Consider the simple integral:
Assume f(x) has a global minimum at 
, such that 
.
If this minimum is well separated
from other minima of f(x) and the value of f(x) at the global minimum
is significantly lower than it is at other minima, then the dominant
contributions to the above integral, as 
will
come from the integration region around 
. Thus, we may expand
f(x) about this point:
Since , this becomes:
Inserting the expansion into the expression for I gives
Corrections can be obtained by further expansion of higher order terms. For example, consider the expansion of f(x) up to fourth order:
Substituting this into the integrand and further expanding the exponential would give, as the lowest order nonvanishing correction:
This approximation is known as the stationary phase or saddle point
approximation. The former may seem a little out-of-place, since there is no
phase in the problem, but that is because we formulated it in such a way as to
anticipate its application to the path integral. But this is only if 
is taken to be a real instead of an imaginary quantity.
The application to the path integral follows via a similar argument. Consider the path integral expression for the density matrix:
We showed that the classical path satisfying
is a stationary point of the Euclidean action 
, i.e.,
. Thus, we can develop a stationary
phase or saddle point approximation for the density matrix by
introducing an expansion about the classical path according to
where the correction 
, satisfying 
has
been expanded in a complete set of orthonormal functions 
,
which are orthonormal on the interval 
andsatisfy
as well as the orthogonality condition:
Setting all the expansion coefficients to 0 recovers the classical path. Thus, we may expand the action S[x] (the ``E'' subscript will henceforth be dropped from this discussion) with respect to the expansion coefficients:
Since
the expansion can be worked out straightforwardly by substitution and subsequent differentiation:
where the fourth and eighth lines are obtained from an integration by parts. Let us write the integral in the last line in the suggestive form:
which emphasizes the fact that we have matrix elements of the
operator 
with respect to the
basis functions. Thus, the expansion for S can be written as
and the density matrix becomes
where 
. 
is an overall
normalization constant.
The integral over the
coefficients becomes a generalized Gaussian integral, which brings down
a factor of 
:
where the last line is the abstract representation of the determinant. The determinant is called the Van Vleck-Pauli-Morette determinant.
If we choose the basis functions 
to be eigenfunctions of the
operator appearing in the above expression, so that they satisfy
Then,
and the determinant can be expressed as a product of the eigenvalues. Thus,
The product must exclude any 0-eigenvalues.
Incidentally, by performing a Wick rotation back to real time according to
, the saddle point or stationary phase approximation to
the real-time propagator can be derived. The derivation is somewhat tedious
and will not be given in detail here, but the result is
where 
satisfies
and 
is an integer that increases by 1 each time the determinant
vanishes along the classical path. is called the Maslov index.
It is important to note that because the classical paths satisfy an
endpoint problem, rather than an initial value problem, there can
be more than one solution. In this case, one must sum the result
over classical paths:
with a similar sum for the density matrix.