DOC.
45 QUANTUM
THEOREM
441
There
are
obviously
two
types
of
closed
curves
on
this double
surface;
they
can
neither be shrunk
to
a
point by
continuous
deformation, nor can one type
be reduced
[p. 90]
to
the other.
Fig.
1
below
depicts
an
example
for each
of
the
two
types
(L1
and
L2).
The
parts
of
a
line
on
the lower leaf
are
drawn with dashes. All other closed
curves
on
the double surface
can
either be contracted into
a
point by
continuous deformation
or, by
the
same
process,
can
be
mapped onto
the
types
L1
or
L2
with
one
or
several
revolutions. The
quantum
theorem
(11)
would here have
to be
applied
to
path types
L1
and
L2.
Obviously,
these considerations
are
to
be
generalized
to
all
movements
that
satisfy
the condition
of
§4.
Phase
space
has to be
imagined split
into
a
number of
Fig. 1.
Fig. 2.
"tracts" that
are
connected
along (l
-
1)-dimensional
"surfaces" such
that-when
interpreted
in the so-constructed manifold the
pi are
univalent and continuous
functions
(also
at transitions from
one
tract to another
tract).
We shall call this
auxiliary geometrical
construction the "rationalized
phase space."
The
quantum
theorem
(11)
shall
apply
to
all lines that
are
closed within the rationalized coordinate
space.
In order
to
attach
a
precise
meaning
to
the
quantum
theorem in this
formulation,
the
integral [Zpidqi,
extended
over
the rationalized
qi-space,
must have the
same
value for all those closed
curves,
which
can
be
continuously
deformed into
one
another. The
proof
for this follows the
customary
scheme
entirely.
In the rationalized
qi-space,
let
L1
and
L2
(see
the schematic in
Fig.
2)
be closed
curves
that
can
continuously
be deformed into
one
another under
preservation
of
the indicated
directional orientation. The contour in the
figure
is then
a
closed
curve
that
can
be
[p.
91]
continuously
shrunk into
a
point.
From this
follows,
due
to
(10),
that the line
integral
extended
over
the entire trail vanishes.
If
one
considers furthermore that the
integrals
extended
over
the
infinitely
close
connecting
lines
A1A2
and
B1B2
are equal,
because
of
the
single-valuedness
of
the
pi
in the rationalized
qi-space,
one
has
the
result that the
integrals
extended
over
L1
and
L2
are equal.
The
potential
J*
is
infinitely
multivalued also in the rationalized
qi-space;
but
according
to
the
quantum
theorem
this multivaluedness
is the
simplest one
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