DOC. 45
QUANTUM
THEOREM 437
considered is
multiply-connected,
then there do exist closed
curves
that cannot be
shrunk
to
a
point by
continuous deformation.
If
then
J*
is not
a
single-valued
(but
an oo-valued)
function of the
qi,
integral
(9)
will,
in
general,
differ from
zero
for
such
curves.
Nevertheless,
there will be
a
finite number
of
closed
paths
in
q-space
into which all closed lines in this
space can
be reduced
by
continuous
processes.
In
this
sense,
the finite number
of
conditions
Zipidqi
=
nih
(11)
can
be
prescribed
as
quantum
conditions. In
my opinion they
have to
replace
the
quantum
conditions
(2).
We will have
to
expect
that the number
of
equations (10)
that cannot be reduced into each other will be
equal
to the number
of
degrees
of
freedom
of
the
system.
If
this number is
smaller,
we are
confronted with
a
case
of
"degeneracy"
(Entartung).
The basic idea
suggested
above
(intentionally
full
of
gaps)
will be
explained
in
more
detail in the
following.
§3.
Descriptive
Derivation
of the
HAMILTON-JACOBI Differential Equation.
When
a point
P of
the coordinate
space
is
given
with the coordinates
Qi
and with the
associated
velocity,
i.e.,
the associated momentum coordinates
Pi,
then its movement
is
completely
determined
by
the canonical
equations (3)
and
(4).2 Every point on
the
orbital
curve
L
thus has
a
certain
velocity, i.e.,
the
pi
on
L
are
distinct functions
of
qi.
If
one
imagines
in
an
(l-1)-dimensional
"surface"
of
the coordinate
space every
point
P
given by
its
Qi
and
Pi,
then to
every point belongs a
motion with
an
orbital
curve
L
in the coordinate
space.
These orbits fill the coordinate
space (or part
of
it)
continuously, provided
the
Pi.
on
the surface
are
continuous functions
of
the
Qi.
There will
be
an
orbital
curve through every point (qi)
of
the coordinate
space;
and, [p. 86]
therefore,
there will be distinct momentum coordinates
pi
associated with this
point.
Thus,
a
vector
field
pi
is
given
for the coordinate
space.
We consider it
our
task to
find
the
law of this vector field.
Considering
the
pi as
functions
of
the
qi
in the canonical
system
of
equations
(3), we
have
to
replace
the left-hand sides
by
dpi
dqk
Zk
dqk
dt
and
this,
according
to
(4),
can
be
written
as
2It
shall be assumed that
H
does not
explicitly depend
upon
time
t.