Talk:Gauge theory

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class High Priority  Field: Mathematical physics
One of the 500 most frequently viewed mathematics articles.
WikiProject Physics This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
B This article has been rated as B-Class on the assessment scale.
Top This article is on a subject of top importance within physics.

Help with this template

Comments:
  • Can we have an intro paragraph for, say, the reader with only first year undergraduate physics, rather than this somewhat last-year mathematical physics student level.... also a roadmap for subjects to read about first (differential geometry, tensors...) ?
IMHO, first-year undergraduate level is rather a tough order -- but I am trying my best to divide this thing into an intro paragraph with the bare physical idea, a short historical note, and a section with a physics example - maybe SU(N) or something like that, followed by the heavy duty diffgeom. Would be nice to get some backup here - Amar 08:01, Jun 12, 2004 (UTC)

Contents

[edit] Help ! Images

  • Just added an image that I drew to the classical theory section. Very evidently, it's kind of clunky - can someone with better artistic skills advise/add a better picture ? - Amar 09:57, Jun 19, 2004 (UTC)
what's a feynman diagram doing in the classical theory section? Feynman diagrams are objects in the quantum theory. and why not use LaTeX feynman diagrams instead of hand-drawn? -Lethe | Talk

The Yang-Mills action is NOT the most general gauge invariant action out there... Phys 05:48, 14 Aug 2004 (UTC)

I would like to rotate the diagram by 90 degrees. As this is a spacetime diagram, the current orientation depicts particles travelling backward in time ("downward" motion = time reversal). If the diagram is re-oriented, then the gauge interaction occurs at some known time (height above the x-axis) which how Feynman would have drawn this. Ancheta Wis 13:13, 28 Sep 2004 (UTC) Thinking about it, the particles need to be bosons, so the straight lines need to be wavy lines. What about the interaction with the gauge boson, now, does it make sense for a collision to occur, or is it simply a decay? Ancheta Wis 21:59, 28 Sep 2004 (UTC)

[edit] Re mathematical formalism

Maybe there should be some mention that the gauge group G is the structure group of the vector or spinor bundle in question, whereas the group of gauge transformations is the group of G-bundle automorphisms, i.e. those bundle automorphisms whose induced isomorphism on the fibres lies within the gauge group G.

I would further welcome some examples of gauge groups and groups of gauge transformations for a few theories, e.g. Relativity or QED...

I have added a parragraph on gauge fixing, I wonder if it should be more detailed, jointly with some expansion on the classical invariances of a force field lagrangian (A.R.)

Maybe the abbreviation "rep" should be replaced by the full word "representation". 140.180.171.121 (talk) 23:21, 4 December 2007 (UTC)

[edit] Correction?

I think the expression for the Noether current(O(n) scalar theory) in the article is wrong. The current should not have the i index.I think the correct expression would be

\ J^{a}_{\mu} = \imath\sum_{ij}\partial_\mu \varphi_i T^{a}_{ij}\varphi_j

-Vatsa Jan 29, 2005

This article isn't consistent. It "defines" A with a factor of 1/g but has a 1/4g2 coefficient for the Yang-Mills term. Phys 06:53, 3 Feb 2005 (UTC)

[edit] Correction

It's not true that the sections of a principal bundle form a group! So, this is not a good definition of the group of gauge transformations. I was confused about this for a long time myself. To get a group, you need the sections of the bundle associated to the principal bundle P by means of the adjoint action of G on itself. I don't have the energy to write a clear explanation of this for people who don't know this sort of stuff. But, I wanted to point out that the description of gauge transformations as sections of a principal bundle is wrong.

(I'll be amazed if this correction actually shows up;I have no clue how Wiki works. Sorry!)

- John Baez, February 3rd 2005

[edit] Definition of gauge symmetry

I followed the Gauge symmetry link from Magnetism, and it redirected here. At the end of the introduction I read Sometimes, the term gauge symmetry is used in a more general sense to include any local symmetry, like for example, diffeomorphisms. This sense of the term will not be used in this article. It doesn't however say what sense of the term is used in the article either here or at the couple of places it is later used. Hv 14:21, 16 July 2005 (UTC)

Also, 'gauge symmetry' should not be a redirect to 'gauge theory', as the two terms have different meanings (regardless of what anyone thinks the terms mean). There should be two separate articles. MP (talk) 15:43, 3 July 2006 (UTC)

Well, gauge "symmetry" is not a symmetry. This is suggested in the article, but then the usual inacurate physical langage is used. There is a great confusion in the physics language, especially in the more elemetary physics books. They speak of global symmetries and local(i.e. gauge) "symmetries". In (quantum) physics, symmetry is a transformation between physical states that preserves the expectation values of all observables O (in particular the Hamiltonian). S: |φ> → |ψ> = S|φ>; |<ψ|O|ψ>|2=|<φ|O|φ>|2 .

The usual formulation of the physics theories uses fields, which sometimes are not physical quantities. Such are the gauge fields (fiber bundle connections for the mathematicians), which provide a redundant but convenient description of the physical degrees of freedom. The gauge (local) "symmetries" are a reflection of this redundancy. The physical quantities are certain equivalence classes of gauge fields.

An analogy can be made with the construction of the real numbers. We can use sequences of rational numbers that have the same limit. Of course, each real number is represented by infinitely many such sequences. We can choose a particular well defined sequence to be a representative of the real number. This corresponds to the procedure of gauge fixing in gauge theories.

The fact that gauge fields are not physical degrees of freedom becomes very clear when we try to quantize them. Then we are forced to work in one way or another with the physical quantities by removing the redundancy (the gauge symmetry).

Another important illustration of the problem with the gauge “symmetries” is when we have anomalies. By definition these are symmetries, which exist in the classical system, but not in its quantum counterpart. Anomalies are something quite usual and also an experimental fact - e.g. the axial anomaly in the strong interactions. However, because gauge symmetries are not symmetries, gauge anomalies are not something that just complicates the quantum theory but something that kills it. I.e. there are no gauge "anomalies", because such theories don't exist. This is why having the exact relation between the number of flavors and quark colors in the Standard model is so important - otherwise there is a gauge anomaly and the theory does not exist. For the same reason string people live in 10 dimensions (26 for the purely bosonic string theorists). Only then the anomalies cancel.

[edit] Correction

The gauge field provided in the O(n) example is trivial (i.e., "pure gauge"). It is *not* the definition of a generic gauge field in the model. One needs some kind of Ansatz or field equation to specify the gauge field in terms of sources (I rather doubt the latter will be invoked), or else avoid saying "the gauge field is defined as" rather than "the gauge field transforms like [the appropriate expr]"...

- SH, September 13/14th 2005

[edit] Frame-dependent claims

A recent edit inserted

we choose a local frame (a local basis of sections) then we can represent this covariant derivative by the connection form A, a Lie algebra-valued 1-form which is called the gauge potential in physics and which is evidently not an intrinsic but a frame-dependent quantity. From this connection form we can construct the curvature form F, a Lie algebra-valued 2-form which is an intrinsic quantity,

Uhh, that's wrong I beleive, or misleading, or something, since I don't know what an "intrinsic quantity" is. For a non-abelian gauge theory, under the action of a gauge transformation φ, F transforms as

F \to \phi F \phi^{-1}

and so naively F is frame-dependent as well. (The trace of F is not; the trace of F is intrinsic, its a Chern class). Can this wording be fixed? linas 00:17, 19 October 2005 (UTC)

linas, think of an instrinsic quantity as one that can be defined without reference to coordinates or bases. For example, a Riemannian metric is an intrinsic quantity, while, say, the trace of a Riemannian metric is not. What about connections and curvatures? Well, yes, they can definitely be defined without reference to coordinates on a G-bundle, so I would say the grandparent post is misleading for that reason: both are intrinsic. As I mention below, it is the local connection form (and connection coefficents) which rely on a trivialisation. Of course, if you're thinking of only the base space as "real" (as you might in physics, where the base space is spacetime), then I agree with the original poster: connection form is not intrinsic, curvature form is. -Lethe | Talk 01:17, 21 October 2005 (UTC)
I hope you can make sense of all the indices. 'a' is an abstract index, the rest are numerical indices. Let V be a vectorbundle and let E^n(V) be the vector-valued n-forms. Covariant derivative D : E^n(V) -> E^(n+1)(V) : s = s^i e_i |-> s^i (e)A_i_a. Now choose a different frame f_i = c_i^j e_j. s = z^i f_i = z^i c_i^j e_j = z^i c_i^j (e)A_j_a = z^i (f)A_i_a. A true vector valued form does not transform under coordinate transformations, only its components do. An invariant is (f)A_i_a f^i = (e)A_j_a c_i^j f^i = (e)A_j_a e^j which is a form, but this probably doesn't capture all information. I hope this explains why I called A "not intrinsic", which I used as a synonym for "not frame-independent". The geometric object which represent the covariant derivative is frame-dependent, not merely the components of that geometric object. This is not true for the curvature. --MarSch 10:38, 19 October 2005 (UTC)
I can't parse the formulas you gave. You seem to be saying that A is not a tensor on the base manifold, and that is correct. What I was trying to say is that F depends on the frame as well. F changes as you slide up and down the fibe. It also changes as you move from one coord patch to another (with φ the transition function as I gave above). Except in the abelian case, F is not frame-independent. linas 00:49, 20 October 2005 (UTC)
I dispute that F changes. Only its coordinates change. This is different for A which changes itself. You cannot specify the connection by giving the array of LAv forms A_i in g. You have to specify a frame for the vector bundle to fix it. Of course if you give A by specifying its coordinates in a given basis plus frame, then you will usually assume that this is the extra frame needed. But changing frames changes the array of geometrical objects A_i, which means that according to preference : either 1) A_i is not an array of LAv forms or 2) A_i is an array of LAv forms, but depends on a choice of frame. --MarSch 10:37, 20 October 2005 (UTC)
I don't know what an LAv form is. Do not confuse coord changes on the base manifold with gauge transformations. Let us for the sake of discussion keep the coords on the base manifold fixed and unchanging. Pick a section of the bundle. This fixes both A and F. To make a "local gauge transformation" is the same thing as saying that you will pick a different section of the bundle. Although picking a different section of a bundle is "kind of like" a "coordinate change" on the fiber, this is dangerous way of thinking about it, as it leads to confusion. A gauge transformation is more correctly visualized as a movement on the fibre, rather than a relabelling of the fiber. (for example: when I rotate a circle, do I rotate it, or the coordinates on it? I prefer to visualize the circle as fixed, and me moving on it. But that's just me.) The movement is given by the action of the gauge group.
If σ(x) was one section of the bundle, and τ(x) was the other section, then \phi(x) \in G is a gauge transformation, valued in the gauge group G, such that \tau(x) = \phi(x) \cdot \sigma (x). Then on has F(τ(x)) = φ − 1(x)F(σ(x))φ(x) and A(τ(x)) = φ − 1(x)dφ(x) + φ − 1(x)A(σ(x))φ(x). (I hope I got these formulas right, they're for "illustrative purposes only"). So as one moves on the fiber, both F and A change. There is no unique way to project F from the fiber to the base space, except in the Abelian case. linas 00:34, 21 October 2005 (UTC)
BTW I'm mystified why A is called a connection form, since it is an array of vector-valued 1-forms or equivalently a matrix of 1-forms in a non-invariant way as connection form also says.--MarSch 11:06, 19 October 2005 (UTC)
Historical usage. A is a form with respect to the base manifold; and not the total space. If you try to think of the fiber bundle as a manifold, then A is most definitely NOT a form on that. (and d+A is most definitly not a connection on that). linas 00:49, 20 October 2005 (UTC)
Linas, I'm not sure that that is correct. The connection form most definitely is a differential form on the total manifold. It is only when you consider local trivialisations that the connection form can be considered a differential form on the base manifold. And this depends on the local trivialisation (i.e. it is not guage invariant). This form is sometimes called the local connection form. -Lethe | Talk 00:51, 21 October 2005 (UTC)
g := T_e G. Perhaps Hom(g, g) is the Lie algebra A is valued in. I always assumed it would simply be g, but Hom(g, g) would make more sense. But is it a Lie algebra? --MarSch 11:45, 19 October 2005 (UTC)
Let me rephrase. A point in the Lie algebra tells you how to move a small distance in the Lie group; it does so by means of group-multiplication. So, when you move from one fiber in a fiber bundle to a nearby fiber, the lie-algebra tells you how to "rotate" the group slightly to get to the next fiber. A point in the lie algebra is an infinitesssimal automorphism of the lie group. Does it make sense now? linas 00:49, 20 October 2005 (UTC)

[edit] Jargon-based assured ignorance

If Wikipedia is to be accessible to reasonably educated people, it must be policed to assure that what passes for knowledge is not buried in the jargon of the priestly class. This article is one in a series of interconnected articles (I began with the Big Bang Timeline) which are completely useless to 99% of the world's educated people. If this practice were extended, for example, to articles on literature, only professors of literature would find them useful.

Jargon is used to exclude people from presumed knowledge, usually with the excuse that it is more efficient in conveying knowledge among experts. An encyclopedia seeks to convey knowledge to non-experts, so jargon must be left far behind (if it is to survive, it should be segregated from the encyclopedia in an "expert" site).

This article (and many of those connected to it) contributes, not to general education (the meaning and purpose of "encyclopedia") but to the preservation of a barrier between specialist and most of the world.

Jargon can always be learnt, if it is defined clearly. Jorbesch 20:29, 2 June 2006 (UTC)
Original poster here is a member of one of the world's most powerful elite classes: the class of people who think that they're incredibly special because they're being excluded and "kept down" by the elite classes, and are willing to display their superiority at every turn by reacting violently to the notion that they might be expected to know anything.


  • * *

Uh? What?

Well, I happen to agree wholeheartedly with the original poster. The article is quite impenetrable - and I'm an astrophysicist! Encyclopaedia entries ought to leave readers with something more than a notion that what's being discussed is far too clever for them. Breathtaking "priestly jargon" is exactly what's on display here, and little attempt has been made to include the interested reader

Jargon, acronyms, and references made without associated context are indeed off-putting to a reader of highly-specialized and abstract information if that person is not immersed in the field of study. I agree with the sentiment of the original poster, but only if the author's intended audience was a group of people not already intimately familiar with gauge theory, supersymmetry, etc. In reading this, I assume that the intended audience was not the generalist but the specialist. For this audience, "jargon" serves as the professional shorthand necessary to communicate efficiently and concisely. Certainly there are individuals who use the jargon of their profession or special interest as an active exclusion of other individuals. But we shouldn't assume this is always the case. Leading-edge thought in Physics is becoming more and more popularized, and a large number of educated people without the requisit knowledge to understand the complete context of advanced topics such as those presented here are being exposed to them nevertheless. All the better, but if one wants to understand all of the context, ramifications, and issues surrounding, say, String Theory, one has a long path of knowledge and discovery to embark upon. This article is clearly meant for those who have "the context" and who rely on the jargon to discuss and advance the topic. It would be wonderful to have an article that provides a high-level treatment of gauge theory and spontaneous symmetry breaking for the educated lay-person - but I do not believe that was the intent of this article.WFN94 16:02, 19 May 2007 (UTC)

[edit] Trivial Links

In the openin paragraph, there's a link to the Yang-Mills action. Yang-Mills action redirects back to Gauge theory. Is it expected that someone will eventually write a Yang-Mills action page, or should this be un-linked?

[edit] hmm

what's "locally" and "globally" refer to in the first sentence? Answering that might begin the road to a more clear article. If I understand this right, which I probably don't, something symetrical should be the same after a symmetry transformation always regardless of what parts are transformed.

[edit] There is too much information on this page

I only want to search for the gauge invariance in classical electrodynamics, but it redirects me to this page of gauge theory. I think it is necessary to separate the information into several entries, each containing more details. One certainly won't like a page titled "Physics" to cover materials from Newtonian Mechanics to Superstrings!

[edit] too general?

"Most powerful theories are described by Lagrangians which are invariant under certain symmetry transformation groups"

that is the first sentence, and i am thinking "WHAT POWERFUL THEORIES?" "no they are not" "i don't even know what lagrangians are..."

i don't think i am one who could fix this, but i think that first sentence needs to be cleaned up a bit. don't you think? —The preceding unsigned comment was added by 69.85.158.29 (talk) 09:05, 7 May 2007 (UTC). - BriEnBest 09:26, 7 May 2007 (UTC) sorry didn't sign before.


[edit] Gravity as a Gauge Theory ??

I think it should be pointed that Gravity is a Gauge theory and how could be applied to GR

[edit] This article needs a complete rewrite / break into pieces

If I have time in the future I will work on this, but several issues need to be addressed:

  • Gauge invariance (gauge symmetry) is a more fundamental concept which is related, but logically distinct from that of a gauge theory.
  • Yang Mills theory is a development significant enough to deserve its own article.
  • The entire article is very disorganized, and the introduction needs to be expanded so that people can find links to articles which might give the required background.

Akriasas (talk) 18:47, 14 December 2007 (UTC)

I'd like to add that I agree this article needs a lot of revision. The ideas being discussed here are of great importance in mathematics and physics, so it would be nice to see a discussion that is more accessible. Obviously, a lot of work has been put into this, but if Akriasas is willing to edit it to provide greater clarity, that would be great. —Preceding unsigned comment added by 194.94.224.254 (talk) 13:21, 6 February 2008 (UTC)

[edit] Intuitive example of local gauge symmetry

To go with the electrical ground example. Consider the apparent sizes of distant objects seen when stood on a plane. These are the differences between the bearings of the sides/ends of the object, but bearings are defined from an arbitrary zero (North) that can be varied from point to point (e.g. Magnetic or True North) with no effect on angular sizes. That's a bit wordy, can anyone shorten it?

172.201.128.223 (talk) 23:34, 9 February 2008 (UTC)SB.

[edit] Big conceptual mistake at the beginning of the article

The following sentence:

"In a gauge theory the requirement of global transformations is relaxed such that the Lagrangian is required to have merely local symmetry. "

is wrong. The requirement of LOCAL symmetry is much more strict than the requirement of GLOBAL symmetry. In fact, a global simmetry is just a local symmetry whose group's parameters are fixed in space-time. My written English isn't very good, so I don't feel confident in editing the article. Is anybody willing to do that?

Ciao,

Guido —Preceding unsigned comment added by Coccoinomane (talk • contribs) 19:30, 20 February 2008 (UTC)

Thank you, I will do it. Masterpiece2000 (talk) 07:45, 23 February 2008 (UTC)