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I am not entirely happy with the terminology here:
to me "matrix mechanics" is an old and outdated historical way to look at quantum mechanics. What is outdated about it is that it was an attempt to understand the structures in QM using naive finite dimensional linear algebra.
But the operators in QM are in general not matrices, of course, but operators on infinite-dimensional Hilbert spaces. To some extent it is useful to think of these as being just matrices, but just to some extent.
I think it would be better to speak about operator algebra linear over an arbitrary rig. In examples it makes sense to restrict to the fin dim case and speak of matrix mechanics there, but for the general case I find it a bit anachronistic.
Even in fin-dim linear algebra: speaking of matrices is always a bit evil, of course. We don't want to be that evil on the nLab, do we?
Extremism in defense of Jacobins may be morally suspect, but Jacobeans are well and good on balance.
Re: A name change is fine, but isn't there something snappier than "operator algebra linear over an arbitrary rig"? How about "rig mechanics"?
Where I come from, a rig mechanic might be regarded as one type of "roughneck". His immediate supervisor would be called a "tool pusher".
But maybe "derrick" (= Der(Rig)?) would do the trick?
I think one could keep the page title (since it's a well-known term) but also echo Urs and start with something like "Matrix mechanics is an old-fashioned name for..." and then work one's way into talking about the new-fangled ways of thinking about what this is all about.
I am fine with Todd's suggestion, but maybe just one more remark to amplify the point even further:
in the field of algebraic quantum mechanics and algebraic quantum field theory (AQFT) there is a strong consensus that QM is all about C-star algebras and in particular von Neumann algebras. Parts of that is presently be absorbed and re-expressed by factorization algebra methods, where people do indeed also consider the transition from classical to qm-systems by changing their codomains. Kevin Costello keeps on his web page the pdf file version of a book he writes on this stuff.
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