 i δb i ds + o(δb), is awkward because i involves i δb i. Inegrae by pars: E h 2 iδb i( ξ(s), s ) h ( = i δb i) ρ dx R n 2 = R δbi u i ρ dx. n Since b u = v, I I = E 1 (v i i S)δb i ds + o(δb). This is rue for all variaions, so ξ is criical for L if and only if he sochasic Hamilon-Jacobi condiion holds: v i = i S. 11
Le R = h 2 log ρ, so i R = u i. If we wrie ou DS = L + and express everyhing in erms of R and S, we obain he sochasic Hamilon-Jacobi equaion: (1) S + 1 2 i S i S + V 1 2 i R i R h R = 0. 2 Expressing he curren equaion v = (vρ) in erms of R and S we obain: (2) R + ir i S + h S = 0. 2 These wo equaions are a sysem of coupled nonlinear parial differenial equaions expressing necessary and sufficien condiions for a Markov process o be criical. How can we solve hem? 12
Le ψ = e 1 h (R+iS). Then he sysem is equivalen o he Schrödinger equaion ψ = ī h ] [ h2 + V ψ. 13
magneic fields Riemannian manifolds spin Bose-Einsein and Fermi-Dirac saisics exisence of finie-energy Markov processes momenum inerference 14
Non-localiy Consider wo paricles on R 1 wih iniial wave funcion ψ(0) = 1 2π e 1 4 σ 1 (0)x x where σ 1 (0) = ( 1 ) 1 1 2 and x = ( x 1 x 2 ). The x 1 and x 2 axes are unrelaed; he paricles may be separaed by an arbirarily large amoun a. A ime 0, urn on a linear resoring force (harmonic oscillaor wih circular frequency ω) for he second paricle. Then he paricles are dynamically uncoupled. 15
Since he paricles are very widely separaed and dynamically uncoupled, we should expec ha does no depend on ω. Eξ 1 ()ξ 1 (0) In fac i does no o fourh order in, bu neverheless he rajecory of he firs paricle is immediaely affeced by he choice of ω in a far disan place. For me his is unphysical, especially since he effec does no depend on he separaion a. 16
A wrong predicion Consider wo harmonic oscillaors, abou wo widely separaed poins a 1 and a 2, wih circular frequency 1, and le X i be he Heisenberg posiion operaor, in unis of disance from a i, wih Heisenberg momenum operaor P i. Then for i = 1, 2. X i () = cos X i (0) + sin P i (0) P i () = sin X i (0) + cos P i (0) Le he Heisenberg sae vecor ψ 0 be a real Gaussian cenered a (0, 0), and wrie A = (ψ 0, Aψ 0 ). Then X i () = P i () = 0 since his is rue for = 0. The operaors X 1 () and X 2 (s) commue. Choose ψ 0 so ha he correlaion X 1 (0)X 2 (0) is 90%. Thus he oscillaors are enangled bu dynamically uncoupled. The quanum mechanical correlaion funcion X 1 ()X 2 (0) is periodic of period 2π since X 1 () is. Hence X 1 ()X 2 (0) =.9 whenever is a muliple of 2π. 17
Bu le (ξ 1, ξ 2 ) be he corresponding Markov process of sochasic mechanics. This is a diffusion process and i evenually loses all memory of where i sared. Thus whereas lim E( ξ 1 (2πn)ξ 2 (0) ) = 0 n X 1 (2πn)X 2 (0) =.9. Here we have an empirical difference beween he predicions of quanum mechanics and sochasic mechanics. Measuremens of he posiion of he firs paricle a ime and of he second paricle a ime 0 do no inerfere wih each oher, and he wo heories predic oally differen saisics. Does anyone doub ha quanum mechanics is righ and sochasic mechanics is wrong? 18