INTRODUCTION TO VOLUME

6

XXV

The

final

paper

on

quantum theory

in this

volume,

Einstein 1917d

(Doc.

45),

deals with

a

totally

different

topic:

the Bohr-Sommerfeld

quantum

con-

dition for

periodic systems.

This condition

can

only

be

put

in

its usual form

\pidqi =

nih

if

a

separation

of coordinates

is

possible;

in

all other

cases

the

condition has the

general

form

£\pidqi

=

nh. In

his

paper,

Einstein discuss-

es a

possible way

to

avoid the need of

a

separation

of variables.

He

gives

the

general

form of

the

condition

a

coordinate-independent

meaning

by

interpret-

ing

it

as a

line

integral

in

(coordinate)

phase space along some

closed

contour.

If

phase space

is

structured in such

a

way

that not

all

closed

curves can

be

contracted

to

a single point,

i.e.,

if

the motion

of

the

system

is

restricted

to

some

invariant

subspace,

there

must

exist

a

finite number of

topologically

in-

dependent

contours.

For

multiple periodic systems

the

integral

has

a

finite

value for these

contours,

each of which

corresponds to

a

separate

quantum

condition. If there is

no

periodicity,

however,

quantization

according

to

this

method

is not

possible.

In

spite

of

its

general

and novel

approach,

Einstein's

paper

was ignored

by

most.

One notable

exception

is

Louis de

Broglie,

who

used Einstein's

phase space

approach

in

1924

in his

dissertation.[42]

Only

many

decades later

was

it

seen

in

retrospect

that

Einstein's

approach

fore-

shadowed

the

use

of the

concept

of invariant tori in

phase space

in the anal-

ysis

of

integrable dynamical

systems.[43]

[42]See

De

Broglie

1924,

chap.

3. In 1926

Erwin

Schrödinger

mentions

Einstein's

paper,

as

well

as

De

Broglie's

dissertation,

in

a

footnote in

one

of

his

papers

on wave

mechanics,

but

only

to

point

out

that the

phase space approach

he takes resembles the

one

of Einstein's

paper

(see Schrödinger

1926,

p.

495,

footnote

1).

[43]See

Gutzwiller

1990, chap. 14.1,

for

more

details.