Blog Hole Memory Rescue and Lost Papers that were Really Lost

There is more than one path to classical mechanics.

So much to do, so little time.  My own lost paper work (i.e. the translation of some of Hilbert’s old papers that are not available in English) is commencing at a snail’s pace, but at Kingsley Jones’ blog we can learn about some papers that were truly lost and misplaced and that he only recovered because throughout all the moves and ups and downs of life, his parents have been hanging on to copies of the unpublished  pre-prints.  Kingsley’s post affected me on a deeper level than the usual blog fare, because this is such a parent thing to do.  Having (young) kids myself, I know exactly the emotional tug to never throw away anything they produce, even if they have seemingly moved on and abandoned it.  On the other hand, the recollection of how he found these papers when going through his parent’s belongings after they passed away, brings into sharp relief the fact that I have already begun this process for my father,  who has Alzheimer’s.  So many of his things (such as his piano sheet music) are now just stark reminders of all the things he can no longer do.

On a more upbeat note: The content of these fortuitously recovered papers is quite remarkable.  They expand on a formalism that Steven Weinberg developed, one that essentially allows you to continuously deform quantum mechanics, making it ever less quantum.  In the limit, you end up with a wave equation that is equivalent to the Hamiltonian extremal principal–i.e. you recover  classical mechanics and have a “Schrödinger equation” that always fully satisfies the Ehrenfest Theorem. In this sense, this mechanism is another route to Hamilton mechanicsThe anecdote of Weinberg’s reaction when he learned about this news is priceless.

Ehrenfest’s Theorem, in a manner, is supposed to be common sense mathematically formulated: QM expectation values of a system should obey classical mechanics in the classical limit.  Within the normal QM frameworks this usually works, but  the problem is that sometimes it does not, as every QM textbook will point out (e.g. these lecture notes).  Ironically, at the time of writing, the Wikipedia entry on the Ehrenfest Theorem does not contain this key fact, which makes it kind of missing the point (just another example that one cannot blindly trust Wikipedia content). The above linked lecture notes illustrate this with a simple harmonic oscillator example and make this observation:

“…. according to Ehrenfest’s theorem, the expectation values of position for this cubic potential will only agree with the classical behaviour insofar as the dispersion in position is negligible (for all time) in the chosen state.”

So in a sense, this is what this “classic  Schrödinger equation” accomplishes: a wave equation that always produces this necessary constraint in the dispersion.  Another way to think about this is by invoking the analogy between Feynman’s path integral and the classical extremal principle.  Essentially, as the parameter lambda shrinks for Kingsley’s generalized Schrödinger equation, the paths will be forced ever closer to the classically allowed extremal trajectory.

A succinct summation of the key math behind these papers can be currently found in Wikipedia, but you had better hurry, as the article is marked for deletion by editors following rather mechanistic notability criteria, by simply counting how many times the underlying papers were cited.

Unfortunately, the sheer number of citations is not a reliable measure with which to judge quality. A good example of this is the Quantum Discord research that is quite en vogue these days. It has recently been taken to task on R.R. Tucci’s blog. Ironically, amongst many other aspects, it seem to me that Kingsley’s approach may be rather promising to better understand decoherence, and possibly even put some substance to the Quantum Discord metric.

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