The short-time action contains the
square distance 
which we have to transform to q-space.
For an infinitesimal coordinate difference 
,
the square distance is obviously given by
. For a
finite 
, however,
it is well known that we must expand up to
the fourth order
in 
to find all terms contributing to
the relevant order 
.
It is important to realize that with the mapping from
to 
not being holonomic,
the finite quantity 
is not uniquely determined by 
.
A unique relation can only be obtained by
integrating
the functional
relation
(65) along a specific path.
The preferred path is the classical orbit,
i.e., the autoparallel in the q-space.
It is characterized by being the image of a straight line in
the x-space. There
const and
the orbit has the linear time dependence
where the time 
can lie anywhere
on the t-axis.
Let us choose for the final time in each interval 
At that time,
is related to
by
It is easy to express
in terms
of
along the classical orbit.
First we expand
into a Taylor series
around
. Dropping the time arguments, for brevity, we have
where 
and 
are the time derivatives at the
final time . An expansion of this type is referred to as a
postpoint expansion.
Due to the arbitrariness of the choice of the time in
Eq. (95),
the expansion can be performed around any other point just as well,
such as 
and 
,
giving rise to the so-called prepoint or midpoint
expansions of 
.
Now, the term 
in (96) is given by the equation of
motion
(21) for the autoparallel
A further time derivative determines
Inserting these expressions
into (96) and inverting the expansion,
we obtain
at the final time expanded in powers of .
Using (94) and (95) we arrive at the
mapping of the finite coordinate differences:
where 
and 
are evaluated at the postpoint.
Inserting this into the short-time amplitude
(93),
we obtain
with the short-time postpoint action
Separating the affine connection into Christoffel symbol and torsion, this can also be written as
Note that the right-hand side contains only quantities intrinsic to the q-space. For the systems treated there (which all live in a euclidean space parametrized with curvilinear coordinates), the present intrinsic result reduces to the previous one.
At this point we observe that
the final
short-time action
(101)
could also have been
introduced
without any reference to the
flat coordinates 
.
Indeed, the same action is obtained by evaluating
the continuous action (3)
for the small time interval 
along the
classical orbit between the points 
and
.
Due to the equations of motion
(21), the Lagrangian
is independent of time (this is true for autoparallels as well as geodesics). The short-time action
can therefore be written in either of the three forms
where 
are the coordinates at the final time ,
the initial time ,
and the average time
, respectively.
The first expression obviously coincides with
(101). The others can be used
as a starting point for deriving
equivalent
prepoint or midpoint actions.
The prepoint action 
arises from
the postpoint one 
by exchanging
by 
and the postpoint coefficients by the prepoint
ones.
The midpoint action
has the most simple-looking appearance:
where
the affine connection can be evaluated at any point
in the interval 
. The precise
position is irrelevant to the
amplitude producing only changes beyond the relevant order epsilon.
In the textbook [12],
the postpoint action turned out to be the most
useful one since it gives ready access to the time evolution of amplitudes.
The prepoint action is completely equivalent to it
and useful if one wants to describe the time evolution
backwards. Some authors favor the midpoint action
because of its
symmetry and intimate relation to
an ordering prescription in operator quantum mechanics which was
advocated by
H. Weyl.
This prescription is, however,
only of historic interest since it
does not lead to the correct physics.
In the following, the action 
without
subscript will always denote the preferred postpoint
expression (101):
