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20
votes
2answers
3k views

Gauge fields — why are they traceless hermitian?

A gauge field is introduced in the theory to preserve local gauge invariance. And this field (matrix) is expanded in terms of the generators, which is possible because the gauge field is traceless ...
14
votes
2answers
2k views

How does non-Abelian gauge symmetry imply the quantization of the corresponding charges?

I read an unjustified treatment in a book, saying that in QED charge an not quantized by the gauge symmetry principle (which totally clear for me: Q the generator of $U(1)$ can be anything in $\mathbb{...
8
votes
1answer
695 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = (\partial_{\mu}+A_{\mu})\...
7
votes
2answers
611 views

Why are the “coupling constants” constant?

The coupling constants (in the gauge theory) fix an inner product on the lie algebra of the gauge group and we use it to define strength of the fields. we are using ad-invariant inner products which ...
6
votes
2answers
123 views

How are quadruple gluon vertices related to $SU(2)$ and $SU(3)$?

I once read that the non-commutativity of the Lie Groups $SU(2)$ and $SU(3)$ is the reason that the weak and strong interactions have Feynman diagrams with quadruple vertices, where four gauge bosons ...
6
votes
2answers
664 views

Interpretation of the field strength tensor in Yang-Mills Theory

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_\nu]...
5
votes
1answer
616 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 = \partial_\mu+i\frac{...
5
votes
2answers
891 views

Calculating an expression for the trace of generators of two Lie algebra

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie ...
5
votes
1answer
224 views

Gauge transformations and Covariant derivatives commute

I would like to understand the statement "Gauge transformations and Covariant derivatives commute on fields on which the algebra is closed off-shell" which was taken from section 11.2.1 (page 223)...
5
votes
2answers
265 views

Gauge covariant derivative on form

Let $e$ be a one-form gauge field that belongs to the adjoint representation of the gauge group, that is SO(1,2). It is defined as \begin{equation} e = e_{\alpha}^{A}T_Adx^{\alpha}. \end{equation} ...
3
votes
2answers
231 views

Is there any $SU(\infty)$ gauge theory in quantum field theory?

The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there ...
3
votes
2answers
498 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: http://www.staff.science.uu.nl/~wit00103/...
3
votes
1answer
998 views

What is the four-dimensional representation of the $SU(2)$ generators?

Recently, I have been learning about non-Abelian gauge field theory by myself. Thanks @ACuriousMind very much, as with his help, I have made some progress. I am trying to extend the Dirac field ...
3
votes
2answers
196 views

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 $SU(N)$ ...
3
votes
1answer
68 views

Gluon have colour-anticolour; what about weak bosons?

Gluons can be red-antiblue, or green-antired, etc. What about weak interaction bosons? (Say before symmetry breaking, to make matters simpler.) Is there a similar "weak charge" structure of charge-...
3
votes
1answer
364 views

The adjoint representation and the gauge boson of $O(n)$

I'm learning "Gauge theory of Elementary Particle Physics Problems and Solutions" by Cheng and Li. In problem 8.4 "$O(n)$ gauge theory" on page 165, Under infinitesimal $O(n)$ representations a ...
3
votes
1answer
639 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field $\...
3
votes
1answer
56 views

Fierz identity for symplectic group

For the fundamental representation of $SU(N)$, there is a Fierz identity: $$ \sum_iT^i_{ab}T^i_{cd}=\frac{1}{2}\left(\delta_{ad}\delta_{bc}-\frac{1}{N}\delta_{ab}\delta_{cd}\right) $$ where $T^i$ is ...
3
votes
0answers
213 views

How do gauge fields transform with an extra inhomogeneous term even though they are Lie Algebra valued in Non-Abelian gauge theories?

I am trying to work out Non-Abelian gauge theories but I couldn't get my head around the fact that gauge fields transform with an extra inhomogeneous term under the adjoint action of a group $G$, that ...
3
votes
0answers
153 views

Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
2
votes
1answer
495 views

Generators of $SU(2)$ and $SU(3)$ Symmetry Groups

I've been reading about gauge field theories, and I keep coming across the generators of the $SU(3)$ and $SU(2)$ Symmetry Groups. I read that each generator corresponds to a gauge boson, but I'm ...
2
votes
1answer
708 views

Commutator of Gauge Covariant derivatives

What is the physical meaning of $$ [D_{\mu}, D_{\nu}] ~\propto~ F_{\mu, \nu}, $$ where $D_{\mu}$ is the gauge covariant derivative and $F_{\mu,\nu}$ is the field strength? Is it just a definition? ...
2
votes
1answer
103 views

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\...
2
votes
1answer
97 views

Factorization of exponential of broken generators in parametrization of scalar multiplet in non-abelian SSB

Describing abelian symmetry breaking in his book on gauge theories, after favouring a vacuum (whose expectation value is $v$) from the symmetric continuum, Quigg parametrize the complex scalar as $$ ...
2
votes
1answer
104 views

What is the physical meaning of the Gell-Mann matrices generating SU(3)?

I understand on a surface level that there are these matrices that generate the group SU(3). However, when reading books on gauge theory they appear to make the jump from SU(3) having 8 generators to ...
1
vote
2answers
88 views

Symplectic group $Sp(2N)$ in Srednicki's book

There is a question in Mark Srednicki's Book (Problem 24.4, p.160) about $Sp(2N)$, but I am not sure I understand the significance (application?) of this group. In that chapter, he talks about $SO(N)$ ...
1
vote
2answers
435 views

Why gauge fields are traceless Hermitian?

So I've had a read of this, and I'm still not convinced as to why gauge fields are traceless and Hermitian. I follow the article fine, it's just the section that says "don't worry about this ...
1
vote
2answers
162 views

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 ...
1
vote
1answer
200 views

Lie algebra decomposition of the gluon field

It is commonly written in the literature that due to it transforming in the adjoint representation of the gauge group, a gauge field is lie algebra valued and may be decomposed as $A_{\mu} = A_{\mu}^...
1
vote
1answer
77 views

Ambiguity with definitions of vector potential

In one of my books (the great Baez & Munian's "Gauge fields, knots and gravity"), the vector potential is defined as a $End(E)$ valued 1-form, with $End(E)$ endomorphisms of the fiber $E$. So, ...
1
vote
1answer
176 views

What exactly are the ADE type of gauge theories?

What exactly are the ADE type of (susy) gauge theories? What exactly we mean, intuitively, the ADE singularities? What are their relation to brane constructions and do you have any references one ...
1
vote
0answers
31 views

Is it always possible to move to the “Cartan Gauge”?

Forgive me for potentially coming up with a new name for what I am about to describe. Let's say we have a scalar field $\phi^a$ which transforms with respect to the adjoint representation of some Lie ...
1
vote
1answer
47 views

Gauge group of Electroweak theory

I am doing a question that asks me to identify the gauge groups of a Lagrangian with the field strength tensors $$\bf{F}_{\mu \nu} = \partial_{\mu}\bf{W}_{\nu} - \partial_{\nu} \bf{W}_{\mu} - g\bf{...
1
vote
1answer
131 views

Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
1
vote
0answers
67 views

Is there a difference of sign conventions of Dirac Index between mathematics and physics?

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by $$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{...
1
vote
0answers
67 views

Gauge group properties VS Particle properties

Let us say that, for simplicity, we are studying gauge theory over 4-dimensional Minkowski spacetime, and that the gauge group is $SU(3)$, which is probably the simplest non-abelian gauge group after $...
0
votes
2answers
69 views

Incorrect proof that all gauge theories are abelian

Consider a gauge field $W_\mu = W_\mu^{a} \tau_a$ where $\tau_a$ are the generators of the Lie algebra and $W_\mu^{a}$ just numbers. Then: $$ W^2 = W_\mu W^\mu = W_\mu^a\tau_a W^{\mu b} \tau_b = W_\...
0
votes
1answer
81 views

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}...
0
votes
1answer
62 views

Lie algebra valued potential vector [closed]

Maybe it is a simple question but I have some difficulty to understand the explicit matrix form of this usual relation: $$A_\mu=A^a_\mu \tau_a$$ where $A^a_\mu $ is the Lie algebra valued potential ...
0
votes
1answer
172 views

Justification of gauge field transformations

I am trying to understand the gauge transformation of gauge fields in a gauge quantum field theory. As an example I considered this wikipedia article, section 'An example: Scalar $O(n)$ gauge theory'....
0
votes
1answer
378 views

Gauge covariant derivative

I have seen distinct definitions of gauge covariant derivative (in Yang-Mills theory) $$ D_\mu \phi = (\partial_\mu + igA_\mu) \phi $$ vs $$ D_\mu \phi = \partial_\mu \phi + ig[A_\mu,\phi] .$$ I ...
0
votes
0answers
20 views

Physics-y resource request on Killing-Cartan forms

Most books that treat nonabelian gauge theory do not contain detailed discussion on Killing-Cartan forms, they'll usually just say that in $\text{SU}(N)$ Yang-Mills theory, one can choose generators $...
0
votes
0answers
27 views

Legal values of spin-1 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

For the spin-1/ boson field $A_\mu$, we may choose it to be a vector which needs to be real $\mathbb{R}$ usually for photon field. The field strength $F= dA$ is also real. Same for the nonabelian ...
0
votes
0answers
101 views

Products of Lie-Groups versus Lie-Group Extensions in Physics

The Standard Model of elementary particle physics is a gauge theory based on the Lie group $U(1) \times SU(2) \times SU(3)$. From the mathematical perspective I read that: Simple Lie groups have ...
0
votes
0answers
16 views

Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-...
-1
votes
1answer
411 views

Representation and gauge transformation of Yang-Mills theory

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 the gauge fields $A_{\mu}^{a}$ ...