One of the things about projective geometry that has fascinated me for more than
thirty years is the fact that all (projective) invariants are cross ratios (or
functions thereof), and that except for special cases there exist six
fundamental cross ratios for any configuration. The numeric values of the six
cross ratios almost always fall into three classes:
1. The two cross ratios bounded by 1 and plus infinity,
The last class of numbers, bounded between zero and one, suggests a connection
between these cross ratio invariants and the concepts of probability and
statistics.
The question that arises is ''What is the physical significance and utility of
the other projective invariant that is bounded between zero and 1, and what sort
of probability does it represent? How are the two probabilities related (if at
all)?
Recent suggestions by Valentini, and Abolhasani and Golshani (quantph/9707052),
concerning the interesting idea of ''heat death'' Variations of the Cross Ratios. In Projective geometry, a particular configuration exemplified by 4 lines in a plane intersecting at some perspective point p, generates 6 usually distinct cross ratios from the 4 points created by a fifth line arbitrarily intersecting all four lines. A similar result can be obtained from four planes intersecting at a common line, and then examining the rays created by a arbitrary plane that intersects the 4 original planes. Other examples can be created. When one cross ratio is evaluated with value k, the other cross ratios are known immediately, and have values (1k), (11/k), 1/k,1/(1k), and 1/(11/k).
A notable idea is that the six distinct cross ratios that can be constructed
sometimes coalesce into pairs. The concept is related to the concept of
degeneracy of eigen values in matrix algebra. That is, there exist certain
configurations for which the cross ratios are degenerate. For example, if k =
1, the other (degenerate) cross ratios are 0 and infinity. In another example,
if k = 1, the other degenerate cross ratios are 1/2 and 2. This case k = 1 is
called a Harmonic cross ratio and has interesting properties. In particular,
the two probabilities are equal at the harmonic limit.
In Figure 1. a schematic variation describes the values of the cross
ratios between the two degenerate configurations where k= + 1 or k = 1. An
arbitrary horizontal intercept yields the usual six values of the projective
cross ratios between the two degeneracy limits, where the cross ratios coalesce
in pairs.
Now suppose that the two xratios bounded between zero and 1 are related
somehow to the Born probability,
Another unanswered question is: What is the physical significance of the k = 1
limit ?
I will try to answer some of these questions in the future.
