I'd like to say a couple of things about "prodigals". A. Countable products in non-abelian groups are also quite interesting and useful. Absolute convergence makes sense: given a left-invariant metric d on the group, you can ask that the sum of d(identity, g_i) converges. With this condition, if the group is complete w.r.t. the metric, then the product converges in any order, but the answer depends on the ordering. The right way to define these products is to allow an index set with **any** countable linearly order, not just the natural numbers. E.g. there are a number of important constructions in topology that depend on the product using the natural order on the negative integers. I've had occasion to use this construction for sets of group elements indexed by an order-dense set, e.g. the rational numbers. You get an uncountable, but not arbitrary, set of values for the product of an absolutely convergent countable set of group elements, depending on the order. (There are probably additional condiitons needed to guarantee convergence in general, but if the metric is say a Riemannian metric on a Lie group I think absolute convergence is enough.) B. I don't want to discourage this nice discussion of prodigals, but just inject a little mainstream mathematical terminology. Another name for the integral-like product in GL(n,C) is linear differential equation (or if you prefer, a system thereof), which may or may not have constant coefficients. The infinitesimal elements of a Lie group constitute its Lie algebra. In the constant coefficient case, the prodigal is also called the exponential map, and it has been extensively studied in many variations. In the variable-coefficient case, these are the same as connections of G-bundles over an interval. The Campbell-Hausdorff formula is a power series expressing corrections to commutativity, as below. The curvature of a connection of a G-bundle over a square is one well-developed differential geometry context to understand non-commutativity in generally. Control theory is another area that addresses many types of questions about this stuff. I think this discussion is partly symptomatic of the malaise in mathematics that things like differential equations are usually taught in a formal way that not many students internalize. On Sat, Nov 09, 2002 at 04:31:33AM -0800, Shel Kaphan wrote:
At 01:52 PM 11/8/2002 -0800, I wrote:
then the product integral P (A(t)B(t))^dt seems to be the continuously combined rotation (at least according to one way of defining it), and seems to be equal to P (B(t)A(t))^dt.
I also see that this doesn't quite work out according to the way of defining things that is most similar to the ordinary integral (because (AB)^dx is not the same as (A^dx)(B^dx), although it would be awfully nice to have some operator that works like this, for "simultaneous" applications of transformations.