Although I have no idea what "matrix product physics" might mean, the problem of rolling a unit sphere (in R^3) along a piecewise smooth path in the plane -- and figuring out its net rotation -- is essentially taking a limit of the product of more and more smaller and smaller matrices . . . and this limit very definitely depends on the path the sphere is rolled along. --Dan Asimov Marc LeBrun wrote: One more ravelet: seeing path invariance as a kind of symmetry, what form of conservation laws might that imply for a matrix product physics? --WDS: In the Feynman path integral (FPI) formulation of quantum physics, everything is the continuum (i.e. "integral" not "sum") version of a matrix (actually operator) product, summed over all possible paths. In most situations, though, the FPI is very dependent upon the path. However, supposing some kind of quantum physics were living in some land where FPI did NOT depend on the path -- either because something fundamentally is that way, or because it's some cheesy model scenario you cooked up -- then you'd have Gosperian PaIn. Or PIMP. Whichever of these two wonderful acronyms you prefer. (How it can be that Gosper's ideas have escaped great attention in spite of having not one, but two, amazing acronyms is truly pathetic.) In which case the FPI would become trivial to evaluate, I guess. Which would be good, since situations so far in which physicists have been able to evaluate FPIs in close form, have been quite rare. A usual rubric in physics ("Noether's theorem") is that "to every symmetry there corresponds a conservation law." What would that tell us in the case? Well, PaIn would really be a humongous symmetry, much larger than the usual symmetries like translation invariance (which corresponds to momentum and energy conservation). I'm not sure what the heck conservation law it would correspond to (if any); it does not seem to obey the usual monkey-see-monkee-do pigeonholing that allows idiots like me to operate? I guess it would constitute some kind of unification, where that sort of physics really does not care what the environment is. It is also possible, Feynman later saw, to formulae statistical physics via a different kind of path-integral.