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nLab > Latest Changes: uncertainty of fluxes

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2023

    starting a page on Poisson/commutator brackets of flux observables in (higher) gauge theory.

    The title of the entry follows the title of Freed, Moore & Segal 2007a because that’s a good succinct description of the subject matter, but I don’t mean the entry to be restricted to their particular perspective (in fact, is their uncertainty relation not ultimately a definition – their Def. 1.29 – rather than a derivation from first principles?)

    The most insightful discussion of the matter that I have seen so far is that in Cattaneo & Perez 2017, which is motivated by application to first-order formulation of gravity (where this has found a lot of attention), but I think the arguments apply verbatim to Yang-Mills theory, too (where however I haven’t seen it find any attention yet(?)).

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2023
    • (edited Dec 1st 2023)

    I have spelled out (here) full details of the computation of Poisson brackets between electric/magnetic fluxes in 𝔤\mathfrak{g}-Yang-Mills for semisimple 𝔤\mathfrak{g}.

    The first Proposition (here), on the bracket among electric fluxes, is the result of Cattaneo & Perez 2017 (7), who left the computation implicit.

    (Of course, it’s a standard-type and straightforward computation — their substantial contribution is not this computation but the identification of the correct form of the flux observables in the first place.)

    The second proposition (here) may be new (but let me know if not):

    It says that an electric flux observable always Poisson-commutes with any magnetic flux observable.

    This is somewhat curious.

    (In the abelian case this is stated in Freed, Moore & Segal 2007b (3.6), p. 19, — though their proof does not seem to take care of the subtlety highlighted by Cattaneo & Perez 2007).

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2023
    • (edited Dec 1st 2023)

    Ah, of course my very last step was wrong, where I use Stokes’ theorem: Since we are integrating over a manifold with boundary, the result is not 0 but is that boundary term. Have fixed it now (here).

    [edit: in fact, I don’t need Stokes at all, can collect terms already before, as shown now]

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2023

    various little amendments and improvements, such as adding more comments on the role of the Gauss law.

    diff, v10, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2023

    I have summed up the computation of the Poisson brackets more succinctly in a theorem (now here)

    stating that the reduced phase space of integrated electromagnetic fluxes through a surface SS in 𝔤\mathfrak{g}-Yang-Mills theory is the Lie-Poisson manifold of the Lie algebra of smooth maps from SS into the semidirect product Lie algebra (𝔤,[−,−])⋉ ad𝔤\big(\mathfrak{g}, [-,-] \big) \ltimes_{ad} \mathfrak{g}.

    diff, v12, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2023

    started a section (here) on the non-perturbative quantization of fluxes in Yang-Mills.

    diff, v13, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2023

    added a remark (here) on how we should really be using differential forms with values in an adjoint bundle 𝔤 P\mathfrak{g}_P instead of just 𝔤\mathfrak{g}, but that in the end it does not matter for the discussion here.

    diff, v17, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2023
    • (edited Dec 9th 2023)

    We now have a write-up on the matter, behind this link:

    \,

    Abstract. While it has become widely appreciated that defining (higher) gauge theories requires, in addition to ordinary phase space data, also “flux quantization” laws in generalized differential cohomology, there has been little discussion of the general rules, if any, for lifting Poisson-brackets of (flux-)observables and their quantization from ordinary phase spaces to the resulting higher moduli stacks of flux-quantized gauge fields.

    In this short note we present a systematic analysis of (i) the canonical quantization of flux observables in Yang-Mills theory and (ii) of valid flux quantization laws in abelian Yang-Mills. We observe (iii) that the resulting topological quantum observables form the homology Pontrjagin algebra of the loop space of the moduli space of flux-quantized gauge fields.

    This is remarkable because the homology Pontrjagin algebra on loops of moduli makes immediate sense in broad generality for higher and non-abelian (non-linearly coupled) gauge fields, such as for the C-field in 11d supergravity, where it recovers the quantum effects previously discussed [I, II] in the context of “Hypothesis H”.

    \,

    Comments are welcome.

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