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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"; http://vixra.org/abs/1908.0223 and is presented in http://www.e-physics.eu/Base%20model.pptx
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 16th 2021

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 16th 2021

adjusted the wording of the Idea-section

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 19th 2022

A basic thought on the infinite tensor product of $L^2$-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 $(X,\mu)$, 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…)

• 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 $C^\ast$-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?)