Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2010

    I just aadded a sentence about Yang-Mills theory to gauge group, but there are some aspects of that article I feel we might want to discuss:

    I don’t think that the statement “gauge groups encoded redundancies” of the mathematical description of the physics is correct. One hears this every now and then, and I suppose the idea is the observation that physical observables have to be in the trivial representation of the gauge group, but there is more to the gauge group than that.

    Notably Yang-Mills theory is a theory of connections on G-principal bundles. No mathematician would ever say that the group G in a G-principal bundle just encodes a redundancy of our descriptins of that bundle. And the reason is because it is true only locally: the thing is that BG={*gG*}\mathbf{B}G = \{* \stackrel{g \in G}{\to} * \} has a single object and hence is connected , but it has higher homotopy groups, and that’s where all the important information encoded by the gauge group sits.

    So I would say that instead of being a redundancy of the description, instead the gauge group of Yang-Mills theory enocedes precisely the homotopy type of its moduli space. This is rather important.

    A different matter are global gauge symmetries such as those that the DHR-theory deals with.

    • CommentRowNumber2.
    • CommentAuthorTim_van_Beek
    • CommentTimeJun 29th 2010

    One hears this every now and then, and I suppose the idea is the observation that physical observables have to be in the trivial representation of the gauge group, but there is more to the gauge group than that.

    Interesting, I did not expect that the entry was already of any interest, therefore I did not start a thread here…

    A different matter are global gauge symmetries such as those that the DHR-theory deals with.

    Yes, that’s all I had in mind. As far as I know there is no concept in AQFT of a local gauge symmetry, and e.g. Haag stresses that this is one of the central challenges and open problems of this approach.

    No mathematician would ever say that the group G in a G-principal bundle just encodes a redundancy of our descriptins of that bundle.

    I think I understand that, but does that contradict the description that gauge transformations are not observable? I mean “redundant” may sound like “uninteresting”, but that’s certainly not meant here, it is just “redundant” = “maybe there is a framework of QFT without gauge symmetries, this could be possible, because gauge symmetries are not observable”. (At least that’s how I always understood that statement).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2010

    My suggestion would be: let’s split the entry into two main subsections: one discussing gauge groups in the sense of Yang-Mills theory, the other global gauge groups in the sense of DHR. These are really two quite different concepts that accidentally go by the same name.

    • CommentRowNumber4.
    • CommentAuthorTim_van_Beek
    • CommentTimeJun 29th 2010

    Ok, are you going to rewrite the idea section to fit the split?

    (If these are really different concepts, maybe they should get their own page?).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2010

    I tried to edit gauge group a bit. But just a rough first attempt.

    I’d says we’ll keep this in one entry for the time being, unless and until we have so much material for each of the two notions that a split is worthwhile.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2011

    added to gauge group a new section Examples with the list of examples that I had posted in reply to an MO-question here

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeSep 20th 2011

    I may be wrong, but my feeling is that there is also a notion of global gauge group at classical level, as the infinitedimensional Lie group of differentiable automorphisms of the principal bundle (rather than at the level of QFT).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2011

    Yes, there are also global gauge groups that should be discussed. For instance also for gravity on (X,g)(X,g) one should mention Iso(X,g)Iso(X,g) as a global gauge group. Maybe I’ll add something about that later.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeSep 20th 2011
    • (edited Sep 20th 2011)

    You did mention the global gauge group. The only thing is that I am emphasising the classical versiom while the entry was saying about QFT and local nets in quantum theory, not about the classical gauge theory.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2011

    Ah, that’s what you mean. Sure, there is also a notion of global gauge group in classical theory. I think the point of mentioning it especially for the AQFT case is that this is a definition of QFT that is independent of quantization from a classical theory, so there it is of special interest if one can still read off the global gauge group from just the information of a local net. And of course it’s one of the main theorems in the whole theory that one can (proved by Doplicher-Roberts reconstruction, as you know).

    • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeSep 20th 2011

    the gauge group of Yang-Mills theory enocedes precisely the homotopy type of its moduli space.

    That is very clear to me, thanks!

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2011

    Thanks for saying this. This made me expand that remark to a new subsection at gauge group: called now Properties – Not a redundancy.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2019
    • (edited Sep 7th 2019)

    added pointer to

    diff, v19, current

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2019

    Given the discussion in #1,2,3 to limit ’redundancy’ talk to the global gauge case, so we have there

    nontrivial gauge groups arise from redundancies of the mathematical description,

    what should we make of

    It is being argued that after embedding into consistent quantum gravity, all global symmetries must become local symmetries ?

    Redundancies are no longer redundant?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2019

    Thanks for highlighting that it says this in the entry. I haven’t actually read the entry text in a long while.

    I do disagree with the slogan that “local gauging is just a redundancy”.

    In the nLab entry this statement originates with Tim van Beek’s rev #1.

    The whole entry needs some love. Right now I am in the sky over, let’s see, Austria, not enough elbow room to edit much. Maybe I find leisure to do some editing tonight in the hotel.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2019

    Heading out to Abu Dhabi? Did you say Vincent was joining you?

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2019

    Yes! Vincent has arrived in AD last week.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2019

    So maybe we’ll get to hear

    how nonperturbative thermal QFT arises as the cohomological quantization of the C-field charge-quantized in Cohomotopy theory.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2019

    Yes, we will be picking up that thread now.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2019
    • (edited Sep 8th 2019)

    added also pointer to yesterday’s

    • Sylvain Fichet, Prashant Saraswat, Approximate Symmetries and Gravity (arXiv:1909.02002)

    diff, v20, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)