The letter is very interesting: it helps to popularise an
intriguing result of Grimov, suggesting a new use which was only
hinted in a previous article of De Gosson. Also, it announces
further advances based in a not very well known result of
Sikorav. Globally it has a high degree of originality and a
scientific quality above average, thus I recommend its publication.
Now, the degree of urgency is not so high for the letter not
to benefit from minor editing, so I'd like to suggest
some corrections to the final version:
- The formula previous to (3) is letter-by-letter identical to (3). Surely it
can we rewritten for clarity. A reference to some article on Maslov
semiclassical quantisation could be added.
-Some results were originally described for complex spaces and they
have been rewritten in terms of real spaces for legibility. So I'd suggest
to double-check all the dimension labels "n" and "2n".
-Same double-check with the subindices "i", "j" in page 2, esp. for
rotated or transformed variables.
-One wonders if there are some restrictions in the dimension of the
subsets of R^{2n} to be admissible quantum blobs.
-To fulfil the divulgative goals, it should be good to add precise
references to theorem numbers in Sikorav and Grimov. Also, the
bibliography could quote the MR numbers of the articles:
[2] has MR 87j:53053
[6] has MR 93f:57033
-The use of Sikorav's result is a new idea and it could be explained
with a little more of detail.
If only such minor modifications are done, I believe the letter does not
need a new refereeing phase and it can be published directly.
Last, let me add that a full article expanding the research should
follow this one. I am specially intrigued about two details:
- The dequantization process, both in the limit h->0 and in the limit
of large quantum numbers. Previous work of the author has been
based on semiclassical mechanics, via Maslov, and it should be seen
if it is possible to reproduce exactly the quantum mechanics results.
- The case of infinite degrees of freedom. As the blob capacity does
not depend of the number of degrees of freedom, the limit case of QFT should
be easier to control than usual. One should research, for instance, what
happens to the symplectic capacities when a (Kadanoff-like) R.G. transformation
decimates the number of degrees of freedom.
If the author has some advances on this, I'd not oppose to preannounce
them at the end of the letter.