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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2014
    • (edited Mar 1st 2014)

    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.

  1. Some hardly known facts about Hilbert spaces are the following.
    Only a subtle difference exists between a vector space and a Hilbert space. In this way it becomes possible that a huge number of separable Hilbert spaces can share the same underlying vector space. Quaternionic number systems exist in many versions that distinguish between the Cartesian and polar coordinate systems that sequence their members. This affects the symmetry of the number system. A Hilbert space selects a version of the number system for specifying its inner product. This selects the symmetry of that Hilbert space. Each separable Hilbert space can manage a private parameter space in the eigenspace of a dedicated normal operator (that I call reference operator) by letting that eigenspace represent by the rational values in the selected version of the number system that is used to specify the inner products of vector pairs. A special category of normal operators can be defined by letting them share the eigenvectors of the reference operators and replacing the corresponding eigenvalues of the reference operator by the target values of a selected function. Each infinite dimensional separable Hilbert space owns in this way a unique non-separable Hilbert space that embeds its separable companion. In this way the special category of normal operators become field operators that combine Hilbert space operator technology with function theory, differential calculus and integral calculus. The continuum eigenspaces of these operators in the non-separable Hilbert space will implement a general field theory that in case of a quaternionic number system treats dynamic fields in a well-defined way.
    In this way a system of Hilbert spaces can act as a structured repository for discrete dynamic geometric data and dynamic continuums that act like (physical) fields.
  2. With the above additions to the concept of a Hilbert space, all elementary particles can be given a private separable Hilbert space. This platform floats with the geometric center of its private parameter space over a background parameter space that belongs to a background platform that is implemented by an infinite dimensional separable Hilbert space and its non-separable companion Hilbert space. The platforms of elementary particles feature a symmetry-related charge that is implemented by a source or sink that is located at the geometric center of the private parameter space. This fact requires restriction of the versions of the quaternionic number system that can contribute to this system of Hilbert spaces that all share the same underlying infinite dimensional vector space. Only versions of which the axes of the cartesian coordinates are parallel to the axes of the cartesian coordinates of the background platform can participate. This means that the sources and sinks require this restriction. If the charges relate to the differences in symmetry between the floating platforms and the background platform, then the resulting short list of charges resembles the short list of charges in the standard model of current physics.

    The fact that the system of Hilbert spaces acts as a repository for discrete dynamic geometric data and dynamic fields opens the possibility to convert the system into a self-creating model of physical reality in which the clock of the universe starts ticking after that the repository is filled with data that tell the complete life stories of the elementary particles. This possibility is explored in "A Self-creating Model of Physical Reality"; and is presented in
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2021

    added pointer to

    also added a parenthetical remark to the end of the entry’s footnote here

    diff, v20, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2021

    adjusted the wording of the Idea-section

    diff, v21, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2022

    A basic thought on the infinite tensor product of L 2L^2-spaces:

    The “grounded” infinite tensor product of Hilbert spaces, which is mentioned in

    • John C. Baez, Irving Ezra Segal, Zhengfang Zhou, p. 126 of: Introduction to algebraic and constructive quantum field theory, Princeton University Press 1992 (ISBN:9780691634104, pdf)

    and discussed in

    • Nik Weaver, Def. 2.5.1 Mathematical Quantization, Chapman and Hall/CRC 2001 (ISBN:9781584880011)

    has, I’ll suggest, a nicely transparent incarnation in the case of the Hilbert space of square-integrable functions on some (X,μ)(X,\mu), because here it is the dual of the limit that gives the infinite product X × X^{\times_\infty}:

    X limSFinSub()X SlimSFinSub()L 2(X) S=:L 2(X) . 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()L^2(-) applied to the projections

    S i S X S X i X S L 2(X S) (X i) * L 2(X S) \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 2L^2-spaces as having as vacuum state the constant unit function.

    This makes one want to say that L 2()L^2(-) takes (these) limits to (these) colimits, so that

    L 2(X) L 2(X ). 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…)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2022

    For what it’s worth, this is appears as Ex. 6.3.11 in

    • K. R. Parthasarathy, Introduction to Probability and Measure, Texts and Readings in Mathematics 33, Hindustan Book Agency 2005 (doi:10.1007/978-93-86279-27-9)
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2022
    • (edited Jan 20th 2022)

    Incidentally, regarding original references on the infinite tensor product in #6:

    Wikipedia here calls this notion the “incomplete-” or “Guichardet-“tensor product and references

    • O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, Springer 1987/2002

    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

    • A. Guichardet, Tensor products of C *C^\ast-algebras Part II. Infinite tensor products, Aarhus Universitet Lecture Notes Series 13 (1969) (pdf)

    where the case of Hilbert spaces is considered explicitly in section 6, while “incomplete” seems to refer to

    • J. von Neumann, On infinite direct products, Compositio Mathematica, tome 6 (1939), p. 1-77 (numdam:CM_1939__6__1_0)

    where something at least similar is considered around Lem. 4.1.2. (But is it the same?)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2022

    (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 B/2\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?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 16th 2022

    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 [i 0 0 i]\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 [0 1 1 0]\left[\array{ 0 & -1 \\ 1 & 0 }\right], i.e. interacting trivially with the anti-linear involution.

    (Notation from CPT symmetry: The complex structure is “iP\mathrm{i} P” and the antilinear involution is “CC” for Type C and “TT” 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.)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2022
    • (edited Oct 17th 2022)

    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 “VV” for the Hilbert space to “\mathcal{H}”, throughout.

    • changed the symbol “xx” for elements of the Hilbert space to “vv”, throughout (the other obvious notation choice would be “ψ\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.

    diff, v24, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2022

    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.

    diff, v24, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2022

    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 xx and yy, both nonzero, let the angle between them be the angle θ(x,y)\theta(x,y) whose cosine is cosθ(x,y)=x,yxy. \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.

    diff, v25, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2022

    Appended the main definition of Hilbert spaces by a remark (here) on the definition(s) of their morphisms.

    diff, v25, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2022

    I have added a proper bibitem for

    and used this to adjust the referencing inside the footnote here

    diff, v26, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2022

    added pointer to

    which seems to be the original publication containing the actual definition

    diff, v26, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2022
    • (edited Nov 8th 2022)

    I have added pointer to today’s preprint

    and while I was at it I have added publication data to the previous one:

    diff, v27, current

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2022

    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.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2022

    I have added pointer to:

    • Miklos Rédei, Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead), Studies in History and Philosophy of Modern Physics 27 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

    diff, v28, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2023

    added pointer to:

    diff, v32, current