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.
683
questions
2
votes
1
answer
71
views
Quantum Mechanics with non-Abelian gauge fields
In "conventional" quantum mechanics, in the presence of electromagnetic fields, we use minimal substitution in the Hamiltonian in the simplest case. I.e.
$$H=\frac{\vec{p}^2}{2m}\rightarrow \...
- 7,482
0
votes
0
answers
66
views
Do we know a reason for why exactly $\rm U(1)$, $\rm SU(2)$ and $\rm SU(3)$? [duplicate]
I always found it a curiousity that in the symmetry groups of the known fundamental forces we find the nice arithmetic progression $1,2,3$: first there is $\DeclareMathOperator{\U}{U}\...
- 514
1
vote
0
answers
41
views
Are non-trivial topologies in the gravitational path integral related to large gauge transformations in Yang-Mills?
While the gravitational path integral is not a well-understood concept mathematically, a number of works (particularly in recent research connected to AdS/CFT) emphasize the importance of integrating ...
- 557
5
votes
1
answer
171
views
Why is there only one coupling constant in Yang-Mills theory? Why are gluon self-coupling and gluon-matter coupling constants the same?
Is it non-trivial that the coupling constant $g$ in gluon self-interaction terms is the same as the coupling constant $g$ in gluon-fermion interaction term in Yang-Mills theory?
Pure Yang-Mills theory ...
- 182
5
votes
5
answers
243
views
Does pure Yang-Mills have a scale?
Consider pure Yang-Mills (YM) in 4 dimensions. The YM mass gap problem (as described in https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf) tells us that this is supposed to have a mass-...
- 674
1
vote
1
answer
63
views
Compactification and gauge choice for instanton solutions
I have several doubts regarding topological solutions in pure YM -- these are related both to less trivial topological misunderstandings as to rudimentary gauge fixing confusions of mine.
What is the ...
- 1,613
0
votes
1
answer
76
views
Why is the gauge field Lagrangian proportional to the square of the strength tensor?
I am studying quantum field theory and was wondering about a mathematically rigorous explanation of why $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}_\alpha F^\alpha_{\mu\nu}$$ for gauge boson fields. I have ...
- 490
0
votes
1
answer
58
views
Theta vacua eigenstates
I have been trying to prove the very simple result that the eigenstates of an operator with matrix elements
$$
\langle n^\prime | H | n \rangle \sim g(|n^\prime-n|),
$$
in a basis $\{|n\rangle\}^{+\...
- 1,613
0
votes
0
answers
52
views
$U(n)$ vs $SU(n)$ Symmetry and Tracelessness - Chargeless Gauge Bosons
I've recently been studying the symmetries that give rise to the bosons of the standard model. I have taught myself just enough group theory to kind of understand what $U(1)$, $SU(2)$, and $SU(3)$ ...
- 35
2
votes
1
answer
46
views
Scale transformation of the scalar field and gauge field
I am reading this paper: "Magnetic monopoles in gauge field theories", by Goddard and Olive. I don't understand some scale transformations that appear in Page 1427.
Start from the energy ...
- 1,347
4
votes
2
answers
253
views
Charges in Terms of Lie algebra Root Decomposition
I would like to clarify the interpretation of the notion of "charge" in therms of theory of Lie algebras. There it is stated that
So, for example, when the symmetry group is a Lie group $G$,...
- 1,253
-2
votes
1
answer
64
views
What is the difference between Yang-Mills theory and ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM) [closed]
I am a beginner trying to learn Quantum Field theories.
I understand the basics of Quantum Electrodynamics and the emergence of Feynman diagrams from the perturbative expansion of the generating ...
- 9
1
vote
0
answers
33
views
Original motivation for promoting global symmetries to local
Assume we are dealing with a Lagrangian $\mathcal{L}$ for matter field $\psi$ which has a global $G$-symmetry and it's possible to promote this global $G$-symmetry to a local symmetry after the usual ...
- 1,253
1
vote
0
answers
33
views
Reference for dual gauge field and magnetic field being canonically conjugate
The gauge field operator in quantum Yang-Mills is canonically conjugate to the electric field.
I know that the dual gauge field is canonically conjugate to the magnetic field, but I can't seem to dig ...
3
votes
1
answer
188
views
Temporal Gauge with periodic boundary conditions
In Yang-Mills theory with periodic boundary conditions in time, is the temporale gauge, i.e. $A_0 = 0$, well defined?
Periodic boundary conditions would be
$$A_\mu(T_2,x) = A_\mu(T_1,x).$$
Naively I ...
- 2,173
2
votes
1
answer
58
views
Invariance of gauge-fixing condition in background field method
In the Peskin & Schröder (chapter 16.6) they use the background field method and spilt the gauge field into an background field $A$ and a fluctuation field $\mathcal{A}$. Next they claim that
the ...
- 480
5
votes
0
answers
65
views
Validity of Lorenz gauge in non-Abelian gauge theory
I understand that this is a long shot, especially because it's such a niche question but: has it been mathematically proven that (under sufficient smoothness conditions, etc.) any field configuration ...
- 557
0
votes
1
answer
48
views
Gauge covariant derivative for fields in tensor representations with multiple indices
In QFT, for fields transforming under some Gauge group, one defines the covariant derivative as
$$
(1)\qquad D_{\mu} \phi = \partial_{\mu}\phi -igA_{\mu}^k \rho(t_k)_{ab}\phi_b
$$
If $dim\rho=dim(\...
- 23
4
votes
1
answer
103
views
Can double copy address the weakness of gravity?
I hope I didn't miss a post that already answers this question. I am simply wondering if there is a natural framework in the double copy approach to explain the weakness of gravity? I don't require ...
- 2,612
2
votes
0
answers
60
views
Gauge invariance and Ward Identity?
I have to work on vacuum polarization and gauge contributions for a given problem.
I have to compute and show that their sum is gauge invariant, which according to the exercise, is equivalent to ...
- 302
6
votes
0
answers
84
views
Topologically interesting non-abelian instantons
Consider a Yang-Mills type theory, with algebra $\mathfrak{g}$, defined over a manifold $M$. The action functional$\newcommand{\tr}{\operatorname{tr}}$ is
$$S[a] = \frac{1}{2g^2}\int_M\tr_\mathfrak{g}(...
- 4,379
1
vote
2
answers
117
views
Covariant derivative of gauge theory in curved space
I am reading Witten's article and have a basic question about gauge theory in curved space.
In ordinary flat space (Euclidean space or Minkowski spacetime), covariant derivative of a gauge field $A_{\...
- 81
1
vote
0
answers
71
views
Yang-Mills-Dirac Lagrangian and Gravity
Let $M$ be a Lorentzian spin $4$ manifold, i.e. admits a spin structure $Spin^+(M)\rightarrow M$, which is just a principal $Spin^+(1,3)$ bundle over $M$, which is compatible with the bundle of ...
- 241
1
vote
0
answers
87
views
Gauge boson self-interaction diagrams
On page 522 of Peskin and Schroeder, we try to calculate the self-energy of gauge boson. Figure 16.7 gives the following diagrams:
However, Peskin and Schroeder says there are three additional ...
- 720
-1
votes
1
answer
51
views
Why is $Tr_R(T_a\{T_bT_c\})=-Tr_\overline{R}(T_a\{T_bT_c\})$ for $SU(N)$ representations?
I'm looking at the chiral anomaly in QFT and the term
$$d_{abc}=Tr_R(T_a\{T_b,T_c\})$$
shows up where $Tr_R$ means the trace in the representation $R$, $\overline{R}$ is the conjugate representation ...
- 228
5
votes
2
answers
413
views
Why is non-abelian gauge theory unique in 4 dimensional spacetime?
On my QFT lecture note there is a comment that says 'Non-abelian gauge theory is extremely unique in 4-dimensional spacetime'. However, I didn't really catch what that means. Why is it extremely ...
- 1,545
1
vote
0
answers
23
views
Proving that the Faddeev-Popov path integral is independent of the gauge choice? [duplicate]
I know that the Faddeev-Popov path integral is gauge invariant. But how does one show that
\begin{equation}
I = \int \mathcal{D}\mathcal{A}_\mu \bigg|\frac{\delta\mathcal{G}}{\delta{\omega}}\bigg|\...
- 529
1
vote
0
answers
47
views
Deriving the ghost Lagrangian in Peskin and Schroeder
On page 514 of Peskin and Schroeder, the book derives
$$\tag{16.31} \det\bigg(\frac{1}{g}\partial^\mu D_\mu\bigg)=\int\mathcal{D}c\mathcal{D}\overline{c}\exp\bigg[i\int d^4x\overline{c}(-\partial^\mu ...
- 720
0
votes
1
answer
39
views
Gauge transformation of vector superfields
I am working my way through Srednicki and am at Chapter 95 which introduces supersymmetry.
$\newcommand\dag\dagger$
In Chapter 95, Srednicki introduces the idea of a vector superfield, which he says ...
- 123
6
votes
2
answers
525
views
Renormalisation of Yang-Mills Breaks Gauge Invariance?
Consider the Lagrangian (renormalised + counterterm) of QED:
$$\mathcal{L} = -\frac{1}{4} F_{\mu \nu}F^{\mu \nu} - \frac{1}{2 \xi}(\partial_{\mu} A^{\mu})^2 + \bar{\psi}(i \displaystyle{\not} D - m)\...
- 1,673
2
votes
2
answers
146
views
Question on Lie algebra indices, spinor indices and gamma matrices indices in QCD and QED lagrangians
The question is written in section $2)$
1) Introduction
1.1) QCD
For a non-abelian group, the connection term on the lagrangian will be written as
$$\mathcal{A}_{\mu}=A_{\mu}^{a}T_{a}\tag{1}$$
This ...
- 2,943
3
votes
0
answers
125
views
Integration by parts of covariant derivative
There already exists posts to discuss this question, but I don't think it's totally done!
We can write the covariant derivative as
$$D_i=\partial_i-igA_i^aT^a \tag{1}$$
There are two kinds of opinions ...
- 1,347
7
votes
3
answers
833
views
How does the absence of quadratic terms in the Lagrangian imply massless quanta?
When studying gauge theory, I often see the statement that gauge invariance does not allow the Lagrangian of the theory to contain terms that are quadratic in the gauge field. For example, to quote ...
- 2,182
1
vote
1
answer
106
views
Transformation of Yang Mills Field Strength
I am confused about the expression $$F_{\mu \nu} \to F_{\mu \nu}' = U F_{\mu \nu}U^{\dagger}.$$ I found related Phys.SE posts How would one show that a nonabelian field strength tensor transforms in a ...
- 31
1
vote
0
answers
61
views
Arbitrary heat kernel coefficients of covariant Laplacian with instanton
The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
- 272
0
votes
0
answers
42
views
Question on the indexes of the lagrangians describing gauge theories
For a gauge group $SU(3)_{C}$ we can construct its principal and associated bundles; we can introduce spinor fields via spin structures and spinor bundles and so on, arriving in a lagrangian theory ...
- 2,943
0
votes
0
answers
20
views
Highest weight generator (HWG) for Higgs branch hilbert series of $U(k)$ gauge group with $N$ flavours
Can anyone help me in calculating the Highest weight generating function for Higgs branch of $U(k)$ with $N$ flavours. I am new to the quiver gauge theory.
I have calculated the Higgs branch Hilbert ...
- 1
0
votes
1
answer
168
views
Vertex factors for Feynman rules in QCD
I am stuck in deriving the vertex factor for the Feynman diagrams for the QCD Lagrangian. For the quantum Yang Mills theory we will have the following interacting Lagrangian.
$$
\begin{aligned}
\...
- 400
4
votes
2
answers
404
views
How do I understand the Hodge $⋆$ operator in Yang-Mills Lagrangian?
The gauge-invariant part in Yang-Mills Lagrangian is
$$
\mathcal{L}_{\text{gauge}} = -\frac{1}{2}TrF_{\mu\nu}F^{\mu\nu} = -\frac{1}{4}F_{\mu\nu}^aF^{a, \mu\nu}.
$$
Sometimes I see the lagrangian ...
- 1,545
0
votes
0
answers
189
views
One-loop renormalization of the gauge coupling
Quoting Yuji Tachikawa, chapter 3 of "${\cal N}=2$ Supersymmetric Dynamics for Pedestrians":
Recall the one-loop renormalization of the gauge coupling in a general Lagrangian field theory $$...
user242231
1
vote
1
answer
38
views
Does the field $\Psi$ have different representation for different generator $T_R^a$?
I'm learning the non-abelian gauge theories. Suppose we have a set of (general) fields $\Psi^\alpha(x)$ transforming in a given representation $R$ of the gauge group, with $\alpha, \beta = 1,..., \dim(...
- 1,545
-1
votes
1
answer
77
views
How do I see the connections between abelian/non-abelian group and their gauge transformations?
I'm learning QED in my QFT class without too much background in group theory. Recently I'm introduced to the Abelian gauge transformation
$$
D_\mu\psi(x)\rightarrow e^{i\alpha(x)}D_\mu\psi(x)\quad
A_\...
- 1,545
1
vote
2
answers
194
views
What is the structure constant in layman's?
In the Yang-Mills field strength tensor, there's this symbol f which is the structure constant. What is the definition of this structure constant in layman's terms?
- 29
1
vote
1
answer
58
views
Bianchi identity contradiction in Abelian case
In non-abelian gauge theory, such as P & S's chapter 15, eq. (15.89), we also have Bianchi identity.
Start with
$$\epsilon^{\mu\nu\lambda\sigma}[D_\nu,[D_\lambda,D_\sigma]]=0$$
and use $[D_\mu,D_\...
- 1,347
3
votes
3
answers
152
views
What is the physical significance of $\mathbf{E}^2-\mathbf{B}^2$ in E and M?
Choosing nice units, i.e $c=1$, the electromagnetic energy density is:
$$u=\frac{1}{2}\left(\mathbf{E}^2+\mathbf{B}^2\right).$$
This is not Lorentz invariant, which makes sense since our ...
- 241
0
votes
0
answers
90
views
Gauge transformation of an adjoint left-handed Weyl spinor in $\rm SU(2)$ fundamental representation
I have a left-handed Weyl spinor field $\Psi_L$ in the fundamental representation of the $\rm SU(2)$ gauge group, which transforms $\Psi_{L,i} \rightarrow \Psi_{L,i} + i\theta^at_{ij}^a\Psi_{L,j}$.
...
- 33
3
votes
0
answers
62
views
Names of the proposals for higher $p$-form non-abelian gauge symmetries
Beyond 1-form non-abelian YM theories with
$$F=dA+A^2$$
What are the alternative names and proposals to generalize YM to non-abelian higher $p$-forms? What is the current state-of-art of those?
Remark:...
- 6,327
2
votes
1
answer
205
views
Non-abelian magnetic field interaction
Consider the magnetic part of a non-Abelian field. I want to know if we can define Hamiltonian for the interaction of this sector with the spin (something such as Hamiltonian of interaction of ...
- 159
2
votes
1
answer
79
views
Magnetic field due to singular gauge transformation
In $SU(2)$ gauge theory over $\mathbb R^3$, consider the following gauge transformation (in spherical polar coordinates)
$$ \Omega=\begin{bmatrix}e^{i\phi}\cos(\theta/2)&\sin(\theta/2)\\-\sin(\...
- 674
1
vote
0
answers
95
views
Large and small gauge transformations
I've a question about the definition of 'large' gauge transformations. There are two competing definitions:
small gauge transformations are equal to the identity at spatial infinity while large gauge ...
- 674