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nLab > Latest Changes: Hermitian form

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2022

    I have added an observation (here) that complex Hermitian inner product spaces ℋ\mathcal{H} may be regarded as (ℤ/2↷ℂ)(\mathbb{Z}/2 \curvearrowright \mathbb{C})-modules of the form ℋ⊕ℋ *\mathcal{H} \oplus \mathcal{H}^\ast in the topos of ℤ/2\mathbb{Z}/2-sets.

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2022

    added (here) a characterization of hermitian operators, in this fashion

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2023
    • (edited Aug 31st 2023)

    I have added some textbook references

    and some historical references,

    such as to von Neumann’s original definition of Hermitian forms (1930, 1932)

    Von Neumann’s is the modern definition – was he the first to state it this way?

    I gather from

    • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke: Integral Binary Hermitian Forms, Ch. 9 in Groups Acting on Hyperbolic Space, Springer (1998) [doi:10.1007/978-3-662-03626-6_9]

    that the concept is named after

    but it is not easy to recognize a Hermitian form in there (admittedly, I have only scanned over the article, so far)

    Similarly for

    • Luigi Bianchi, Forme definite di Hermite, §24 in: Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî, Mathematische Annalen 40 (1892) 332–412 [doi:10.1007/BF01443558]

    which may be the earliest coinage of the term “Hermitian form” (?)

    diff, v9, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2023

    I have now written out more text in the section on reagarding Hermitian/unitary complex modules as complex Euclidean/orthogonal “Real complex modules” internal to the category of real vector spaces with involutions.

    Also adjusted some of the terminology, for better flow and thus renamed the sub-section:

    As Real complex modules.

    diff, v13, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2023

    added pointer to:

    • Max Karoubi, §1 in: Le théorème de périodicité en K-théorie hermitienne, Quanta of Maths 1, AMS and Clay Math Institute Publications (2010) [arXiv:0810.4707]

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2023

    I see now that the embedding of fin-dim Hermitian spaces into self-dual ℤ/2↷ℂ\mathbb{Z}/2 \curvearrowright \mathbb{C}-modules which I described (here) is essentially what is known in the context of Hermitian K-theory as the hyperbolic functor

    Or rather, what I consider here is the hyperbolic functor equipped with ℤ/2×ℤ/2\mathbb{Z}/2 \times \mathbb{Z}/2-equivariance. I see now that this is known as establishing an equivalence between KR-theory and topological Hermitian K-theory.

    The two references for this equivalences that I am (now) aware of (here) are:

    and

    both of which are rather terse, each in its own way.

    So I am not surprised that my construction here is known, on the contrary, I was surprised that I couldn’t find something that elementary discussed in the literature. And with the above two references it still takes some squinting to see what’s going on.

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2023

    added (here) original references expressing hermitian forms via dagger-compact structure

    diff, v23, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2023
    • (edited Nov 10th 2023)

    added (here) statement and proof of a more pronounced way of phrasing the expression of Hermitian forms as inner products internal to Real vector bundles:

    Passing a real inner product space (𝒱,g)(\mathscr{V}, g) equipped with an isometric complex structure JJ through the equivalence of real vector spaces with Real vector bundles over the point yields the corresponding Hermitian form ⟨−|−⟩≡g(−,−)+ig(J(−),v)\langle - \vert - \rangle \,\equiv\, g(-,-) + \mathrm{i} g\big(J(-), v\big), as shown here.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2023

    added pointer to:

    diff, v27, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2023

    added pointer to:

    diff, v28, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2023

    We now have a (very) brief note on the story I was getting at above, at Quantum and Reality.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTime6 days ago
    • (edited 5 days ago)

    …interestingly tying the foundations of quantum theory to homotopy theory.

    Interesting, indeed.

    That point about the right 0-truncation of spectra being the heart, and

    the proper notion that replaces nn-truncation in stable ∞\infty-categories are “t-structures”,

    what happens next? Is it easy to say what these t-structures are for higher nn for say HℂH \mathbb{C}?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTime5 days ago

    Are you asking if there is a useful generalization of t-structures from 1- to nn-excisive functors?

    I don’t know.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTime5 days ago
    • (edited 5 days ago)

    For some context, strangely I’m back to thinking about linear HoTT having the need to deal in my day job with bunched dependent types. So I gave a talk at the OASIS seminar in Oxford last week on it. I was just thinking that it’s odd that even when just dealing with finite dimensional Hilbert spaces in quantum computing that it’s worth treating them via a language that suited to parameterized spectra. But then one sees how important the larger context is with: groupoid dependence for topological protection; the KRK R-spectrum for density matrices; equivariance to avoid the dagger construction; etc.

    So then I was wondering if we see some use for the bigger picture in this current case, e.g., if the heart corresponds to 0-truncation (so fin-dim Hilbert spaces for HℂH \mathbb{C}-modules), there might be some use for 1-truncated things. So presumably that would concerns parameterized spectra limited to dimensions 00 and 11. Perhaps in the case of HℂH \mathbb{C}-modules a kind of 2-vector space, such as the Baez-Crans form?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTime5 days ago
    • (edited 5 days ago)

    The role of “higher quantum state space” that I am aware of so far is intermediate – they appear as an intermediate step in the construction of anyon state spaces in the main theorem of Topological Quantum Gates in Homotopy Type Theory.

    In the main formula there, (224) p. 76, inside the 0-truncation [−] 0[-]_0 we have the “higher complex lines” B n(ℂ)\mathbf{B}^n(\mathbb{C}), playing the role of something like higher prequantum line bundles for the anyons — the formula constructs the space of sections of these and in the end 0-truncates it all to retain a bundle of ordinary vector spaces – the Knizhnik-Zamolodchikov bundle of conformal blocks, being the Hilbert spaces for 𝔰𝔲(2)\mathfrak{su}(2)-anyons.

    Even if just intermediate, the higher state spaces here are absolutely crucial for the construction. That’s why we write, in the abstract and introduction of the article, something to the extent that topological quantum computing may be the first real-world use case of higher homotopy computing.

  16. Interesting!

    I see Quantum and Reality explicitly speaks of those “higher quantum state spaces”

    While higher structures are profoundly interesting – and while we have here at our fingertips a definition of higher “(∞,1)-Hilbert spaces” (of finite type) which is worth exploring further – for the scope of this note we want to restrict attention to ordinary quantum state spaces.

    On a separate (or not so separate) note, if the constructions of this paper are such as to “absorb the dagger-structure into the type structure”, is there a (related) strategy for this in the case of other appearances of dagger-structure? Could it be that some kind of ℤ 2\mathbb{Z}_2-equivariant approach would allow us to avoid any explicit appeal to dagger-structure?

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