> Name of first author: JM Isidro
> Title: Quantum-mechanical dualities on the torus

The letter is basically an example of a theory of transformations developed
by the author in previous letters. It is a very pertinent example because it
is easy to understand and it illustrates well the author intentions. Now, 
beyond pertinence, the appropriateness of publishing examples in a journal 
is not a question of referring but of editors. To this referee, it seems
that a letter journal al PLA is adequate for it, just signalling this fact in
the title or the abstract of the article. On other hand, I would 
encourage the author to join all its previous work and to try to form
a full exposition article aimed, say, to the Annals of Physics or a 
similar publication.  I am afraid that publishing and curriculum pressures
are being prioritised by the author, instead of pure science issues. This
guess is somehow supported by inspection of the website of his institution, 
where it appears to hold a non-extendable two-years position.

If the letter is to be published in  its present form, it still needs of major 
rewriting. This is because the work has drifted from a theory of dualities to 
a theory of transformations. The main equation in the paper, eq. 29, can be 
only considered a duality for $\beta=-1$, the only case where it could be 
claimed that the dual of the dual contains the original. The whole 
denomination of "duality" should, then, be reconsidered or at least 
re-explained.

Nomenclature sloppiness spreads also across all the paper. The terms in the
Hamiltonian are divided in "kinetic" and "potential" without considering
if there are related to kinetic and potential energies. The torus T^2
and the cylinder, Y^2, are used to label specific Hamiltonians as if there
were a canonical way to associate a unique Hamiltonian to a given topology. 
Perhaps the author has some concrete mechanism in mind, but it is not
clear from the paper.

Also the expression "limit of large n" is unfortunate in section 3.4, as it
usually means "the classic limit", but here it is related to some subspace
where the vacuum lives. On other hand, this referee is unable, from the paper 
contents only, to grasp the motivation, goals and results of this section. If 
I am missing some point, it is the author mission to show the point clearly.

The rewriting could be used, too, to correct some minor style problems in the
quotation architecture. There are an excess of unneeded, out of gentilesse, 
references. Starting with one towards the journal editor, in the first line of
the letter. Such references are, if not irrelevant, at most marginally 
relevant, and it one should to keep this kind of references at a number
strictly smaller than the relevant ones. The same could be said of the 
theoretical allusions in the concluding part.

And still, some interesting references are lacking. It occurs to me that 
the reader, for instance, could want to get a proof of the monotonical 
increase of the dimension of H (start of section 3.1), specially considering 
the possibility of degeneracy of states in the Hilbert space and non trivial 
topologies in the phase space.

In the case of references to books, it can be made more relevant by using the 
optional field in LaTeX "cite" command to give a clue of the section or 
chapter when it is not obvious.

I notice also that in some cases the ArXiV number of a paper is not given,
even if it is obviously known to the actor. As the goal of complete references
is to help people to peruse the articles, it is a good idea to give always
the numbers, and let the editorial team to decide if and when remove them.