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there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics:
John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.
George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963
E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.
added pointer to
also added a parenthetical remark to the end of the entry’s footnote here
A basic thought on the infinite tensor product of $L^2$-spaces:
The “grounded” infinite tensor product of Hilbert spaces, which is mentioned in
and discussed in
has, I’ll suggest, a nicely transparent incarnation in the case of the Hilbert space of square-integrable functions on some $(X,\mu)$, because here it is the dual of the limit that gives the infinite product $X^{\times_\infty}$:
$X^\infty \;\coloneqq\; \underset{ \underset{ \mathclap{ S \in FinSub(\mathbb{N}) } }{\longleftarrow} }{\lim} \; X^{S} \;\;\;\;\;\;\; \mapsto \;\;\;\;\;\; \underset{ \underset{ \mathclap{ S \in FinSub(\mathbb{N}) } }{\longrightarrow} }{\lim} \; L^2(X)^{\otimes_S} \;\;\;\; =: \; L^2(X)^{\otimes_\infty} \,.$Here the “groundedness” of the product is just the contravariant functoriality of $L^2(-)$ applied to the projections
$\array{ S' &\xhookrightarrow{ \;\;\;\; i \;\;\;\; }& S \\ X^{S'} &\xleftarrow{ \;\;\;\; X^i \;\;\;\;}& X^S \\ L^2\big( X^{S'} \big) &\xrightarrow{ \;\;\; (X^i)^\ast \;\;\; }& L^2\big( X^{S} \big) }$which implicitly regards each $L^2$-spaces as having as vacuum state the constant unit function.
This makes one want to say that $L^2(-)$ takes (these) limits to (these) colimits, so that
$L^2(X)^{\otimes_\infty} \;\; \simeq \;\; L^2\big( X^\infty \big) \,.$Is this the case? (Just firing off this question, maybe it’s evident once I start to really think about it…)
For what it’s worth, this is appears as Ex. 6.3.11 in
Incidentally, regarding original references on the infinite tensor product in #6:
Wikipedia here calls this notion the “incomplete-” or “Guichardet-“tensor product and references
where the notion is briefly mentioned on p. 144 (and only there, it seems), without, however, calling it anything, neither “incomplete” nor “Guichardet”.
The attribution to Guichardet must refer to
where the case of Hilbert spaces is considered explicitly in section 6, while “incomplete” seems to refer to
where something at least similar is considered around Lem. 4.1.2. (But is it the same?)
(almost back from an offline week…)
On the topic of category-theoretic characterizations of finite dimensional complex Hilbert spaces (i.e. fin dim Hermitian inner product spaces), I am wondering about the following:
In the topos of involutions (i.e. presheaves on $\mathbf{B}\mathbb{Z}/2$), consider $\underline{\mathbb{C}}$ to be the ring of complex numbers equipped with its involution by complex conjugation.
Then self-dual $\underline{\mathbb{C}}$-modules are of the form $\mathcal{H} \oplus \mathcal{H}^\ast$ for $\mathcal{H}$ a finite dimensional complex Hermitian inner product space and for the involution given by the anti-linear isomorphism $\mathcal{H} \overset{\sim}{\leftrightarrow} \mathcal{H}^\ast$ which reflects the sesquilinear form. A morphism between such objects is a unitary map if its tensor square preserves (is sliced over) the evaluation map.
Has this perspective been discussed anywhere?
More in detail, on a $\underline{\mathbb{C}}$-module (in the sense of #9) we may still ask for an actual complex structure (i.e. a compatible $\mathbb{C}$-module structure for $\mathbb{C}$ the complex numbers equipped with the trivial involution). There are two types of these:
Type C acts by a linear map of the form $\left[\array{ \mathrm{i} & 0 \\ 0 & - \mathrm{i} }\right]$, i.e. interacting non-trivially with the anti-linear involution;
Type T acts by a linear map of the form $\left[\array{ 0 & -1 \\ 1 & 0 }\right]$, i.e. interacting trivially with the anti-linear involution.
(Notation from CPT symmetry: The complex structure is “$\mathrm{i} P$” and the antilinear involution is “$C$” for Type C and “$T$” for Type T.)
Now: Finite dimensional complex Hermitian inner product spaces are equivalently the $\underline{\mathbb{C}}$-modules with complex structure of Type C which are self-dual via (co-)evaluation maps that are real with respect to this structure.
(The point of this formulation is that Chern characters of equivariant KR-theory come equipped with such kind of structures.)
Have touched the Definition-section at Hilbert space (here) (up to and excluding its first sub-section).
In particular I have:
introduced Definition/Remark-environments and cross-links between them;
moved the actual definition of Hilbert spaces from the end to the beginning of the Definition-section
added the adjective “hermitian” to all or most occurrences of “inner product”
changed the symbol “$V$” for the Hilbert space to “$\mathcal{H}$”, throughout.
changed the symbol “$x$” for elements of the Hilbert space to “$v$”, throughout (the other obvious notation choice would be “$\psi$”, of course).
The Definition-section used to allude to other choices of ground fields besides the complex numbers, without saying more about this. I have added a remark (here) making explicit the case of real numbers and mentioning that under mild conditions this is the only alternative ground field, with a pointer to MO:a/4184099.
I feel uneasy about the comment on “physicist’s conventions” versus “mathematician’s convention” (now this remark). Better would be to point out concrete prominent textbooks which use one or the other convention.
In the Idea-section (here) I have expanded the two main paragraphs,
(a) making more explicit what the two axioms (inner product and completeness) “mean” in the application to quantum physics and
(b) mentioning the dagger-structure on the category of Hilbert spaces as reflecting the presence of the inner product.
The Definition-section used to have no less than four subsections, which however in total amounted to little more than a comment on the definition. To adjust this I have:
introduced subsections “Standard formulation” and “Alternative formulation”
merged the previous subsections “Hilbert spaces as Banach spaces” and “Hilbert spaces as metric spaces” into one sub-subsection “As Banach spaces and metric spaces” under “Alternative formulations”.
deleted the subsection “Morphisms of Hilbert spaces” which in its entirety consisted of little more than a pointer to the entry Banach space
deleted – for the time being – the subsection “Hilbert spaces as conformal spaces”, because it just said the following:
Given two vectors $x$ and $y$, both nonzero, let the angle between them be the angle $\theta(x,y)$ whose cosine is $\cos \theta(x,y) = \frac { \langle x, y \rangle } { \|x\| \|y\| } .$ (Note that this angle may be imaginary in general, but not for a Hilbert space over $\mathbb{R}$.) A Hilbert space cannot be reconstructed entirely from its angles, however (even given the underlying vector space). The inner product can only be recovered up to a positive scale factor.
Maybe this deserves to be kept as a remark, though then I would suggest to expand on what the point being made here is meant to be.
I have added a proper bibitem for
and used this to adjust the referencing inside the footnote here
added pointer to
which seems to be the original publication containing the actual definition
I have added pointer to today’s preprint
and while I was at it I have added publication data to the previous one:
One thing that would be good to know is the physical “axioms” that correspond to the axioms in the new paper, to fulfill the promise in the intro, to better axiomatise quantum theory without putting in Hilbert spaces and the reals/complex numbers by hand.
I have added pointer to:
added pointer to:
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