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.