path integral formulation of general diffusion processes

Z. Physik B 26, 281-290 (1977) Zeitschrift Physik B

© by Springer-Verlag 1977

Path Integral Formulation of General Diffusion Processes

Robert Graham

Universit~t Essen- Gesamthochschule, Fachbereieh Physik

Received October 14, 1976

A method is given for the derivation of a path integral representation of the Green's c3P

function solution P of equations ~ = L P , L being some Liouville operator. The method

is applied to general diffusion processes. Feynman's path integral representation of the Schr6dinger equation and Stratonovich's path integral representation of multivariate Markovian processes are obtained as special cases if the metric of the general diffusion process is flat. For curved phase spaces our result is a nontrivial generalization and new. New applications e.g. to quantized motion in general relativity, to transport processes in inhomogeneous systems, or to nonlinear non-equilibrium thermodynamics are made possible. We expect applications to be fruitfull in all cases where (continuous) macroscopic transport processes in Riemann geometries have to be considered.

1. Introduction

In the study of the dynamics of macroscopic systems one is led, in a more or less cogent way, to the wellknown concepts of a complete set of macrovari- ables q~{q,,}, v = l ... n and the multivariate continuous Markovian random process in time which is realized by their motion. Two wellknown descriptions of such processes exist: the Langevin equations (and their mathematically refined versions: the Ito equations), and the stochastically equivalent Fokker Planck equation [1]. Both descriptions have in common, that they allow, at least in principle, to determine the stochastic properties of the system at time t, if those properties are known at some earlier time t o . However, a third equivalent description of such processes and such systems exists, which is now receiving an increased a t ten t ion-par t ly due to the recent de- velopment of new analytic methods. In that description the dynamics of the system is formulated by writing down a probability density for observing a complete path of the system in the phase space spanned by the macroscopic variables. Such a description was pioneered by Onsager and Machlup 1-2] who derived an expression for the probability density

of a path. Their analysis and subsequent work [3,4] was restricted to the special case, where the system undergoes a linear Gaussian process. For this case, Onsager and Machlup showed, that the probability density for a path q(t) could be written in the functional form [2, 3]

W([q(~)])~exp - S d'c L(gl(r), q(z)) (1.1)

where equality holds in (1.1) apart from a normali- zation factor. The function L(0('c), q(~)) is often called "Onsager-Machlup function". For a linear Gaussian process it is of the general form (summation over repeated indices is always implied)

L " 1 - 1 - A (q, q) = ~ Q~, (q~ - ~ q.~)(glu - Au,, q,~) (1.2)

where Av). and Q~I are constant matrices. We will refer to L in Equation(1.1) as a Lagrangian in the following, for reasons explained below. Among the virtues of the description (1.1) of fluctuat- ing macroscopic systems are the following: i) An expression like (1.1) gives information about complete paths of the system in the interval-0o < t < c~. E.g. the condition for the most probable path

282 R. Graham: Path Integral Formulation of General Diffusion Processes

between two given end points is given by

t2

~ L(0(~), q(~)) d~ =0 (1.3) t l

where q(tl), q(t2) are kept fixed during the variation of paths. Equation (1.3) immediately leads to a set of Euler Lagrange equations for the most probable path with L as a Lagrangian. Equation(1.1) allows to take into account the fluctuations around the most probable path.

ii) Equation (1.1) gives a formulation of the macroscopic stochastic dynamics which is at least formally very similar to the formulation of the macroscopic stochastic equilibrium state in the form W(q)~exp[S(q)] where W(q) is the probability density to observe q and S(q) is a thermodynamic potential. Thus one may hope to found on the Lagrangian L(gl, q) a rational formulation of non-equilibrium thermodynamics. L there takes the place of a non- equilibrium thermodynamic potential. For systems close to equilibrium L is closely related to the en- tropy production, and the principle of minimum en- tropy production can be derived from the extremum principle just mentioned. A report on this theory, which has also interesting implications for systems far from thermodynamic equilibrium is in prepara- tion and will be published [5].

iii) By means of (1.1) all averages, correlation functions, cond i t i ona l -o r multi-time probability densities etc. can be expressed in terms of path integrals. Hence, all the approximative techniques, which have been developed for such integrals are put to one's service (cf. e.g. [4]; in the case (1.2), of course, the exact evaluation of these path integrals is possible).

iv) A functional like (1.1) can be used to generate the diagrammatic expansions for correlation and re- sponse functions for classical systems recently given by Martin, Rose and Siggia [6] in a very concise way [7].

There have been many attempts, recently, to obtain a representation of the form (1.1) for more general diffusion processes than just linear Gaussian ones. An early partial solution of this problem by Stratonovich [8] was completely overlooked in the literature. Stratonovich's solution makes obsolete most of the recent work in this area, including some of my own (cf. below). However, the limits of validity of his result for the Lagrangian L of general diffusion processes are not given in his work. In fact, they seem to be derived here for the first time: We find below that

L in (1.1) is given correctly in Stratonovich's work if the phase space, spanned by the variables q, has a flat metric. The definition of the metric tensor of phase space used here will be given later. Here it may be sufficient to note that the results in [8] for flat metric were obtained from (1.1). (1.2) with A~,=0 by a set of transformations of both W and q. The Lagrangian for stochastic nonlinear continuous Markovian processes in curved phase spaces cannot be derived from (1.1) and (1.2) by a set of transformations. The derivation of L for such processes is the main purpose of the present paper*. By obtaining L for curved phase space we hope to open up to the methods of functional integration a new class of transport processes in Riemann geometries. Another class of applications is made possible by the observation that, for imaginary t = - i u (u real) Equation(1.1) just gives Feynman's path integral representation of a probability amplitude [4]. After a trivial generalization Stratonovich's result then be- comes equivalent to Feynman's results for wavefunc- tions in external electromagnetic fields represented in curvilinear coordinates.

Knowing the form of L also for curved spaces allows to give a consistent quantization of motion in external gravitational fields by the application of the equivalence principle. A generalization of the Klein Gordon equation to allow for external gravitational fields has already been given elsewhere [9] along these lines.

The present paper is organized as follows: In the next section the equation of motion and its formal solution are given in a form which is convenient for the following analysis. In Section 3 we give a method for deriving the Lagrangian from a solution of the equation of motion for small time intervals. Although this method is independent of the special form of the equation of motion, it is limited by an assumption on the analytic properties of the process q(t): we assume that q(t) is continuous and may be non differentiable, but can be approximated by continuous differentiable q(t) arbitrarily well. (A mathematical refinement of this somewhat loosely stated assumption, e.g. by methods used in [8, 10], would clearly be desirable). It is believed, but not proven here, that the diffusion processes considered explicitely, satisfy this requirement. It is clear, that all discontinuous processes are excluded by our assumption. In Section 4 the method given is applied to a general diffusion process. The main labor consists in obtaining a suitably accurate solution of the equation of motion for small times. This solution is obtained with

* The form of L for curved phase space was reported in Ref. 9, where an application to general relativity has been given

R. Graham: Path Integral Formulation of General Diffusion Processes 283

the help of an operator ordering technique. Some of the necessary quite extensive algebra is given in the Appendix. At the end of this section the main result of this paper, the Lagrangian L for curved phase space, together with the accompanying integration measure in function space, is obtained. Finally, in Section 5 various simple special cases of our result and its relation to some of the recent work are discussed.

2. Equations of Motion and Formal Solution

We want to study properties of continuous Markoff processes, described by an equation of motion of the form

W = L W . (2.1)

Here W is any member of the hierarchy of joint probability densities (if L acts on the "latest" variable)

W(ql t l ) , W(2)(q2 t2, ql tl), .-.,

w(N)(q N tN, ... , q~ tl) (2.2)

which characterize the stochastic process; they are densities in the n-dimensional phase space spanned by the macroscopic variables q. The Liouville operator L acts on probability densities satisfying a given set of boundary conditions. L may or may not depend on the time t. Later L will be taken in the form

L(p, q) = - l p v p~ Qv.(q) - ipvKv(q) + V(q) (2.3)

with

0 Pv =- - i c? q .

Knowledge of the single time probability density W(qt) at a given time and of the Green's function solution P(qtlqoto) of (2.1) which satisfies the initial condition P(qtolqoto)=g~(q-qo) is sufficient to con- struct the complete hierarchy (2.2) recursively by

W(N)(qN tN . . . . , q l t l )

= P(qNtNIqN- 1 tN- i) W(N-1) (qN- 1 tN- 1 , ' " , q~ tl). (2.4)

P(qtlqoO) is closely related to the operator S(t) which solves (2.1)

W(t)=S(t , to) W(t0); t

S(t, to) = T exp ~ L(z) dz (2.5) to

(T is the time ordering operator)

since, by definition

P(q tlqo 0) = S(p, q, t, 0) 6 ( q - qo). (2.6)

The solution (2.5), (2-6) of Equation (2.1) is, of course, purely formal. One way to transform it into a solution of practical value consists in ordering the non- commuting operators p and q in the operator function S(p, q, t, 0). In the following we will call a function of p and q "normally ordered" if all p operators appear on the left, i.e. act only after the multiplication with the functions of q on the right has been carried out. Let S(p, q, t, 0) be the function which we obtain by putting S(p,q,t, 0) into normal order observing the commutation rule [q,.,p~]=i~,. u. The order of p,q in g is made explicit by writing it as the normally ordered Fourier integral

S(p,q,t, t o )=~d~exp( - ip .4 ) f2 (~ ,q , t , to). (2.7)

Let us formally associate a C-number function with each operator in normal order by replacing the operators p in such functions by C-numbers p. Equation (2.7) then holds for such C-number functions and can be inverted to yield

dp f2(~,q,t, t o ) = ~ ( ~ ) ~ exp(ip.4)S(p,q, t , to). (2.8)

Let us see what would be achieved by a knowledge of

S(p,q,t, to) or O(4, q,t, to)

by looking at a two-time correlation function

( f (q(tl)) g(q(to)) )

= ~ dq l dqo f (q O W(2)(q l tl , qo to) g(qo). (2.9)

Using Equations(2.4), (2.6) and carrying out the integration over qo, we obtain

( f (q(t~)) g(q(to)) )

= ~ dq f (q) S(p, q, t~, to) W (qto) g(q). (2.10)

Inserting the normally ordered form (2.7) of S in (2.10) we let the operator e x p [ - i p . 4] act to the left on f (q) by partial integration; the appearing boundary terms are assumed to vanish due to the boundary conditions on W(q I to) and its derivatives. Using the relation exp[i~-p] f (q) = f (q + O. Equation (2.10) reduces to

( f (q(tl)) g(q(to)) ) = ~ d ~ d q f ( q + ~ ) f 2 ( ~ , q , t , , t o ) W ( q t o ) g ( q ). (2.11)

Using, in particular, f ( q ) = 6 ( q - q ~ ) , g(q)=l and W(q to) = ~5(q- qo) we obtain

P(ql tl I q0 to) = f2(qa - qo, qo, tl, to)

dp =.f , ~ ; , , exp [ ip . (q l -qo ) ]S (p , qo, tl,to). (2.12)

284 R. Graham: Path Integral Formulat ion of General Diffusion Processes

is thus just the characteristic function associated to f2({,qo, t, to), the probability density to observe an increase ~ of q at time t, if q = qo is given at time t o. Putting S(p,q,t, to) into normal order thus reduces the general solution of (2.1) to quadratures. The strategy how to achieve that is described in the following section.

3. Propagation of Normal Order from Infinitesimal to Finite Time Intervals

Putting the operator S(t, to) into normal order for arbitrary time intervals is an arduous task. What can be done, however, for our explicit form of L, Equation (2.3), is to establish the normally ordered form S(t, to) for small 8 = t - t o as an expansion in e. This expansion will be worked out in Section 4. In the present section we address ourselves to the question of how to use our knowledge of the normally ordered operator S(t, to) for small ( t - t 0 ) = e to obtain the normally ordered form at arbitrary time intervals. The answer is given in terms of a path integral. Let us iterate the right hand side of Equation (2.4) N times and use P(qlqlqoto) as initial probability density W(q~t~). Then, integrating out all intermediate variables q~¢_ 1... ql, we obtain

P(qN tN[qo to) N - 1

=~ l~ {dqjP(qj+~ tj+llqjtfl} P(ql tllqoto) (3.1) j = l

or, making use of (2.12)

f2(qN - qo, qo, tN, to) N--1 N - 1 ]

----~yIdqjexp[~_olnf2(qk+l--qk, qk,tk+l,tk) . (3.2) j = l k -

Let t=tN and t o now be the end points of a finite time interval separated into N equal time intervals of length e by the intermediate times t l , . . . , tN_l , and consider (3.2) in the limit e ~ 0

N--1

f2(q - qo, qo, t, to) = lim ~ [ I dqj e~0 j = l

N--1

"exp [k~_olnf2(qk+ l--qk, qktk+l , tk) ] . (3.3)

The expression on the right hand side is used to define the path integral in the expression

q(t) = q

f2(q-qo,qo, t, to) = ~ D#([q(z)]) ~([q(z),q(z)]). q (O) -qo

(3.4)

The integral is over all paths q(z) between q(0)=qo and q(t)=q, f2([O, q]) is a C-number functional on the space of all continuous functions q(z), and D#([ql) is

a measure of integration in that space. In fact, the function space which has to be considered is determined by the form of L. We assume that we are dealing with a continuous Markoff process. More specificly, we will assume that the processes considered may be approximated by continuous, first o r d e r - differentiable functions arbitrarily well. Actually, only the object ~([~(~),q(z)])D#([q(z)]) is defined by (3.3), (3.4). A separate definition of f2([0(z), q(z)]) is possible, if the limit e ~ 0 in (3.3) can be taken separately for the sum over k in the exponent. Generally, the limit e -~ 0 in the exponent of (3.3) will only be finite if we first subtract that part of

in O(q k + 1 - - qk, qg, tk + ~, tk),

called irregular in the following, which does not vanish at least like e in that limit. The remaining part we will call the regular part of In ~?, [ln ~Jreg" In that way we obtain the definition

In f2([~(z), q(z)])

N - 1 1 =-lim ~ e . [ln~(qk+l--qk, qk, tk-J-e, tk]reg. (3.5)

e ~ O k . _ O E

The irregular part [ln f2]irrcg is used to define the measure D#([q(z)])

N - 1

D#([q(z)])=lim [ I [dqj] ¢ ~ O j = l

"exp{kN~=i[lnf2(qk+l--qk, qk,tkq-g~,tk~irreg}. (3.6)

When carrying out the limit ~ ~ 0 in (3.5) the discrete times t~, become a continuous time z and qk turns into q(z). If the functional f2([0(z), q(z)]) would be defined on the space of all continuous and differentiable functions q(z) it would be clear how to deal with qk +1 --qk in (3.5) in the limit e ~ 0 , since then clearly

qk + l -- qk ~ gJ(Z). (3.7) g

In fact, that space is contained as a subspace in the space of all continuous functions which we really have to consider, so that Equation (3.7) at least holds for that subspace. Moreover the set of all continuous differentiable functions is supposed to be everywhere dense in our space of continuous functions ( i .e . -we can approximate a continuous, nondifferentiable function arbitraryly well by a continuous differentiable function). We therefore are allowed to require, that the functional O([O('c), q(r)]) takes the same form in the space of all continuous differentiable functions and the space of continuous functions which we


consider*. Hence, we can write (3.5) in the form

0([~(z), q(z)3 )

= exp [~ dz l im- [ln f2(eO(z), q(z), z + e, z)]~gj. (3.8)

We can now insert Equation (2.12) on the right hand side of Equations (3.8) and (3.6) and obtain

t

in f2([~(z), q(z)]) = ~ dr lim _1 to e ~ O

N - 1

Dg([q(z)])=lim ~I [dqjJ ~ 0 j _ 1

exp_2 ° In

• exp(ip'(qk+l--qk))S(p, qk, tk+g, tk)}] I (3.10) i r eg"

Defining the "Lagrangian"

L(il(z),q(z))--lim 1 [ln{~ dp

.exp(ip.gl(z)OS(p,q(z),'c + e, z)}] (3.11) reg

so that t

In ~([c)(~), q(r)]) = - j" '/~ ~4(~), q(r)) (3.12)

is the negative "action", we obtain from Equations (3.4), (2.12) the solution of (2.1) in the path integral form

q (t)- q

P(qtlqoto)= [. D#([q(ffl) q ( t o ) = q o

• exp - ~ dz L(O(z),q(z)) . (3.13) to ..i

Let us now see what we have achieved: Equations (3.10), (3.11) allow us to evaluate the measure D#([q(z)]) and the Lagrangian L(gl(z),q(z)), once we have determined the normally ordered operator function S(p, qk, tk + e, tk) asymptotically for small e. S has to be determined in sufficient accuracy to know the irregular part in (3.10) exactly and the regular part in (3.11) to the order e. After having determined D#, L, (3.13) yields the Green's function solution of (2.1) or its Fourier transform for finite time intervals in terms of a path integral. Except for the last step, this program will be carried through for the Liouville operator L given by (2.3).

* The results of Section 4 remain unchanged if, instead of (3.7), we adopt the.rule (qk--qk-1)/e "~ 0(2.) (cf. note added in proof).

Even though the final path integral can be carried out exactly for trivial cases only (Ornstein Uhlenbeck processes and special cases of such) it is very valuable to have a method for determining the Lagrangian L and D#, as was discussed in the introduction.

4. Path Integral Representation of General Diffusion Processes

We now want to evaluate (3.10), (3.11) for S(t)= exp(Lt) where L is given by (2.3). Since L is here independent of time we have rigorously in e

S(p, q, e) =exp[-e(lp~puQ~u(q)+ipvKv(q)-V(q))J. (4.1)

We find it convenient to eliminate e from the term with the leading order of p by defining a new oper-

ator O = ~ - p with the commutation relation

[P, g(q)] = - i l ~ ~g(q) (4.2) 0q

in terms of which (4.1) takes the form N

P - , q , e e)

=exp [-½p~ Pu Q v u - i l ~ p ~ K~ +e V]. (4.3)

In the Appendix we put the right hand side of (4.3) into normal order up to terms inclusive of the order e. The result can be written in the form

X = J((0, q, e) = ~/°{exp(~)(1 + ] ~ Y1 + e Y2)} (4.4)

where Y is the normal ordering operator (cf. Appendix); e=-½pvp~,Qv~,, and Y1, Y2 are operator functions of p, q which are determined in the appendix,Equations (A.12), (A.13). We have

S ( ~ , q , e ) =X(~,q,e), (4.5)

simply by replacing all operators p in (4.4) by C- numbers /~. We are now in a position to evaluate Equations (3.10), (3.11), if we substitute there p = f/l~e for the variable of integration. This is very usefull, since it brings out the singular e-dependence of the p- integrals in these equations as a factor in front of the integral, which allows us to identify the regular part, and the irregular part of the logarithm more easily. We have

dp ~ ( 2 ~ exp(ipO~) S(p, q, 8)

=(27~]~) -"~d5exp( i /~ .0 ]~)g ( ~ , q, e). (4.6)


Inserting Equations (4.5), (4.4) on the right hand side of (4.6) we are left with a number of Gaussian integrals to be carried out. They are easy to do, in particular, since the answer for the/5-integral is only of interest to the order e. We obtain, after combining similar terms in rather extensive algebraic manipu- lations

~ , q , ~ ~clfiexp(ip'g!]/e)S(l/~ )

= [ / ~ e x p \ - z

e 01nQ c~-eF(c),q)], 1 2 ~?q~ (4.7)

where the function F(O, q) stands for

F(gl, q) = Q;~( - gl~ h. + ½ h. h~)

+½ l~_Oaq~ l/Qh, _ V + ~2 R (4.8)

We have introduced here the following abbreviations

Q v ~ .

h~=K~-½l/Q ~q u 1/@-, Q = I[Q~.[[

R _ 3 a2Q~ 3 2 Q~u

3 ~?lnQ OlnQ k¼Q,uQ[~ I a2Q~

~Q~z c?Q.,~ 3 ct Q (~lnO~ (4.9) -X Q~u~ 3q,~ ~q~ 2 ~q~ ~\ 3q u ].

From Equations (4.6), (4.7) we obtain for the irregular part of the bracket in (3.10)

["" ]irreg = -- 1 In (Q (qg) (2 rce)") (4.10)

and for the regular part

[...]reg=8 ( ½c?lnQ3q, 0~-½Qf'~ (I~O~,-F(gt, q)). (4.11)

Applying the method of §3 literally, we would determine the measure D/~ from (4.10) and the Lagrangian L from (4.11). However, as will become clear below, it makes much more sense to transfer the first term in (4.11) to the irregular part, even though it vanishes like e. Doing this, we obtain from (3.10)

Dl~([q(z)])=lim NFII ~ dqj ~ o ~=, (1/(2~z~)" Q(q~)J

• ( ( 2 ~ &Q(qo))- ~/2 \~q ) !

=lim ~[i~ f dqj }.((2~e).Q(q))_,/2" (4.12)

The manipulation of the irregular part of (4.10) mentioned above leads to the cancellation of Q]/~qo)0) in (4.12) and ensures, that Dl~.dq transforms like a scalar* under general coordinate transformations in the phase space q. The remainder of the regular part (4.11), inserted in (3.11), gives the Lagrangian

/ 4 0 , 1 - , • • • q)=~Q~. Gq,+F(q,q)

0 h~ V+~R.

(4.13) The result (4.13) is still rather unattractive, because of the very complicated form (4.9) of R, containing first and second order derivatives of Qv, or its inverse. However, we will show now, that R must have a very simple meaning. To this end we determine the transformation be- haviour of L under general coordinate transformations*. Knowing, that

e(q tlqoO) dq q(t)=q ( t )

=dq ~ Dp([q(z)])exp-~/4(l(z),q(~))dz (4.14) q(0) = qo to

has to transform like a scalar (being a physical probability), and knowing that DI~. dq also is a scalar, we infer that L(c~,q) is a scalar. Again this property

e c31nQ depends on our transfer of 2 ~?q~ ~, from (4.11) to

(4.10). The quantity h~ defined in (4.9), transforms like c~,. Defining c~ as a contravariant vector, Qyu ~ transforms like a covariant tensor. We use Q;-1 to define a covariant metric tensor in physe space. Then the first three terms in (4.13) are manifest scalars• Hence R has also to transform like a scalar. Due to the linear dependence of R on second order derivative terms of the metric tensor Q~u z it is clear, that R must be proportional to the Riemann curvature scalar [12] obtained from the metric tensor Q;-). In fact, it is only an algebraic exercise to show the identity of our R, Equation (4.9), with the Riemann scalar of the metric tensor of Q ~ , defined by [12]

with _1_ [a2Q;) a2Q#)

R x . ~ - = [c~q~ 3q~ aq~ c~qx

+ Q#) [cd G - r d G] and

~ = 1 e v a [~Qff ¢ .j_ ~ e ~-vl a Q~-,al 1 [ 3q.~ ~q~ gq~ J"

02 Q)a~ 2 - 1

(4.15)

* The transformation [11].

(4.16)

(4.17)

properties used here have been derived in

R. Graham: Path Integral Formulation of Genera1 Diffusion Processes 287

By this identification of R we have succeeded in transforming a result, which looked rather repelling, into one of considerable beauty. Equation (4.13), with R understood as Riemann curvature scalar of the metric tensor Q;-,~, is the central result of this paper.

5. S p e c i a l C a s e s and C o m p a r i s o n wi th O t h e r R e s u l t s

Correct results for the Lagrangian L have been given earlier only for flat phase spaces, in which R = 0. We list the most important special cases and give the associated Lagrangians i) Wiener process, Q~, = const, K~ = 0, V= 0:

1 - I Lw =7Q~, 0v0,. (5.l)

The path integral representation of this simplest process has been given by Wiener [13], who thereby became the inventor of path integrals. ii) Ornstein Uhlenbeck process [14], Q~=cons t , K~ =Av, q~, V=0:

LoM=½Qf~I(Ov-A~zq;,)(~-A~Kq~)+½A~,. (5.2)

Apart frgm a constant, which can be absorbed into the measure D#, Equation (5.2) is the Onsager Machlup function, already given in (1.2). By means of this result Onsager and Machlup succeeded in applying thermodynamical concepts to time dependent processes [2]. iii) Nonlinear process with constant diffusion, Q~, = const, V=0:

- 1 . ,, . (5.3) L3=gQ~, (%-K,)(O,-Kf,)+½ OK~ oq~

This Lagrangian was derived in [15], [16] directly from the Langevin equation of the stochastic process by a discretization method. The last term is due to a change of a phase space volume element during its non-linear drift through phase space. The discretization method is not sufficiently accurate to give L for the most general case with variable Q~ [17]; our earlier result [15] is not correct for the general case [17]. iv) One dimensional process, most general case, V =0:

Oq ]

+½]/Q 0q ] /Q K ~qq • (5.4)

In one-dimensional spaces curvature is not possible, hence R = 0 always. The result (5.4) was given by Horsthemke and Bach [17].

It can be derived from (5.3), specialized to one dimension, by a transformation of the coordinate q. v) General diffusion process in flat phase space, V= 0, R = 0 :

1 c? h~ Ls =½QY~l(O~-h~)(Ou-hu)+gl/Q ~-q~ ~Q (5.5)

where h~ is defined by (4.9). This result was derived by Stratonovich [8], however without noting the restriction of its validity to flat phase spaces. This result was obtained earlier than iii) and iv) and contains these as special cases. It seems to have been discovered by the physics community only recently [18]. There is also a field of applications of (4.13) in quantum mechanics [4]. In this case V~ 0 is of great interest. The translation of (2.1), (2.3), (4.12), (4.13) into quantum mechanics is achieved by the following dictionary:

t = - - iu , qv = x~

1 e - iQ~ h~,=cA ~

-½Q~ h,.hu 0 hv

-½,]/-Q 0% ]/-Q

+ V(q) = q) (q) L=L

u = time x v =particle coordinate Mv~ =mass matrix

A~. = vector potential

R

12

q)(q) = scalar potential L = classical Lagrangian.

It is easy, to solve this dictionary for the statistical quantities in terms of the quantum mechanical ones. Let us list again the most relevant special cases. The cases listed here correspond to those with the same numbers in the list of statistical examples. We do not write down the associated Lagrangians, which are all well known.

i) free particle,

ii) linear harmonic oscillator or free particle in homo- geneous constant magnetic field (Landau levels).

iii) particle in time independent but spatially varying electric and magnetic field in cartesian coordinates.

iv) particle with space dependent mass in one dimension; this case can always be transformed to case iii) in one dimension.

v) same case as iii) in general coordinates.

Again, all these examples are for flat spaces only. In a recent paper [9] we have applied our Lagrangian (4.13) for R@0 in order to derive the generalization


of the Klein G o r d o n equation needed in general relativity. Since the knowledge of the Lagrangian (4.13) opens up a completely new method for quant izat ion in general relativity, with results differing from those obtained by the hitherto known, methods, we believe that this branch of applications is very promissing and only in its beginnings now. A similarly fruitfull branch of applications is in non-equi l ibr ium thermodynamics. At the end of this paper, we would like to clear up some of the confusion in the literature associated with the uniqueness of Lagrangians in path integral representations. In a recent preprint [19] the disentangling theorem of F e y n m a n n [20] is used in order to put the operator S in (2.5) into normal order. However, as F e y n m a n n himself remarks [21], his method is not sufficiently accurate for dealing with variable Qvu or even nonlinear K~. Thus, the result of [19] depends on the particular way in which p and q are ordered in (2.3) and does not agree with Stratonovich 's result for flat phase space. In another recent paper [22], the path integral representation was derived from a solution of (2.1) to first order in e; a Lagrangian was not calculated in that paper, since the final limit e ~ 0 was not taken. However it was pointed out that the discrete version of the Lagrangian contained some undetermined parameters, even for constant diffusion matrices Quv. As an example we consider the result of [22] for 1- dimensional phase space and constant diffusion matrix

1 L=~2~k~= 1 eQ-1%--qk-le ~zK(qk)_~K(qk_l)

+2c~e OK(qk- 1)~ (5.6) ~qk-1 )

where

a + f i = l , 0_<~_<1, 0<~__<1. (5.6)

and ~ are not fixed in (5.6). The question we want to ask is whether the ambiguity in c~ and fi survives the limit e--*0. In fact, this question has to be an- swered in the negative. When we carry out the limit e--*0 we have to replace (qk--qk'02/~ by c~2~, accord- ing to (3.7). There also occurs the term (qk--qk-1) " (eK(qk)+ fiK(qk-1)) in (5.6). In order to be correct to order e in replacing this expression by / / ( e + / / )K(q)e while qk--qk_l~l//~ [22] we have to take c~=fl= 1.

This fixes c~ and ]~ uniquely and the Lagrangian (5.6) goes over into case iii) listed above, specialized to one dimension. The ambiguity in the discrete Lagrangian therefore arises only from the different, but equivalent ways in which a cont inuous function can be

approximated by discrete ones.* It was easy, in this case, to obtain the cont inuous Lagrangian from the discrete one. However, we want to point out that this is no longer easy, if it is possible at all, for the case of variable diffusion matrix. The same difficulties with the transition from the discrete representat ion to the cont inuous one, which also occur when the Langevin equat ion is taken as a starting point [15], reappear in this case. There is also no refuge, contrary to what is suggested in a recent preprint [23], in using Simpson's rule or some other discretization technique with more than two points per interval, since the Markovian proper ty of the paths wipes out all infor- mat ion concerning points in phase space which are passed earlier to the latest one. Hence approxi- mations of this kind will introduce spurious memory effects not present in the cont inuous r andom process without adding to the accuracy of the approximation. (Note, that the error estimates for Simpson's rule contain derivatives of the process, which do not exist here). We consider it as one of the prime advantages of our method, described in section 3, that ambiguities in the discretization process never have a change to appear.

Appendix We want to display here some of the algebra which is necessary in order to put the right hand side of (4.3) into normal order, using the commuta t ion relation (4.2). Let us start by expanding the e-function in (4.3) in a power series

l__x, x(i,, q, ~) =~ n! (A.1)

Xn@,q,~)=(-½pvpuQ~u-i F/~p~K,,+~V) ".

* Homer [24] has looked at the consequences of (5.6) in perturbation theory. Taking the continuous limit in the perturbation expansion he found that ~, ]~ appear only as factors of undefined expressions like O(z) 3(z). If these distributions are defined as limits of continuous differentiable functions, c~=/~=½ turns out to be necessary, If this requirement is relaxed, it is necessary to let the definition of these distributions depend on c~ in order to obtain c~- independent results for correlation functions. However, in the latter case the most probable path (and other differentiable paths occuring with measure zero) is still e-dependent. We conjecture, that there is a close relationship between Horner's condition for e=/~=½ and our own assumption that the paths q(r) are continuous and non-differentiable with probability 1 but can be approximated by continuous differentiable paths arbitrarily well. In our opinion it is necessary to consider such a restricted class of continuous functions in order to give any meaning to expressions involving ~ where the derivative obeys the usual rules of calculus. The process of extending the rules of calculus from the subclass of differentiable functions of measure zero to the full function space which we consider is then entirely analogous e.g. to extending the rules of algebra from the rational numbers to all real numbers, even though the former are contained in the latter only with measure zero.


We now want to put each term in this series into normal order up to terms of order s and then resum the series. Let us write the normally ordered form of X, in the form

X, = X . - ~/~ {A, + B= l f e+ C,e + O(e3/2)} (A.2)

where ~ is an operator acting on operator functions of ~ and q such that it places all ~ left from the q regardless of the commutation relations. A,, B,,, C, are operator functions of ~, q which are independent of s. The operators X. satisfy the recursion relation

X , = X , _ 1 .(-½O~,Q,u-i]fsO~K,+eV). (A.3)

Equation(A.3) is easily put into normal order, since X,_ ~ is in normal order already. We make use of the relations

-a~ [ L k ] = i 1/~ - - ~q~

= p~ ~q,-+ p,~q~ ~ + s - - ~qv~qs (1.4)

where iv is a normally ordered operator function. If we use the form (1.2) of X,_ 1 in (1.3) and keep only terms up to e we reobtain, after carrying out the necessary, commutations in (1.3), the original form (A.2) for X,,, where now the coefficients A,,, B,, C, are expressed in terms of A,_ 1, B = _ 1, C . _ ~. These recursion relations appear under the ~ - operator and take the form (with ~-= - ~ pv p~Q~u)

N~{A.} = ~/{~A,_ ~} (A.3)

~/{B.} ( .- ~?A._ 1 =~A/'~-~p,,~q~qu Q~u-'p~A~_~K~+~B=_~} (A.4)

~{c.} =~{~ a~A.-~ aA.-~ K

• ~ a B n 1 _ ~ C n i t ' - ~ p ~ Q ~ - i p ~ B , _ ~ K ~ (A.5)

We have made use of the symmetry Q~u--Qu~ in writing these equations. These recursion relation can be solved successively, starting with the relation (A.3). We obtain

~4/" {A,} == ~V'{~"} (A.6)

~{B.}=Y {-2 ~(n-~)~"-~

~ - in~" ~p~K~; (A.7)

Y {C,} = X { - ~ n ( n - 1 ) (n -2 ) (n - 3) ~"-4

p~PuQ~Qu~, aq= aq z - - - - + ~ n(n - 1)(n - 2) ~ = - 3

m

oqu c~q'̀ ~q~ ~qT

~ \ 1 - 3~p~,Q~,~ K~ - - ! + 2 n ( n - 1) 0~ n - 2 cOq~ /

a2 ~ a(Kv~ ) " ½Q~uaq~aqu + Oq~ (p~KO ~) + n~"- l V}. (A.8)

We now determine X in the form

x = Zo +1/~ Zl +* Z; + o ( E ~) (A.9)

by carrying out the sums

n! J (A.10)

Lc. Z2=x{2 n! j

We obtain

Zo =~ / ' {expe} ;

~l=JV.{(exp~)[_i- K i- O~I)

~?qu ] ]

+~K~ 0q~+ V •

(A.1I)

In (A.11) e = - ½ 1 ~ P u Q ~ has to be inserted. The partial derivatives acting on • will then act only on Qv,(q). Equations (1.11), (A.10) inserted in (1.9) yield the result (4.4) immediately, with

Yl=-ipv (K~+½Qv~ ~q,) (A.12)


Y2 = --½ [Pv / 1 00~ \ ] 2

+ 2 Kv ~q~ + (A.13)

References

1. Stratovonich, R.L.: Topics in the Theory of Random Noise, Vol. 1, New York: Gordon and Breach 1963

2. Onsager, L., Machlup, S.: Phys. Rev. 91, 1505 (1953); 91, 1512 (1953)

3. Hashitsume, N.: Progr. Theor. Phys. 8, 461 (1952); 15, 369 (1956); Tisza, L., Manning, L: Phys. Rev. 105, 1695 (1957)

4. Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Phath Integrals, New York: Mc. Graw Hill 1965

5. Graham, R., to be published 6. Martin, P.C., Siggia, E.D., Rose, H.A.: Phys. Rev. A8, 423

(1973); Deker, U., Haake, F.: Phys. Rev. A l l , 2043 (1975); A12, 1629 (1975)

7. Janssen, H.K.: Physik B23, 377 (1976)

8. Stratonovich, R.L.: Sel. Transl. Math. Stat. Prob. 10, 273 (1971) (translation of russian paper 1962)

9. Graham, R.: Phys. Rev. Letters 38, Jan. 10 (1977) 10. Bach, A., Dtirr, D., Stawicki, B.: preprint 1976 11. Graham, R.: Z. Physik, to be published 12. cf. Weinberg, S.: Gravitation and Cosmology, New York: John

Wiley 1972 13. Wiener, N.: Acta Math. 55, 117 (1930) 14. Uhlenbeck, G.E., Ornstein, L.S.: Phys. Rev. 36, 823 (1930) 15. Graham, R.: Springer Tracts in Modern Physics, Vol. 66, New

York: Springer 1973 16. Graham, R.: in Fluctuations, Instabilities and Phase

Transitions (ed. Riste) New York: Plenum 1976 17. Horsthemke, W., Bach, A.: Z. Physik B22, 189 (1975) 18. Horsthemke, W.: private comunication 1975 19. H~inggi, P., Nettel, S.J.: Feynman Path-Integral Formulation

for the Fokker-Planck Equation, preprint 1976 20. Feynmann, R.P.: Phys. Rev. 84, 108 (1951) 21. ref. 20, p. 127, first paragraph 22. Haken, H.: Z. Physik B24, 321 (1976) 23. Dekker, H.: On the Functional Integral for Generalized Wiener

Processes and Nonequilibrium Phenomena, preprint 1976 24. Horner, H.: private communication 1976

Robert Graham Fachbereich Physik UniversitS.t E s s e n - Gesamthochschule Postfach 6843 Unionstrage D-4300 Essen Federal Republic of Germany

Notes Added in Proof.

1. As an explicit proof for the absence of discretization ambiguities in our procedure we derive here the results of section 4 by taking qk - qk 1 ~ E0 instead of (3.7). Then ,~(p, q, z + ~, z) in (3.9), (3.10) has to be replaced by

S(p ,q-a O,'c,'c-~)=S(p,q,z + 8, z ) -~ ~'~{p'q' ~ + e''c) z 0v+o(e ) 8qv

and the sum in (3.10) now runs from k = 1 to N (because we substitute k + l - - , k as a summation index before a-*0). The change He in

1 81nQ . leads to the replacement of F(~, q) by F(O, q ) -~ ?Iv ~ in (4.7),

i.e. the additional term cancels the opposite term in (4.7) and L remains as given in (4.13). The only effect of the cancelled term, before, was to make the measure (4.12) a proper scalar density. However, due to the above mentioned change in the boundaries of summation in (3.10) D# is now given by (4.12) automatically (i.e. the transfer of a term from the regular to the irregular part in (4.11), (4.10) is now unneccessary), which finishes the proof. 2. As a simple application for R4:0 we derive the Schr6dinger equation for curved 3-space with the classical Lagrangian:

L=#° K~ kSx~ 8xk - - - - - - g . 2 ~3u c3u

Comparison with (4.14) shows that we have t ~ - iu, Q~J --* I% Kik, h,=0 , R(QG1)~-#8IR(K~k), V - ~ R ~ U ; in order to make

IOj2d3x a scalar we take P ~ tp(K) 1/4. With these translations the Schr6dinger equation may be read off (2.3). It can be brought into the form

St) 1 K . _ I / 4 ~ _ _ 3 1/K K ~k K i/4 i 8u 2~o ~x ~ _ _ ~ O

1

+CO-l~-~o~o Rq'. The same result is obtained from 1-9] in the v --, 0 limit, if the metric

C (1° t tensor is of the form g,~ = 0 --gik

After this paper had been completed, a paper of Ringwood, G. A., J. Phys. A9, 1253 (1976) (and other papers quoted there dating back to an early paper by B. S. DeWitt, Rev. Mod. Phys. 29, 377 (1957)) has been drawn to our attention in which two forms of a general relativistic Schr6dinger equation have been derived by different methods (note a different sign convention for the metric tensor and R in that paper). Our result seems to disagree with (and, in fact, lies just in the middle of) his two results. We believe the discrepancy to arise from the use by these authors of Feynman's [41 discretization technique for the evaluation of the short time propagator from the functional integral, a method which we believe not to be directly applicable in curved phase space. We hope to return to this point in future work.

path integral formulation of general diffusion processes

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