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3
votes
1answer
69 views

Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
2
votes
1answer
68 views

Are the Yang-Mills equation and its generalization gauge invariant?

I have derived the Yang-Mills equation and its generalization coupled to a current of a scalar field $\phi$ by extremalizing the action describing a $\mathrm{SU}(2)$ scalar field gauge theory: ...
8
votes
1answer
159 views

Why are non-Abelian gauge theories Lorentz invariant quantum mechanically?

I seem to be missing something regarding why Yang-Mills theories are Lorentz invariant quantum mechanically. Start by considering QED. If we just study the physics of a massless $U(1)$ gauge field ...
2
votes
1answer
65 views

A Variation on Laplace's equation (context: Yang-Mills N-Instantons, Rajaraman's book)

Statement of the problem I need to solve the equation \begin{align} 0 = \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi \hspace{20mm} (1) \end{align} where $\phi$ is a scalar field and ...
3
votes
2answers
198 views

Quantum Yang-Mills Theory and AdS/CFT

I just read the first chapter of Becker-Becker-Schwarz. To quote: A remarkable discovery made in the late 1990s is the exact equivalence (or duality) of conformally invariant quantum field ...
4
votes
1answer
106 views

Is there a method which quantizes non-abelian gauge theories without path integrals formalism?

In the most QFT books there is a method of quantization of non-abelian theories through path integral methods. But I want to learn also the other methods without using of this formalism. Does anyone ...
2
votes
1answer
101 views

Mass term in the Lagrangian

I have read that the mass term appearing in the electroweak Lagrangian stops it (the Lagrangian) from becoming gauge invariance. Can someone explain where and why this term is creating the problem?
1
vote
0answers
40 views

On self-duality of N=4 super Yang Mills theories

I am looking at S-duality a bit, and was wondering if anyone had the answer to the following question. It is known that the supersymmetric Yang-Mills theory in 4 space dimensions is self-dual with ...
1
vote
0answers
25 views

How to show isotropy of $SU(2)$ Yang Mills stress energy tensor?

When I vary the action of the YM Lagrangian density $$L = -\frac{1}{4} F^a_{\mu \nu}F^{\mu \nu}_a + J_a^\mu A^a_\mu$$ with respect to the metric, I obtain: $$T_{\mu \nu} = \frac{-2}{\sqrt{|g|}} ...
0
votes
1answer
49 views

Yang Mills theory and SU(N) groups [duplicate]

Trying to get a better understanding of the relation between a SU(N) Yang Mill theory and its number of "color" space. Most of the description I've found so far are either way to complex/specific. ...
2
votes
1answer
153 views

Lagrangian depends on second derivative of field

In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu ...
3
votes
1answer
84 views

Yang-Mills Lagrangian invariant under BRST

In equation 16.47 in Peskin & Schroeder, it is claimed that $$ -\frac{1}{2}g^2f^{abc}f^{cde}\left(A_{\mu}\,^{b}c^{d}c^{e}+A_{\mu}\,^{d}c^{e}c^{b}+A_{\mu}\,^{e}c^{b}c^{d}\right) ~=~ 0 \tag{16.47}$$ ...
4
votes
0answers
98 views

Gauge Invariance of Yang Mills Lagrangian

I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$ under the following gauge transformation ($\theta$ being a ...
3
votes
1answer
105 views

Deriving field equation in Yang Mills theory

Trying to show that $$D_\mu\vec{F^{\mu \nu}} = \partial_{\mu}\vec{F^{\mu \nu}} + g \vec{A_\mu} \times \vec{F^{\mu \nu}} = 4 \pi \vec{J^\nu},$$ or (correct me if I'm wrong) $$ \partial_{\mu} F^{\mu ...
1
vote
0answers
81 views

Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four ...
2
votes
1answer
148 views

Rigorous QFT on a Torus

The problem description for the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf) says in its "Mathematical Perspective" section that Some ...
10
votes
2answers
203 views

Why isn't Quantum Yang-Mills Rigorous?

Obviously one of the major components of the Yang-Mills existence and mass gap problem of the Clay institute is the proof that 3+1d quantum yang-mills theory has rigorous foundations. This (I believe) ...
5
votes
1answer
95 views

The 6-j symbol and intersecting Wilson loops, redux

This is a quite specific question continuing the problems I have with computing the expectation value of intersecting Wilson loops I laid out here. Using the tools from the answer there, I quite ...
7
votes
1answer
167 views

Intersecting Wilson loops in 2D Yang-Mills

I am currently trying to understand 2D Yang-Mills theory, and I cannot seem to find an explanation for calculation of the expectation value of intersecting Wilson loops. In his On Quantum Gauge ...
22
votes
1answer
2k views

What does it mean that there is no mathematical proof for confinement?

I see this all the time* that there still doesn't exist a mathematical proof for confinement. What does this really mean and how would a sketch of a proof look like? What I mean by that second ...
6
votes
1answer
57 views

How many coupling constants if my gauge group has many factors?

I am reading a review article where $U(1)\times{}SU(2)\times{}SU(3)$ gauge transformations are considered. It says that when such a gauge transformation is done the gauge fields $A^{\alpha}_{\mu}$ ...
0
votes
2answers
134 views

Reasons for choosing $SU(3)$ as the color group vs. $SO(4)$

What are the reasons that $SU(3)$ is used for QCD? Why wouldn't the simpler & smaller group $SO(4)$ make a better candidate?
6
votes
2answers
110 views

Yang-Mills existence and mass gap

In the Clay institute problem description of the Yang-Mills existence and mass gap problem it states that the quantum Yang Mills needs to be formulated in $\mathbb{R}^4$ space. I was wondering whether ...
2
votes
1answer
168 views

Equations of motion for the Yang-Mills $SU(2)$ theory

I have an exercise for Yang-Mills theory. I can't find answer anywhere. Derive equations of motion for the Yang-Mills theory with the gauge group $SU(2)$ interacting with $SU(2)$ doublet of scalar ...
10
votes
2answers
456 views

Yang-Mills CP violation

Why does a term proportional to $\left(F,\,\tilde{F}\right)\propto Tr\left[ F_{\mu\nu}\tilde{F}^{\mu\nu}\right]$ in the Lagrangian of the pure Yang-Mills theory violate CP?
4
votes
1answer
95 views

The properity of $\mathbb{R}^4$ that has infinitely many differential structures is related to Yang-Mills field?

I heard a saying that $\mathbb{R}^4$ having infinitely many differential structures which are not diffeomorphic to each other has a relationship with Yang-Mills field. Does anyone can explain it, and ...
2
votes
0answers
44 views

Transformation Law for Covariant Derivative in $SU(2)$ Yang-Mills

In page 488 of Peskin and Schroeder, it is stated (emphasis mine): It is not difficult to check using (15.27) and (15.21) that, even for finite transformations, the covariant derivative has the ...
5
votes
1answer
116 views

Why is the Yang-Mills Comparator unitary?

In chapter 15.2 of Peskin, the comparator is defined, as some object $U\left(y,\,x\right)$ which transforms as: $$ U\left(y,\,x\right) \mapsto V\left(y\right) U\left(y,\,x\right) ...
2
votes
1answer
117 views

$SU(2)$ gauge symmetry

Take the Lagrangian with one fermion: $$ \mathcal{L} = -\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi$$ where the gauge covariant derivative $D_\mu = ...
4
votes
1answer
222 views

The BRST construction for YM with or without auxiliary field

I'm learning BRST symmetry for Yang-Mills theory and I see that there are two ways of writing BRST differential. In some books (for example Ryder's and Ramond's textbooks) BRST differential acts as ...
6
votes
1answer
192 views

SU(N) Yang-Mills $gg \to ggg$ scattering at tree level

When talking about the spinor-helicity formalism in his new textbook on quantum field theory, Matthew D. Schwartz claims as a highly nontrivial example, it is quite easy to use the Parke-Taylor ...
1
vote
1answer
68 views

Gauge field with flat connection

Consider a gauge field $A_z^a$ with a flat connection $$F_{z{\bar z}}^a = \partial_z A_{\bar z} ^a - \partial_{\bar z} A_z^a + f_{bc}{}^a A_z^b A_{\bar z}^c = 0$$ where $f_{bc}{}^a$ is the structure ...
7
votes
1answer
78 views

Large-$N$ Yang Mills

I've bumped into the study of the $SU(N)$ theory in the large-$N$ limit. I'm wondering in which way the study of this Yang-Mills theory, can give contribution to QCD with gauge group $SU(3)$, i.e. ...
4
votes
2answers
184 views

How many physical degrees of freedom does the $\mathrm{SU(N)}$ Yang-Mills theory have?

The $\mathrm{U(1)}$ QED case has two physical degrees of freedom, which is easy to understand because the free electromagnetic field must be transverse to the direction of propagation. But what are ...
2
votes
2answers
151 views

Why do we require the generators of $\mathrm{SU(N)}$ gauge theories to be $N \times N$ matrices?

I have often read that the generators for $\mathrm{SU(N)}$ gauge theories must be $N \times N$ matrices; see for instance these notes at the top of page 3: ...
3
votes
1answer
124 views

Are “confinement” and “asymptotic freedom” two sides of the same coin?

On Wikipedia it says that the two peculiar properties of quantum chromodynamics (QCD) are: confinement and asymptotic freedom. Asymptotic freedom is the idea that at low energies we cannot use ...
4
votes
1answer
204 views

Field strength vanishes iff $A_{\mu}$ is pure gauge

Is it true that the field strength $F_{\mu\nu}$ in a non-Abelian gauge theory with gauge group $G$ vanishes if, and only if, the gauge field $A_{\mu}$ is a pure gauge? I can show one implication. ...
2
votes
1answer
113 views

How to get a $\mathcal{N}=2$ SuperYang-Mills Lagrangian from a quiver

How can one write down the $\mathcal{N}=2$ SuperYang-Mills Lagrangian given a quiver graph? For concreteness consider the quiver $$(2)-(4)-[6]$$ where the node $(2)$ corresponds to a $U(2)$ factor ...
5
votes
0answers
103 views

(coordinates) Invariance/Covariance of Chern-Simons theory and Yang-Mills theory

It is known that 3D Chern-Simons(C-S) theory has no explicit metric involving in the Lagrangian density: $$ A \wedge dA + (2/3) A \wedge A \wedge A $$ while the 4D Yang-Mills(Y-M) theory has the ...
6
votes
1answer
241 views

Wightman axioms and gauge symmetries

I have a basic understanding of the Wightman axioms for QFT. I was reading the about the Mass Gap problem for simple compact gauge groups and was wondering how the gauge group is supposed to be ...
1
vote
2answers
134 views

Fermion field structure in non-abelian gauge theories

I am trying to understand the structure of the fermions in non-abelian gauge theories. Disclaimer: my question might be very trivial (I suspect the answer could simply be "a change of basis"), but I ...
5
votes
1answer
144 views

Why do three dimensional gauge theories flow to conformal theories in the infrared?

What is meant with the fact that Super Yang-Mills flows to a conformal field theory in the infrared? Also, is this a general fact or does this depend on the fact of considering a certain class of ...
4
votes
1answer
165 views

Conserved topological charge for d=3 Yang-Mills. G=U(2)

Consider a pure Yang-Mills lagrangian density $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu}$$ with gauge group $U(2)$. Take the generators for $U(2)$ to be $t_0$, $t_i \ i=1,...,3$ with ...
2
votes
2answers
480 views

Mass dimension of coupling constants in various dimensions

Just a quick question: Suppose I want to consider QED or YM in 4 dimensions we always say that the coupling constants are dimensionless and that the field then has a specific mass dimension. What ...
1
vote
1answer
41 views

Question on field strength tensor in YM

just a quick question on $F_{\mu\nu}^a$. I'm correct to think $F_{\mu}^{\mu,a}$ vanishes, aren't I? (Just want to make sure...) My reasoning is as follows: The derivative terms cancel anyways - ...
8
votes
1answer
192 views

Boundary currents for Asymptotic Symmetry Group (ASG)

In the context of asymptotic symmetry groups, what is a boundary current? Why is it called a "current"? Context: I'm reading Strominger's recent paper on Asymptotic symmetry group of Yang-Mills ...
5
votes
2answers
174 views

How do people historically have come to use the Yang-Mills theory in physics?

There are many books, in which Yang-Mills theory is introduced "just like that". But I didn't find some book with set of historical arguments, which had led people to using it in quantum field theory. ...
7
votes
1answer
319 views

Hodge star operator on curvature?

I've a question regarding the Hodge star operator. I'm completely new to the notion of exterior derivatives and wedge products. I had to teach it to myself over the past couple of days, so I hope my ...
2
votes
0answers
111 views

From Berry's phase to artificial Gauge potential

How a nonzero geometric phase in a loop is used to generate artificial gauge potentials? If possible, can you also tell how to generate the non-abelian artificial gauge potentials.
6
votes
1answer
182 views

Proof that we can always find a gauge transformation such that $A_0=0$?

I'm trying to follow Coleman's proof from his lectures "Aspects of Symmetry" on page 200-201. He proofs it is always possible to work in the temporal gauge for a general Yang-Mills-Higgs theory. I ...