The van Vleck biscalar is defined by
Equations (63) and (80
) imply that at
coincidence,
and
. Equation (89
), on the other hand,
implies that near coincidence
We shall prove below that the van Vleck determinant satisfies the differential equation
which can also be written asTo show that Equation (95) follows from
Equation (94
) we rewrite the
completeness relations at
,
, in
the matrix form
, where
denotes the
matrix whose entries
correspond to
. (In this translation we put tensor and
frame indices on equal footing.) With
denoting the
determinant of this matrix, we have
, or
. Similarly, we rewrite the completeness
relations at
,
, in the matrix form
,
where
is the matrix corresponding to
. With
denoting its determinant, we have
, or
.
Now, the parallel propagator is defined by
, and the matrix form of this equation
is
. The determinant of the parallel
propagator is therefore
. So we have
To establish Equation (98) we differentiate the
relation
twice and obtain
. If we replace the last factor by
and multiply both sides by
we find
In this expression we make the substitution , which follows directly from
Equation (94
). This gives us
and taking the trace of this equation yields
We now recall the identity , which relates the variation of a
determinant to the variation of the matrix elements. It implies, in
particular, that
, and we finally obtain