Questions tagged [gauge-theory]

A gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of *continuous symmetries* of these internal degrees of freedom, *i.e.* the predicted physics under any transformation in this group on the degrees of freedom. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.

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The integer eigenvalues for a generator of $SU(N)$

I am studying gauge theory by V.P. Nair's QFT textbook. He explains in p. 455 that all components of all fields which are $\mathbb{Z}_N$ invariant will have integer eigenvalues for $Y=diag(1/N,1/N,\...
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Deriving Yang-Mills Equations

On the same spirit of this unanswered question I am proposing this question which I have been trying for some time now. Here I'm working with dimension $n = 4$ (identifying $\mathbb H = \mathbb R^4$) ...
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How to prove $\Lambda_\mathrm{weight}(\mathfrak{g})/ \Lambda_\mathrm{root}(\mathfrak{g}) = \mathbb{Z}_N$ for $\mathrm {SU}(N)$?

I have a question about $\mathrm {SU}(N)$ Lie group from David Tong's Gauge Theory notes (p. 92). He considers $\Lambda_w(\mathfrak{g})$ and $\Lambda_\mathrm{root}(\mathfrak{g})$ as the weight lattice ...
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Gauge fixing in the classical $U(1)$ gauge theory

My question concerns the gauge fixing in classical v.s. quantum $U(1)$ gauge theory. I will ask about the gauging fixing in quantum $U(1)$ gauge theory in a separated Phys-SE post. For the classical $...
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Formalism of Non-abelian Gauge theory?

Depending on the source I have seen two different definitions/formalisms for Non-abelian Gauge theories and was wondering how the two were related. The first one is the more common where the gauge ...
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Darboux variables for $SU(3)$

In principle, for any $SU(N)$ group it is possible to construct Darboux variables (which are desired since they obey canonical commutation relations) corresponding to the color charges $Q_a$ (which ...
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Do primary first class constraints change the electric field in the Hamiltonian form of Maxwell's theory?

In my understanding of Dirac's theory of constrained Hamiltonians, the primary (and also the secondary) first class constraints are generators of canonical transformations that do not change the ...
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Chern-Simons term for a non-abelian gauge multiplet

In equation (20.9) of Freedmann and Van Proeyen's Supergravity, it is stated that for the following Chern-Simons term: $$S_{\mathrm{CS}} = C_{IJK}\int A^I\wedge F^J \wedge F^K$$ to be invariant under ...
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Landau Pole and gauging of Dirac fermions

Let us focus on 4d spacetime dimensions. It is well known that $N$ free massless Dirac fermions is conformal invariant. This theory has a $U(1)$ symmetry under which all the fermions have charge 1, i....
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What does it mean to be “gauging” a symmetry?

I read this and other similar questions, but they all address the problem of gauging a global symmetry (implying that one could also gauge a local one). This confused me a lot: in my mind gauge and ...
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Local gauge transformation in Fock space of charged particles

I'm currently fiddling around with gauge-phase-transformations in Fock space. Especially, I'm trying to write a local gauge-phase-transformation as an operator in a basis-independent way. Here is what ...
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Local gauge transformation in Fock space of photons

I'm currently playing around with gauge transformations in the Fock space of photons. Let's consider a local gauge transformation with gauge function $f(\mathbf{r})$. So the magnetic vector potential ...
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Why are gauge anomalies characterised by the non-triviality of $\pi_5(\mathcal G)$?

The folklore in 4-dimensional gauge theories is that the existence of potential gauge anomalies from the triangle diagrams that need to be cancelled are characterised by the non-triviality of the ...
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Reference request - derivation of $\mathrm{ind}\,D_+=-\frac{1}{8\pi^2}\int\text{tr}\,F\wedge F$

Let $D$ be the Dirac operator. The equation \begin{equation}\tag{1} \mathrm{ind}\,D_+=-\frac{1}{8\pi^2}\int_M\text{tr}\,F^2=-\frac{1}{8\pi^2}\int_MF^a\wedge F^b\ \mathrm{tr}(T_aT_b) \end{equation} is ...
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As a physicist, why are associated bundles important?

I have a good grasp on principal bundles as providing a lie group on some fibers of our field. So for example, the wavefunction tells us the phase of a particle in space and time, and this can be a ...
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Why and where Poincaré gauge theory fails?

I would like to post this question because I have seen no one post it in this explicit way. From what I have seen, Poincaré gauge theory uses connections (gauge fields) like the spin connection and ...
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Exponential of operators not defined on a Hilbert space (Dirac operator)

Let $D$ be the Dirac operator. In his proof of the Atiyah Singer index theorem, Fujikawa considers the operator $\exp{D^2}$. However, as far as I know, $D$ is not defined on a Hilbert space (the ...
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Lattice Gauge and Spin Network

I see the similarity between the Lattice Gauge and Spin Network. (For example, the both theories depict the node part as quantum (the latter is explained as spin).) Are there any other mathematical, ...
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Anomaly, symmetries, and Ward identity

I'm trying to bring together and understand the concepts of anomaly, quantum symmetries, and Ward (or Ward-Takahashi, or Slavnov-Taylor) identity in QFT. I think I know what the ideas mean, but I'm ...
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Could a theory that looks like an effective field theory also be fundamental?

In the context of the standard model, I got the impression that gauge field theories are considered fundamental, and effective field theories can be derived from them for certain energy scales. But ...
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How does gauge invariance protect the SM gauge boson masses in SUSY from divergent radiative corrections?

The W and Z gauge bosons receive radiative corrections in loop from the heavy SUSY scalars. There is an argument using gauge invariance which explains how the masses remains protected. I am not able ...
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Mathematical structure of gauge theories

What physical motivations lead to the mathematical structure of gauge theories? (principal bundles, connections, ...). I know it's a very vague question, I'm looking for some concrete examples to try ...
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When and why can the spin connection term of the Dirac Operator be omitted?

The Dirac Operator $D$ is defined by \begin{equation}\tag{1} D=i\gamma^a\nabla_a=i\gamma^a\nabla_{e_a}=i\underbrace{\gamma^a{e_a}^\mu}_{=\gamma^\mu}\nabla_{\partial_\mu}=i\gamma^\mu\nabla_\mu=i\gamma^\...
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Conformal coordinate change: gauge symmetry or global symmetry

While reading the fourth chapter "Introducing Conformal Field Theory" of D. Tong's string theory notes, I read that A transformation of the form $\sigma^\alpha\to\tilde{\sigma}^\alpha(\...
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Why is the group of gauge transformations on the frame bundle isomorphic to $\text{Diff}(M)$?

Consider the frame bundle $LM \to M$ for given Lorentzian manifold $M$. The group $\mathcal{G}$ of gauge transformations of the second kind are automorphisms $\phi:LM \to LM$ covering the identity $\...
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Massless gauge field quantisation in spherical polar coordinate

Can anybody share a link or paper regarding massless gauge field quantisation in spherical coordinate flatspacetime?? Actually I have seen in the book "Field Quantisation" by Greiner , ...
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Why is Kitaev's toric code a $Z_2$ gauge theory?

I am reading Kitaev's 2003 paper. In the literature, it is often said that the model proposed in this paper is a $Z_2$ gauge theory. I don't quite see why it is the case. Where is the $Z_2$ gauge ...
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Quantisation of gauge field in temporal gauge

Whenever we use temporal gauge and quantise gauge field we implement Gauss law. I have seen some papers but the point is not cleared to me that why we implement Gauss law there. Please explain this if ...
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Diffeomorphisms and coordinate changes

In this question: Simple conceptual question conformal field theory, an answer states that the invariance of the theory under the change of coordinates is a particular case of invariance under ...
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Why Faddeev-Popov ghost cannot exist in external line?

I was studying the path integral quantization of non-abelian gauge field. After the path integral quantization, the action becomes $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\frac{1}{2\zeta}(\...
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Why can we not observe Faddeev-Popov ghost fields? [duplicate]

I know that ghost fields de-couple from the gauge fields in abelian QED. But my question is how does decoupling prove that we cannot observe ghost fields?
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Gauge fixing, invertibility and Green's functional

consider the photon in QED and the corresponding EOM of its Green's functional in k-space: $$(k^\mu k^\nu-k^2g^{\mu\nu})\Delta_{\nu\rho}(k)=i\delta^\mu_\rho.$$ Now, I understand that $U^{\mu\nu}(k):=...
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Consistency condition in the context of gravitational waves

Reference: archive.org/details/GeneralRelativity/page/n250/mode/1up I am studying the phenomenon of gravitational waves as one of the conseguences of the linearized Einstein field equation. In order ...
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Faddeev-Popov method to scalar fields

So I was learning about the Faddeev-Popov method and my instructor briefly told to use it on the following action: $$S = \int d^dx\left( \frac12 \partial_{\mu}\phi_1 \partial^{\mu}\phi_1+ \frac12 \...
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Vector Potential of a rotating Spherical Shell

I have done all the calculation in finding the vector potential leading up to the equation 5.68. But then Prof. Griffiths goes back to the original figure and mentions that the coordinate of the point ...
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Is there a sistematic way to verify if some gauge group embedding is possible?

Consider the breaking pattern of the well known 331 models: $SU(3)_C\otimes SU(3)_L \otimes U(1)_X \to SU(3)_C\otimes SU(2)_L \otimes U(1)_Y$. In this case the subgroup $SU(2)_L \otimes U(1)_Y$ is ...
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Explanation of $\sum_n\langle\psi_n(x)|(O\psi_n)(x)\rangle=:(\mathrm{tr}\,O)(x)=\mathrm{tr}\int\frac{\mathrm{d}k}{(2\pi)^4}e^{ikx}Oe^{-ikx}$

Let $D$ be the Dirac operator, $O_N:=e^{-(D/N)^2}$ for $N\in\mathbf{N}$ and $\{\psi_n\}$ a complete set of eigenfunctions of $D$. On page $69$ and $78$ of Path Integrals and Quantum Anomalies and in ...
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2 coinciding D-branes leads to a $U(2)$ gauge theory

I'm having trouble understanding how two D-branes leads to a $U(2)$ gauge theory from David Tong's notes, chapter 7, pages 191-192. I am learning group theory and I understand that a 'charge' is a ...
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I get an extra term when I try to derive $\nabla\!\!\!\!/\ ^2=\nabla_\mu \nabla^\mu+\frac{1}{4}[\gamma^\mu,\gamma^\nu]F_{\mu\nu}$

The equation in the title can be found in Nakahara's book, page 507 (chapter 13, Anomalies in Gauge Field Theories) and in Fujikawa's book, pages 69 and 78 (chapter 5, The Jacobian in Path Integrals ...
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Interpreting the conserved charge in scalar QED

In scalar QED, applying Noether's theorem for internal global symmetries results in a Noether current that is dependent on the gauge because of the presence of the covariant derivative. $$j_\mu=-i(\...
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Trajectory of a particle moving in a non-abelian gauge potential

Particle moving in a abelian Maxwell gauge potential can be understood from Lagrangian $$ L=(1/2) m \dot{\vec{r}}^2- q\phi +q \dot{\vec{r}}\cdot \vec{A} $$ or Hamiltonian $$ H=\frac{(\vec{p}-q \vec{A}...
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Transverse-traceless gauge: Why the traceless condition?

I'm right now following a course on GR and I arrived to the gravitational waves part. Letting the metric be that of the plane Minkowski space with a small perturbation: $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\...
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Does the structure constant of Yang-Mills field change sign under time reversal?

The time reversal of Abelian (electromagnetic) field strength is pretty straight forward. The electric field $F_{0i}$ is even under time reversion. The magnetic field $F_{ij}$ is odd under time ...
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Topological meaning of the integral of the trace of Cartan-Maurer forms in Anthony Zee's book on QFT in a nutshell

I learnt in S. Sternberg's book "Curvature in Mathematics & Physics" over the Maurer-Cartan form that if there is a tangent vector $v \in TG_g$ at a point $g \in G$ ($G$ is supposed to ...
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Do gauge fields violate causality?

For example, take $$V(\textbf{r},t)=\frac{1}{4\pi \epsilon}\int\frac{\rho(\textbf{r},t)}{|\textbf{r}-\textbf{r}'|}d^3r'.$$ This integral is purely spatial, the time of the density corresponds to the ...
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Is the equation $[\nabla_{\mu},\nabla_{\nu}]=F_{\mu\nu}$ correct? If yes, how does it have to be interpreted?

It seems like simply using the equation \begin{equation} \nabla_{\mu}=\partial_{\mu}+A_{\mu} \end{equation} isn't enough: One obtains \begin{equation} [\nabla_{\mu},\nabla_{\nu}]=\underbrace{[\...
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Bianchi identity of a non-Abelian gauge theory?

How can one prove the Bianchi identity of a non-Abelian gauge theory? i.e. $$ \epsilon^{\mu \nu \lambda \sigma}(D_{\nu}F_{\lambda \sigma})^a=0 $$
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Section of adjoint bundle in gauge theory

I have a couple of questions on gauge theory. Considering a principal $G$-bundle $P\xrightarrow{\pi} M$, we have a connection, a local Lie algebra-valued 1-form $A$. This is the photon field, correct? ...
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Faddeev-Popov Ghosts in the canonical formalism

In the Lorenz gauge in electrodynamics, the timelike and longitudinal components can be eliminated by prescribing the Gupta-Bleuler condition $\partial^{\mu}A_{\mu}|\Psi)$ on physical states. This ...
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Gauge theory - definition of the trace

I am currently reading Nakahara's book and starting from chapter $10$, some sort of trace constantly shows up in the equations. For example, \begin{equation} S=-\frac{1}{2}\int\mathrm{tr}(F\wedge*F) \...

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