We let b— be the mean backward velocity, u = (b ’ b— )/2, v = (b + b— )/2.

By (13.5) and (13.6) of Chapter 13,

‚u h

¯

=’ grad div v ’ grad v · u,

‚t 2m (15.2)

‚v h

¯

=a’v· v+u· u+ ∆u,

‚t 2m

where a is the mean acceleration.

Suppose that the particle is subject to an external force F . We make

the dynamical assumption F = ma, and substitute F/m for a in (15.2).

This agrees with what is done in the Ornstein-Uhlenbeck theory of Brow-

nian motion with friction (Chapter 12).

Consider the case when the external force is derived from a poten-

tial V , which may be time-dependent, so that F (x, t) = ’ grad V (x, t).

Then (15.2) becomes

‚u h

¯

=’ grad div v ’ grad v · u,

‚t 2m (15.3)

‚v 1 h

¯

= ’ grad V ’ v · v + u · u + ∆u.

‚t m 2m

If u0 (x) and v0 (x) are given, we have an initial value problem: to solve the

system (15.3) of coupled non-linear partial di¬erential equations subject

to u(x, 0) = u0 (x), v(x, 0) = v0 (x) for all x in ‚3 . Notice that when we

do this we are not solving the equations of motion of the particle. We are

merely ¬nding what stochastic process the particle obeys, with the given

force and the given initial osmotic and current velocities. Once u and v

are known, b, b—, and ρ are known, and so the Markov process is known.

It would be interesting to know the general solution of the initial

value problem (15.3). However, I can only solve it with the additional

assumption that v is a gradient. (We already know (Chapter 13) that u

is a gradient.) A solution of the problem without this assumption would

BROWNIAN MOTION IN THE AETHER 107

seem to correspond to ¬nding the Markov process of the particle when

the containing ¬‚uid, the aether, is in motion.

1

Let R = 2 log ρ. Then we know (Chapter 13) that

m

grad R = u. (15.4)

h

¯

Assuming that v is also a gradient, let S be such that

m

grad S = v. (15.5)

h

¯

Then S is determined, for each t, up to an additive constant.

It is remarkable that the change of dependent variable

ψ = eR+iS (15.6)

transforms (15.3) into a linear partial di¬erential equation; in fact, into

the Schr¨dinger equation

o

‚ψ h

¯ 1

∆ψ ’ i V ψ + i±(t)ψ.

=i (15.7)

‚t 2m h

¯

¯

(Since the integral of ρ = ψψ is independent of t, if (15.7) holds at all

then ±(t) must be real. By choosing, for each t, the arbitrary constant

in S appropriately we can arrange for ±(t) to be 0.)

To prove (15.7), we compute the derivatives and divide by ψ, ¬nding

‚R ‚S h

¯ 1

∆R + i∆S + [grad(R + iS)]2 ’ i V + i±(t).

+i =i

‚t ‚t 2m h

¯

Taking gradients and separating real and imaginary parts, we see that

this is equivalent to the pair of equations

‚u h

¯

=’ ∆v ’ grad v · u,

‚t 2m

‚v h

¯ 1 1 1

∆u + grad u2 ’ grad v 2 ’ grad V.

=

‚t 2m 2 2 m

Since u and v are gradients, this is the same as (15.3).

Conversely, if ψ satis¬es the Schr¨dinger equation (15.7) and we de¬ne

o

R, S, u, v by (15.6), (15.4), and (15.5), then u and v satisfy (15.3). Note

that u becomes singular when ψ = 0.

108 CHAPTER 15

Is it an accident that Markov processes of the form (15.1) with the

dynamical law F = ma are formally equivalent to the Schr¨dinger equa-

o

tion? As a test, let us consider the motion of a particle in an external

electromagnetic ¬eld. Let, as is customary, A be the vector potential,

• the scalar potential, E the electric ¬eld strength, H the magnetic ¬eld

strength, and c the speed of light. Then

H = curl A, (15.8)

1 ‚A

= ’ grad •.

E+ (15.9)

c ‚t

The Lorentz force on a particle of charge e is

1

F =e E+ V —H , (15.10)

c

where v is the classical velocity. We adopt (15.10) as the force on a particle

undergoing the Markov process (15.1) with v the current velocity. We do

this because the force should be invariant under time inversion t ’ ’t,

and indeed H ’ ’H, v ’ ’v (while u ’ u) under time inversion. As

before, we substitute F/m for a in (15.2). Now, however, we assume the

generalized momentum mv + eA/c to be a gradient. (This is a gauge

invariant assumption.) Letting grad R = mu/¯ as before, we de¬ne S up

h

to an additive function of t by

m e

grad S = v+ A,

h

¯ mc

and let

ψ = eR+iS .

Then ψ satis¬es the Schr¨dinger equation

o

‚ψ i e ie

2

=’ ’i¯ ’A ψ’

h •ψ + i±(t)ψ, (15.11)

‚t 2m¯h c h

¯

where as before ±(t) is real and can be made 0 by a suitable choice of S.

To prove (15.11), we perform the di¬erentiations and divide by ψ,

obtaining

‚R ‚S h

¯

∆R + i∆S + [grad(R + iS)]2

+i =i

‚t ‚t 2m

ie2

e 1e ie

A2 ’ • + i±(t).

A · grad(R + iS) + div A ’

+

2m¯ c2

mc 2 mc h h

¯

BROWNIAN MOTION IN THE AETHER 109

This is equivalent to the pair of equations we obtain by taking gradients

and separating real and imaginary parts. For the real part we ¬nd

m ‚u h

¯ e

=’ grad div(v + A)

h ‚t

¯ 2m mc

h

¯ mm e e m

’ grad 2 u · A· u

v+ A + grad

2m h

¯ h

¯ mc mc h

¯

1e

+ grad div A,

2 mc

which after simpli¬cation is the same as the ¬rst equation in (15.2).

For the imaginary part we ¬nd

m ‚v e ‚A

+ =

h

¯ ‚t mc ‚t

h

¯ m mm m e m e

grad div u + grad u· u ’ grad A·

v+ v+ A

2m h

¯ hh

¯¯ h

¯ mc h

¯ mc

e2

e m e e

A2 ’ • .

A· A’

+ grad v+

mc h

¯ mc 2m¯ c

h h

¯

Using (15.9) and simplifying, we obtain

‚v e h

¯ 1 1

grad div u + grad u2 ’ grad v 2 .

= E+

‚t m 2m 2 2

Next we use the easily veri¬ed vector identity

1

grad v 2 = v — curl v + v · v

2

and the fact that u is a gradient to rewrite this as

‚v e h

¯

= E ’ v — curl v + u · u’v· v+ ∆u. (15.12)

‚t m 2m

But curl (v + eA/mc) = 0, since the generalized momentum mv + eA/c

is by assumption a gradient, so that, by (15.8), we can substitute eH/mc

for curl v, so that (15.12) is equivalent to the second equation in (15.2)

with F = ma.

110 CHAPTER 15

References

There is a long history of attempts to construct alternative theories to

quantum mechanics in which classical notions such as particle trajectories

continue to have meaning.

´

[46]. L. de Broglie, “Etude critique des bases de l™interpr´tation actuelle

e

de la m´canique ondulatoire”, Gauthiers-Villars, Paris, 1963.

e

[47]. D. Bohm, A suggested interpretation of the quantum theory in terms

of “hidden” variables, Physical Review 85 (1952), 166“179.

[48]. D. Bohm and J. P. Vigier, Model of the causal interpretation of

quantum theory in terms of a ¬‚uid with irregular ¬‚uctuations, Physical

Review 96 (1954), 208“216.

The theory that we have described in this section is (in a somewhat

di¬erent form) due to F´nyes:

e

[49]. Imre F´nyes, Eine wahrscheinlichkeitstheoretische Begr¨ndung und

e u

Interpretation der Quantenmechanik, Zeitschrift f¨r Physik 132 (1952),

u

81“106.

[50]. W. Weizel, Ableitung der Quantentheorie aus einem klassische

kausal determinierten Model, Zeitschrift f¨r Physik 134 (1953), 264“285;

u

Part II, 135 (1953), 270“273; Part III, 136 (1954), 582“604.

[51]. E. Nelson, Derivation of the Schr¨dinger equation from Newtonian

o

mechanics, Physical Review 150 (1966).

Chapter 16

Comparison with quantum

mechanics

We now have two quite di¬erent probabilistic interpretations of the

Schr¨dinger equation: the quantum mechanical interpretation of Born

o

and the stochastic mechanical interpretation of F´nyes. Which interpre-

e

tation is correct?

It is a triviality that all measurements are reducible to position mea-

surements, since the outcome of any experiment can be described in terms

of the approximate position of macroscopic objects. Let us suppose that

we observe the outcome of an experiment by measuring the exact position

at a given time of all the particles involved in the experiment, including

those constituting the apparatus. This is a more complete observation

than is possible in practice, and if the quantum and stochastic theories