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Questions tagged [yang-mills]

Yang–Mills theory is a QFT, a *gauge theory* normally symmetric under a compact non-Abelian Lie group relying on (originally massless) gauge vector fields. YM theories describe the strong and electroweak interactions of elementary particle physics, the Standard Model.

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Wilson loop and Polyakov loop

As I understand, the Wilson line is the operator $W(x) = P\exp(i\int_{xi}^{xf} A.dx)$, where $P$ is path ordering. The Polyakov loop $P(x)$ on the other hand is the trace of the Wilson loop $W(x)$ ...
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Prerequisites on Yang Mills Theory [duplicate]

I was wondering of what kind of subject are necessary to study the Yang - Mills theory from basic university level. Can you suggest some good book for subject?
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Invariance of Yang-Mills Lagrangian under charge conjugation

The Yang-Mills Lagrangian gauge invariant under an $SU(N)$ tranformation can be written as $${\cal L} = -\frac{1}{4}F_{\mu\nu}^i F^{i\ \mu\nu} \tag1$$ (Sum over $i$ implicit) This Lagrangian ...
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Why do like charges repel in a spin-1 gauge theory, while they attract in spin-2 theories?

A presenter recently said this at a colloquium, but they didnt explain why. I know a fair bit of GR and Yang-Mills theory, so dont shy away from the details (unless they are cumbersome and don't aid ...
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Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$

Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that $$D_\mu = \partial_\mu - igA_\mu, \tag1$$ you can built the kinetic term for the gauge potential $A_\...
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Notation and concepts of Yang Mills Theory

I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about ...
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Checking modularity-like transformation property

Assume $M$ is a 4 manifold. Let $Z_v$ be partition function of fixed magnetic flux $v$ with all instanton configuration summed over where $v\in H^2(M,Z/nZ)$. $\tau$ denotes complex parameter on upper ...
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Differentiating D3 brane worldvolume theories with NS5 brane and NS5 antibrane boundary conditions

In 'Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory', Gaiotto and Witten derive boundary conditions for the worldvolume theory of the D3 brane. In particular the boundary conditions (...
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Is color charge quantized?

I was reading this stackexchange question, and found the answer to my question not totally answered. Clearly there is color and anti-color in analogy to electric charge, and color charge clearly ...
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Why is Lattice QCD done in Euclidian 4-space?

This could be a really naive question (and honestly I just don't want to dig through ArXiV review papers on lattice QCD), but my question is simple: Why exactly is Lattice QCD done in $\mathbb{R}^4$ ...
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Gauge fixing while preserving supersymmetry

In supersymmetric gauge theories, the vector potential is a part of a vector supermultiplet which is represented by a real superfield $V$. Expanded out in components, the Lagrangian for such a field ...
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Why is color confinement a difficult problem?

Assuming color force follows a constant rule of force instead of an inverse square rule of force. And that red, green and blue are all attracted to each other. Why is color confinement considered a ...
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Another way to write the Einstein-Hilbert action?

Let's take a look at the equation for the Riemann tensor in terms of an arbitrary 1-form: $$\nabla_{\mu}\nabla_{\nu}A_{\alpha}-\nabla_{\nu}\nabla_{\mu}A_{\alpha}=R_{\mu\nu\alpha}^{\quad\:\delta}A_{\...
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Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
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What is the relation between the Gribov problem and color confinement?

I have heard that the Gribov problem is in some way related to color confinement (For instance: Gribov copies and confinement). Although I understand what both the Gribov and confinement problems are, ...
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Understanding Measure in Path integrals

I know this is still a current topic of study, so my question is less about mathematical underpinnings, and more about the transition from the "sum-over-all-possible-intermediate-points" measure $$\...
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Where do the color indices come back in $SU(3)$ Yang-Mills Quantization?

Can the partition function of $SU(3)$ (the Generic Partition function for a yang-mills theory found on the linked wiki page below), be split into a sum of 8 functional integrals for each gauge field? ...
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Feynman rules out of the Lagrangian

Accordingly to chapter 10, section 10.6 Feynman Rules of 'Introduction to Elementary Particles' by David Griffiths, there is a way to extract the vertex and propagators just by inspection of the ...
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Is it enough to assume $F_{\mu\nu}\to 0$ at infinity but not $A_\mu$ to derive the equation of motion?

Suppose the the Lagrangian $\mathscr{L}$ of the free electromagnetic field is augmented with the term $$F_{\mu\nu}\tilde{F}^{\mu\nu}=\partial_{\mu}(\epsilon^{\nu\nu\lambda\rho}A_\nu F_{\lambda\rho}).$$...
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Isn't there a unique vacuum of the Yang-Mills quantum theory?

The theta vacua$^1$ of a Yang-Mills quantum theory are given by $$|\theta\rangle=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}|n\rangle.$$ In Srednicki's Quantum Field Theory, he claims that the ...
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Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
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Commutator of Gauge Transformations for Yang-Mills Theory

Following the conventions of "Quantum Field Theory and the Standard Model" by Schwartz, we have that for Yang-Mills Theory, an infinitesimal gauge transformation acts like $$\delta_{\alpha} A = d\...
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Conserved currents in Yang-Mills theory: gluon current vs. quark current

In Yang-Mills theory there are two currents we can construct. There is the well-known quark current related to the global $SU(3)_C$ symmetry, $$j{}^{\mu\,A}_\text{quark} = \overline{\psi}{}^i \gamma{}^...
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What is the difficulty in extending geometrodynamics to non-abelian fields?

In an attempt to widen my own horizons I've decided to educate myself in Wheeler's Geometrodynamics. In the so-called "already unified theory" one can essentially reproduce an electromagnetic field ...
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Covariant Derivative in QCD: How does it act on gluons?

Let the covariant derivative be $$D_\mu = \partial_\mu + \text ig\ A_\mu^a t^a,\quad a=1,\ldots,8$$where $g$ is the bare QCD coupling, $A_\mu^a$ are the eight gluon fields and $t^a=\tfrac{1}{2}\lambda^...
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What is the Noether charge associated with the the color $SU(3)$ symmetry of QCD?

A version of the Noether's theorem applies to local gauge symmetries. What is the Noether's charge associated with a non-abelian gauge symmetry such as the color $SU(3)$ and how is that derived? I ...
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Infrared divergencies in Yang-Mills theory

I'm trying to better understand the nature of infrared divergencies in YM theory; for now, I'm only interested in soft divergences. The usual explanation one is given about the origin of IR ...
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Can Yang-Mills field strength be defined as covariant derivative squared?

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as $$ \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\...
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Local charge current for gauge field and conservation of charge

Motivation: It is a well known fact that the gravitational field (in General Relativity and direct generalizations of it) has no local energy-momentum density. Usually there are two reasons stated, ...
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Are the nonphysical degrees of freedom in Yang-Mills theory analogous to the worldsheet metric in the Polyakov formalism?

The Polyakov string action on a flat background (in the Euclidean signature) $$S_{P}[X,\gamma]\propto\int_{\Sigma}\mathrm{d}^2\sigma\,\sqrt{\text{det}\gamma}\,\gamma^{ab}\delta_{\mu\nu}\partial_{a}X^{...
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What are chromoelectric and chromomagnetic fields?

Are they the normal elctric and magnetic fields from Maxwell fields? Or are they just the corresponding components from $G_{\mu\nu}^a$ (the gluon fields), say chromoelectric fields are simply $G_{0i}^...
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Is Yang-Mills theory as a topological quantum field theory a good alternative?

Suppose we know a classical solution for gauge Boson fields $A_{\mu,c}$ and Fermion fields $\psi_c$ and now we want to consider ist Quantum fluctuations. These fluctuations arise from loop corrections,...
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What does that means? “QCD is a non-linear and non-trivial field theory?”

I know QCD is represented by the $SU(3)$ group and is non-abelian. Then, as a consequence QCD is a non-linear and non-trivial field theory. I would like to know why? and what does that means?
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Which of the Wightman axioms are not incorporated by four dimensional quantum Yang-Mills?

I am trying to understand the quantum Yang-Mills existence problem but the best I have seen so far is the statement that there is no known interacting relativist field theory in four dimensions which ...
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$SU(5)$ representation and higher anti-symmetric traces

In Zee QFT book v2 p.411 eq.16-17, he shows the SU(5) gauge theory anomaly cancellation by the following: The 1st line in fundamental of SU(5) $$ tr(T^3)=3(+2)^3+2(-3)^3=30, $$ is easy to follow, ...
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Why not regard all large gauge transformations as genuine ones?

A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to ...
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Updating link variables in lattice $SU(N)$ gauge theory

I'm currently writing a basic program in python to simulate a 1 + 1 dimensional yang mills gauge theory with symmetry group $SU(2)$. On the lattice you work with link variables, which are $SU(N)$ ...
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What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
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A doubt in Yang-Mills procedure

My question is this: I saw the next relation in a Yang Mills theory paper: $$L_{i}P^{\mu \nu}_{i}=P^{\mu \nu}$$ With $L_{i}$ a generator of su(2) and for any $P^{\mu \nu}_{i}$. But I can't ...
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Proof of the expression of field strength in a non-abelian theory

I want to prove that the expression for the field strength in a non-abelian theory is \begin{equation} G_{μν}Ψ=[D_{μ},D_{ν}]Ψ=(∂_{μ}A_{ν}-∂_{ν}A_{μ}+[A_{μ},A_{ν}])Ψ \end{equation} where the covariant ...
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Pathway to Yang-Mills theory [duplicate]

I am a probabilist, having majored in Statistics. I am trying to learn Yang Mills theory. I have seen there is a recent excitement among probabilists in understanding Yang Mills theory. I know nothing ...
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Is there a name for symmetry in which fermions and bosons are in identical adjoint representations?

In a Yang-Mills theory, with gauge group $G$, if the Fermions are in an adjoint representation then for every Fermion with "charge" $Q$ there is a boson with charge $Q$. i.e. there is no difference ...
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Super-gauge transformation in two dimensional $\mathcal{N}= (0,2)$ superspace

I'm trying to couple matter to $\mathcal{N}=(0,2)$ SYM in 2d using superfield formalism. There are some paper (this on Sec. 6, or this on Sec. 3 [whose notation will be used here]) that construct what ...
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Formalization of the concept of a topological charge

I want to write precisely in mathematical terms what a topological charge is. This is what I have, but I am not sure of how correct it is. Let $M$ be spacetime. Quantization of $M$ in some QFT will ...
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Yang-Mills action - a potential mistake in Wikipedia

Currently at the Wikipedia page on Yang-Mills theory, you see that [a screenshot], Isn't that an obvious mistake? Based on this normal convention: $$ F^2 \equiv F \wedge \star F $$ Shouldn't ...
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Motivation for Non-Abelian Gauge Invariance

I have a very similar question to the one asked below: Why are non-Abelian gauge theories Lorentz invariant quantum mechanically? In particular, the setup to my question is essentially the same: ...
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Comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of: Chern class (1st, 2nd), and ...
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Transformation of fields in non-abelian gauge theories

Let us consider a gauge group, e.g. $SU(N)$. One usually says that a fermionic field $\psi$ belongs to the fundamental representation of the group. As far as I understand, the fundamental ...
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Going from full non-Abelian gauge transformation to its infinitesimal version in component notation

Let $A_\mu^a(x)$ be a non-Abelian gauge field, with $\mathrm{SU}(N)$ generators $T_a$. We can write the field as a Lie-algebra-valued object $$ \mathbf{A}_\mu \equiv A_\mu^a T_a.$$ The full local ...
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2D ${\cal N}=(2,2)$ Super Yang-Mills with Superspace

I'm reading this famous paper by Witten. There is the expression of field strength for the abelian vector multiplet (eq. (2.16)): $$\Sigma = \frac{1}{\sqrt{2}}\bar{D}_+D_- V\;.\tag{2.16}$$ I'm ...