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 , 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):