What is Yang-mills: Definition and 51 Discussions

In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(N), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.

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  1. M

    A Uncovering the Combinatorial Origins of Yang-Mills Theory?

    For many years now, the theorist Nima Arkani-Hamed has lent his prestige and energy to a research program that aims to transform our understanding of quantum field theory, by using symmetries in the sums of Feynman diagrams to uncover perspectives on the theory not based in ordinary space-time...
  2. ergospherical

    I Yang-Mills Stress-Energy Tensor Explained

    It's given as ##T_{\mu \nu} = - \mathrm{tr}(F_{\mu \lambda} {F_{\nu}}^{\lambda} - \frac{1}{4} g_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta})##. Can somebody explain the notation, i.e. what is the meaning here of the trace? (usually I would interpret the trace of a matrix as the number...
  3. BohmianRealist

    A Bootstrapping quantum Yang-Mills with concrete axioms

    From: https://en.wikipedia.org/wiki/Axiomatic_quantum_field_theory But, that seems like a fairly abstract place to begin the kind of QFT construction that was asked of us by Witten in 2012: At the bottom of that page on axiomatic QFT are the "Euclidean CFT axioms": Are there any examples of...
  4. John Greger

    A What are some example Feynman diagrams in Yang-Mills theory?

    Hi! So I have just been studying Yang-Mills theory advanced quantum field theory. In chapter 72 of Srednicki's book Quantum Field Theory they list the Feynman rules for non-abelian gauge theory. I was asked if I could show some sample allowed diagrams but I could not.. In standard particle...
  5. Antarres

    A Is My Localized Yang-Mills Theory Solution Correct?

    While revising Yang-Mills theory, I have stumbled upon a certain problem, which I solved in a somewhat trivial way so I would like to check whether my reasoning is correct. Let's say we have a multicomponent matter field ##\{\phi^m(x)\}## which transforms according to some Lie group ##G## of...
  6. takunitoche

    A Derivation of the Yang-Mills 3 gauge boson vertex

    Hello everyone, I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is$$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)} $$ I have rewritten this term using...
  7. takunitoche

    Three gauge boson Yang-Mills vertex

    Hello everyone, I am stuck in deriving the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is $$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)} $$ I have rewritten this term using the...
  8. O

    A Verifying the Relation in Yang-Mills Theory with a Scalar Field

    I'm trying yo verify the relation \begin{equation} [D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi, \end{equation} where the scalar field is valued in the lie algebra of a Yang-Mills theory. Here, \begin{equation} D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi], \end{equation} and \begin{equation}...
  9. A

    I Yang-Mills Lagrangian: Is ##F^{\mu \nu}F_{\mu \nu}## a Number?

    I am sorry for asking this stupid question, but in the Yang-Mills lagrangian, there is a term ##Tr(F^{\mu \nu}F_{\mu \nu})##. Isn't ##F^{\mu \nu}F_{\mu \nu}## a number?
  10. A

    I Real function instead of spinor field in Yang-Mills field

    An old thread (https://www.physicsforums.com/threads/state-observable-duality-john-baez-series.451101/) triggered a lively debate on whether complex functions are necessary for quantum theory or real functions (but not pairs of real functions) can be sufficient for it. I argued that one real...
  11. L

    A Yang-Mills 3 boson Lagrangian term in Peskin and Schroeder

    Hi all, I'm not certain if this is the correct section of the forum for this thread but I'm trying to understand ghosts and BRST symmetry and my starting point is chapter 16 of Peskin and Schroeder where I've found a nagging issue. My issue is regarding the derivation of equation (16.6) on...
  12. CantorsLuck

    A Momentum Energy tensor and Wilson Loop in Yang-Mills Theory

    Hello Everyone. I Was Wondering how excatly the Gauge invariance of the trace of the Energy-momentum tensor in Yang-Mills theory connects with the trace of an Holonomy. To be precise in what I'm asking: The Yang-Mills Tensor is defined as: $$F_{\mu \nu} (x) = \partial_{\mu} B_{\nu}(x)-...
  13. MathematicalPhysicist

    What is ##G(2,1)## in the Sliders episode mentioning Yang-Mills?

    Does someone know which episode from Sliders did they mention Yang-Mills? I want to watch this episode for nostalgia sake. If you want to move this post to other subforum then it's fine by me. Cheers!
  14. S

    A How Does Gauge Invariance Shape Classical Yang-Mills Theory with Scalar Fields?

    Let us consider a classical field theory with gauge fields ##A_{\mu}^{a}## and a scalar ##\phi^{a}## such that the Lagrangian is gauge-invariant under the transformation of 1. the gauge fields ##A_{\mu}^{a}## in the adjoint representation, with dimension ##D_{\bf R}##, of the gauge group...
  15. S

    A Coupling constant in Yang-Mills Lagrangian

    The Yang-MIlls Lagrangian is given by ##\mathcal{L}_{\text{gauge}} = F_{\mu\nu}^{a}F^{\mu\nu a} + j_{\mu}^{a}A^{\mu a}.## We can rescale ##A_{\mu}^{a} \to \frac{1}{g}A_{\mu}^{a}## and then we have ##\frac{1}{g^{2}}F_{\mu\nu}^{a}F^{\mu\nu a}.## How does the second term change? Does the...
  16. S

    A N=2 Super Yang-Mills: Auxiliary Fields Question

    I was wondering since the Vector supermultiplet in N=2 SUSY can be built from a Chiral and a Vector supermultiplet from N=1, in order to make up the off-shell degrees of freedom, would you include the two auxiliary fields from the N=1 theory (traditionally F from the Chiral and D from the vector...
  17. Andrea M.

    Yang-Mills theory, confinement and chiral symmetry breaking

    I was thinking about hadrons in general Yang-Mills theory and I have some doubts that I'd like to discuss with you. Suppose that we have a Yang-Mills theory that, like QCD, tend to bind quarks into color singlet states. So far nothing strange, even QED tend to bind electromagnetic charges to...
  18. Breo

    Yang-Mills, QED, QCD and Yukawa theories

    What is the difference between Yang-Mills and QED theories? Yukawa and QCD? specially in terms of the lagrangians. I really want to get into this subject with a previously first sight.
  19. C

    Showing that Yang-Mills equations transform homogeneousy

    Hi! I'm trying to show that the differential from equation $$D \star F = 0$$ transform homogeneously under the adjoint action ##F \mapsto gFg^{-1}## of the lie group ##G##, where ##D## denotes the covariant exterior derivative ##D\alpha = d \alpha + A \wedge \alpha## for some lie algebra valued...
  20. N

    Is Yang-Mills scale invariant?

    I remember hearing this, but not sure if it's true.
  21. B

    Differential Geometry for General Relativity and Yang-Mills Theories

    I have been teaching myself QFT and General Relativity. The mathematics of those fields is daunting, and I find that what I have come across is very difficult to master. Of course it will take work, but can someone recommend a good text for self-leaning differential geometry with application...
  22. Q

    Yang-Mills Theory Explained: SU(N) Symmetry & Gauge Theory

    What exactly is a Yang-Mills Theory? Is it a general theory, based on SU(N) symmetry, which can then be applied for particular cases (ElectroWeak, Chromodynamics) ? Is it like a general mathematical model of gauge theory ?
  23. michael879

    Gauge Fixing in Yang-Mills gauge theory

    I've come across countless sources that gauge fix SU(N) Yang-Mills fields using the typical U(1) gauges (e.g. Lorenz gauge, coulomb gauge, temporal gauge, etc). However, I can't find a single one where they prove that this gauge fixing is valid for all field configurations... I've tried to...
  24. O

    (Simple) Derivation of Yang-Mills Equations

    Hello all, my teacher assigned a problem related to the yang-mills equation in my general relativity class and I just wanted to ask a couple of questions about this problem. I believe it is a simplified version of the Yang-Mills you encounter in particle physics. Basicly assuming that...
  25. C

    Feynman rules from Yang-Mills lagrangian

    In reading Ryder's book on quantum field theory he advocates reading off the Feynman rules directly from the Lagrangian in the path integral quantization method. I can sort of do this in phi-four theory, but it is not obvious in for example Yang-Mills theory, so I wondered if someone could...
  26. C

    Gauge transformation of Yang-Mills field strength

    Hi. I'm reading about non-abelian theories and have thus far an understanding that a gauge invariant Lagrangian is something to strive for. I previously thought that the Yang-Mills gauge boson free field term ##-1/4 F^2 ## was gauge invariant, but now after realizing that the field strength...
  27. D

    Terms in the Yang-Mills Lagrangian

    I've been doing some self-study in Peskin and Schroeder and been struggling a bit in Part III. Right now, I am stuck on the last two terms in 16.6 (Lagrangian for Yang-Mills). Presumably these come from (-1/4) (F^{a}_{\mu\nu})^2, but I am getting stuck on getting the indices to work out...
  28. E

    Consequences of Yang-Mills and Mass Gap?

    What are the consequences of a possible solution to Yang-Mills existence and mass gap problem? Are the consequences only physical or they will also have applications to other branches of science or life? If yes, what are they?
  29. C

    Deriving the Yang-Mills-Higgs Equations of Motion

    Homework Statement 1. Derive the Yang-Mills-Higgs equations of motion (D_{v}F^{\mu, v})^{a} = - \epsilon^{abc}\phi^{b}(D^{\mu}\phi)^{c} and (D_{\mu}D^{\mu}\phi)^{a} = - c(|\phi|^{2} - v^{2}) \phi^{a} from \mathcal{L} = - \frac{1}{4} F^{a}_{\mu v}F^{\mu v a} +...
  30. P

    Is the glueball the only stable physical particle in pure Yang-Mills theory?

    By "physical particle" I mean color-singlet particles which have asymptotic T=\pm \infty states. How many stable particles exist in the theory? Only one? SU(2), SU(3), and SU(N) gauge groups can all be discussed.
  31. S

    Gravity/Electroweak unification based on generalized Yang-Mills

    http://arxiv.org/abs/1106.2121 Abstract: Gravitational and electroweak interactions can be unified in analogy with the unification in the Weinberg-Salam theory. The Yang-Mills framework is generalized to include space-time translational group T(4), whose generators $T_{\mu}(=\p/\p x^{\mu})$...
  32. R

    Reviewing Yang-Mills Gauge Field: Symmetries & Path Integral Methods

    Just to review a little bit: In general, for a gauge field with Yang-Mills Lagrangian \mathcal L=-\frac{1}{4}F^{c}_{\mu \nu}F^{c \mu \nu} for each c it is impossible to find the resulting free Green's function G(k) in momentum space: (g^{\mu \nu}k^2-k^{\mu}k^{\nu})G_{\nu...
  33. C

    Trace of this yang-mills operator

    Hi there, I'm trying to compute the trace of an operator found here: http://inspirebeta.net/record/360247 (eq 7.5) I'm not going to make you read the article, so i state the problem: I have the following operator in a Yang-.Mills theory, using the background field method...
  34. H

    Understanding mass gap in Yang-Mills theory

    Here and there some rumors come out about this relevant problem. I have read the following article in Wikipedia http://en.wikipedia.org/wiki/Yang–Mills_existence_and_mass_gap and the talk page http://en.wikipedia.org/wiki/Talk:Yang–Mills_existence_and_mass_gap but people cited there...
  35. G

    A question about the Exterior Product in Yang-Mills theory

    Right, so in Yang-Mills theory, the vector potential is modified from: F = dA To: F = dA + A\wedge A However, it is my understanding that the exterior/wedge product is anticommutitive, so that for a given exterior algebra over a vector space, V: \omega \wedge \omega = 0, \forall...
  36. M

    Exploring Gauge Bundle Breaking in Yang-Mills Theory

    Hello, suppose you start with Yang Mills theory with some gauge group G, for example SU(5). Then you turn on a gauge bundle, say a U(1) bundle, and the group breaks down. I know that from hearsay but I wonder how would you describe that explicitly in formulas? meha
  37. O

    What're the superspace formulations of SUSY Yang-Mills of D=10, N=1,2,4,8,16,32?

    When N=1, is the superspace formulation of 10D SYM the same as the 4D SYM, except the differences of the coefficient and gamma matrices? For the extended SUSY in 10D, are the superspace formulation available for N=2, 4? How about N=8, 16, and 32? Thanks for any tips you may tell me.
  38. C

    Classical Yang-Mills Force Law ?

    Even though classical (as opposed to quantized) non-abelian gauge theories do not have any physical applications at this time, it is mathematically valid to say that these classical Yang-Mills fields generalize Maxwell's equations of E&M in some sense i.e. the Yang-Mills equations reduce to the...
  39. Q

    Yang-Mills Geometry: An Explainer for Simpletons

    Since it appears (so far) I am infringing no rule, here is another shameless copy/paste of a thread I started on another forum, where I didn't get too much help - rather, folk tried, but confused me even further! See if you guys can do better. (Note:I am not a physicist) The mathematics here...
  40. Phrak

    Yang-Mills covariant derivative

    In developing the Yang-Mills Lagrangian, Wikipedia defines the covariant derivative as \ D_ \mu = \partial _\mu + A _\mu (x) . Is A_mu to be taken as a 1-form, so that \ D _\mu \Phi = \partial _\mu \Phi + A _\mu (x) or an operator on \Phi, such that \ D _\mu \Phi = \partial...
  41. L

    Pure Yang-Mills theory in D=1+1

    I was aware that this theory can be solved exactly and is trivial, i.e. there is no dynamics. I was also convinced that this result is due to 't Hooft. Could you confirm this view and put out some relevant refs about? Thanks beforehand. Jon
  42. K

    What is Yang-Mills theory? What is its significance?

    I just have one question, what is it (yang-mills theory)? Could someone explain this to me please
  43. K

    Yang-Mills Theory: Unsolved Mathematics and String Theory

    Hello, I am curious to know what yang-mills theory is? What does it say? Clay mathematics institute has, in the year 2000, issued a $1,000,000 dollar prize for who ever solves some sort of proof for it. Does anyone here work on this? I noticed, in the documentary "the elegant universe"...
  44. L

    C-Parity Oddity of Magnetic Current in SU(2) Yang-Mills Theory

    Why a magnetic current obviously must be C- parity odd in the (exact) SU(2) theory? (C^{-1} J_M C=-J_M). Best regard. Leonid
  45. N

    Yang-Mills Field Strength Tensor

    I was wondering why for [tex] F_{\mu \nu} = [D_{\mu},D_{\nu}] = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}] [\tex] the term [tex] A_{\mu}\partial_{\nu} - A_{\nu}\partial_{\mu} [\tex] vanishes.
  46. K

    Trees to Loops: Computing SUSY Yang-Mills Amplitudes

    From Trees to Loops and Back Andreas Brandhuber, Bill Spence, Gabriele Travaglini 49 pages, 17 figures http://www.arxiv.org/abs/hep-th/0510253 We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with...
  47. B

    Yang-Mills with light-cone coordinates

    Can someone give me some references on Yang-Mills theories formulated with light-cone coordinates? (preferably on-line) Thank you very much in advance.
  48. humanino

    Mass gap in Yang-Mills theories

    Could someone rephrase in a short and casual manner the famous Millenium Problem of the Clay Math institute ? http://www.claymath.org/millennium/Yang-Mills_Theory/ Thank you for help !
  49. K

    Yang-mills and standard model

    What in simple terms is a Yang-Mills field? What has it got to do with the standard model of particle physics and why are all particles in the standard model massless - is this something to do with getting the standard model to be consistent with the Higgs field? Do we need the standard...
  50. J

    Photon-Pair Productionand Yang-Mills Equations

    (1) Is it true that when a pair of particles is created from a Photon, that the Photon Energy (h x nu) is equal to 2(msub0)c^2 ? If say an electron and positron are created does the photon Energy equal (at least) twice the mass of one electron 2(9.109 x 10^-31 kg)times c^2? (2) Could anyone...