Why and whence the Hilbert space in quantum theory?
- URL: http://arxiv.org/abs/2110.05932v3
- Date: Wed, 1 Mar 2023 13:27:57 GMT
- Title: Why and whence the Hilbert space in quantum theory?
- Authors: Yu. V. Brezhnev
- Abstract summary: We explain why and how the Hilbert space comes about in quantum theory.
An issue of deriving the norm topology may no have a short-length solution but is likely solvable in the affirmative.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We explain why and how the Hilbert space comes about in quantum theory. The
axiomatic structures of vector space, of scalar product, of orthogonality, and
of the linear functional are derivable from the statistical description of
quantum micro-events and from Hilbertian sum of squares
$|\mathfrak{a}_1|^2+|\mathfrak{a}_2|^2+\cdots$. The latter leads
(non-axiomatically) to the standard writing of the Born formula $\mathtt{f} =
|\langle\psi|\varphi\rangle|^2$. As a corollary, the status of Pythagorean
theorem, the concept of a length, and the 6-th Hilbert problem undergo a
quantum `revision'. An issue of deriving the norm topology may no have a
short-length solution (too many abstract math-axioms) but is likely solvable in
the affirmative; the problem is reformulated as a mathematical one.
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