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Ian,
maybe you should mention the word "Hilbert space" somewhere.
I am a bit unsatisfied with what you have under "Definition". I think that under that section name we should have stuff thatt really qualifies as a definition in the mathematical sense. What you write there is more like an extended Idea-section.
Also, I don't understand the remark about states and objects in a category. And the link to 2-vector spaces is a bit unmotivated. You might want to tell the reader what you think 2-vector spaces have to do with quantzum states.
If you don't want to write down precise statements yourself, maybe you can point the reader to some literature where details are given.
Despite what we, as physicists, may publicly claim, there is some disagreement on what that means.
Well, there are different proposals for formalizations of quantum theory, which are known in parts to be related, in parts not, and they have different precise notions of what a quantum state is.
One should start with plain quantum mechanics, where states are rays in a Hilbert space. One should mention the special case where that Hilbert space is square integrable functions on a Riemannian space, or square integrable sections of a Hermitian vector bundle on that space, more generally.
Then one could talk about AQFT, where states are certain linear functionals on the operator algebra.
Then one could talk about extended TQFT, where quantum states are the generalized elements of whatever the TQFT n-functor assigns in some codimension.
All these definitions are a little different, but there are some known and some conjectured ways of how they relate. We don't have to give the full picture in one go, but should try to describe th aspects that are known.
But an incomplete entry is okay, of course. But le't not write "Definition" if then no definition of anything is given. I think that gives the wrong impression.
I added some words at the beginning of the idea-section in an attempt to set the scene.
Then I moved what was under "Definition" to the end of the "Idea"-section and instead put at "Definition" an empty template list of subsection headlines that indicate the types of definitions that I think should eventually be included here.
One should start with plain quantum mechanics, where states are rays in a Hilbert space.
This is true in quantum mechanics of anything including in QFT. Of course sometimes there is more fine structure, but still at the bottom is a projectivization of a Hilbert state or an equivalent formalism (e.g. stated in terms of C-star algebra without a concrete representation fixed, what can be done via GNS-construction).
Of course, in statistical mechanics one can have in addition states which are not pure states, but are rather weighted sums of pure states.
Urs, I do not like that pejorative referal you use about general quantum mechanics calling it "plain quantum mechanics". Quantum mechanics is about a Hilbert space and a Hamiltonian operator. QFT is a special case, at least from the physicist' point of view. There is a finite-dimensional case (QM of finitely many degrees of freedom) when the mathematical problem with well-definedness etc. are typically simpler than for the quantum field theories. When you talk about specifical features of finitely many degrees of freedom, then why not just saying QM with finitely many degrees of freedom. Talking about Hilbert space does NOT confine to this setup but is rather the feature of general QM. Substationally different formalisms are when one allows non-pure states in statistical (quantum) mechanics and in study of open quantum systems.
Another thing is axiomatic quantum field theory. One should distinguish axiomatic quantum field theory as a general subject dealing with rigorous approach to QFT to the biggest subset of it usually called algebraic QFT which is based on C-star algebras.
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