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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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Gauge transformation [on hold]

We consider a particle with charge $q$, the Hamiltonian $H$, potential function $V(x,t)$ and the vector potential $\vec{A}(x , t)$. The evolution of the wave function $\Psi (\vec{r}, t )$ is given by ...
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Compactification of space in Hamiltonian formulation of Yang-Mills theory

I am reading David Tong's lecture notes on Gauge Theory where he talks about Hilbert space interpretation of Yang-Mills theories in Section 2.2 of Chapter 2. When discussing the gauge dependence of ...
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Conservation of magnetic charge

It is well known that the electric charge of a system can be thought of as the Noether charge associated with isotropic large gauge transformations. That is, given Einstein-Maxwell theory $$S=\frac{1}...
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Gauge fixing in Ginzburg-Landau simulation

I am developing a computer simulation of the Ginzburg-Landau model of superconductivity. In a few words, I have discretized the domain with finite differences and I am using Nonlinear Conjugate ...
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Covariant derivative in field theory

I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-half - spin-1 field: $$ ...
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Fields transforming under an exceptional Lie group

We may think of tensors as sections of an associated vector bundle to a principal $\mathrm{GL}(n,\mathbb R)$ bundle, with a fibre chosen to be $\mathbb R^m \times (\mathbb R^*)^n$ - these play a role ...
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Unitary gauge for spontaneous symmetry breaking

I'm given a lagrangian $$ \mathcal{L} = \partial_{\mu} \Phi^{\dagger} \partial^{\mu} \Phi + m^2 \Phi^{\dagger} \Phi - \lambda (\Phi^{\dagger} \Phi)^2 $$ where $m^2 > 0, \lambda > 0$. This ...
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Can we choose polarization vectors freely in any given gauge theory?

When quantizing a gauge theory, we obtain spin-1 particles propagating in space-time. When we want to count the degrees of freedom of the theory or, equivalently, when we are trying to decompose the ...
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Is the supercurrent gauge invariant?

If we consider ${\cal N}=1$ renormalizeable chiral gauge theories, specifically discussed in section 27.4 of Weinberg's Quantum Theory of Fields, Supersymmetry book, should the supercurrent be gauge ...
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A Naive Question about Gauge Theory

I am suffering from a question I encountered from the lecture notes of gauge theory by David Tong. The problem comes from page 67 on the gauge fixing in back-ground gauge method. In David Tong's ...
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What classifies gaugings?

Gauging a global symmetry $G$ introduces several free parameters to the theory. For example, In $d=4$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, ...
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How can I compute the spin texture for a $SU(2)$ gauge model?

I am trying to determine the helicity of 4 Dirac cones in my model, and one way I want to approach it is by plotting the spin-texture. However, I am unsure of how one would calculate the spin-texture ...
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Why quarks in the fundamental and gluons in the adjoint?

I have been told that in gauge theories “fermionic matter goes in the fundamental rep of $SU(N)$, while gauge fields go in the adjoint rep”. I understand how this works, and for instance, in QCD,...
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Gauge singlet under SM

What does it mean when they say something is a gauge singlet under the Standard Model group? I would like to understand this concept of Singlets and Doublets. Thanks in advance.
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Lagrangian for fermions

I was trying to understand last term in the Lagrangian. $$\mathcal{L} =- \frac{1}{4} F_{\mu \nu}(x)F^{\mu \nu}(x) - \frac{1}{2} \alpha\Big ( \partial_\mu A^\mu(x)\Big)^2 +\sum_{f} \overline{\Psi}(x) ...
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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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Dependence of BRST Quantization on the Choice of Gauge-Fixing Function

There is a point which confuses me about BRST procedure. One shows that, if we define physical states as the ones that are annihilated by BRST charge $Q$, the scattering amplitudes don't depend on ...
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Is there a connection between these two results on soft hair on black holes?

In 2016 Strominger, Hawking and Perry published the paper "Soft Hair on Black Holes" proposing new results that could have importance to the study of the black hole information problem. One ...
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Path integral measure in Chern-Simons/WZW correspondence

The relationship between 3d Chern-Simons theory on the product of the disk and the real line ($D\times \mathbb{R}$) and the chiral WZW model on $S^1\times \mathbb{R}$ was shown in Elitzur et al Nucl....
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QCD vs. QED gauge invariance

Trying to understand the difference between QED and QCD gauge invariance treatment I found the following paper: https://arxiv.org/pdf/1101.3425.pdf I have the following questions: 1) I understand Eq....
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Gauge invariance in GR perturbation theory

I have been following this video lecture on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the ...
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Are all solutions of Maxwell's equation related by a gauge transformation?

Consider Maxwell's equation (without source): $$ \partial_\mu F^{\mu \nu} = 0 \implies \partial_\mu \partial^\mu A^\nu = \partial_\mu \partial^\nu A^\mu.$$ Can we find a pair of classical field ...
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QCD gauge invariant amplitude?

In order to keep the correct degrees of freedom (which are 2) for massless gauge fields one imposes, $$p^\mu \epsilon_\mu = 0 \tag1$$ Together with the gauge redundacy/equivalence relation, $$\...
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Does the Vector Potential Coil and Transformer Violate Special Relativity?

"Vector Potential Coil and Transformer" by M. Diabo et al reports the induction of voltage by a time varying curl free vector potential. A refinement reported in "Vector-Potential Coil and ...
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Beta function in the Standard Model

In Srednicki's textbook "Quantum Field Theory", Problem 89.4 asks us to compute the leading terms in the beta function for each of the three gauge couplings of the Standard Model. These gauge ...
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Hodge dual and QED

I was studying the paper Topological massive gauge theories in three dimensions by Deser, Jackiw and Templeton. In the paper, they use Hodge dual for some reason which I don't understand at all. So I ...
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Higgs-Mechanism: Why are gauge boson masses not protected by gauge symmetry

In non-spontaneously broken QFT like QED the gauge bosons cannot have a mass due to gauge symmetry (follows from Ward identity). Also they have only 2 polarizations. However in a spontaneously broken ...
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Duality transformations, such as between a massless scalar field and the Kalb-Ramond field

There is a kind of duality transformations between antisymmetric tensor fields which I learnt from a series of lectures by Gia Dvali on quantum field theory. I have not been able to locate a source ...
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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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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 ...
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Naive question about lepton/quark energy states

I understand there are 3 energy states for quarks and leptons (electron, muon, tauon... Up, charm, top etc.) And we have 3 forces (not including gravity, em, strong and weak). This naively seems ...
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How do we determine who eats who (or which field gets massive) in a Higgs phase transition?

Consider the following theory in which a $2$-form field $B_{\mu\nu}$ with field strength $P_{\alpha\mu\nu}=\partial_{[\alpha}B_{\mu\nu]}$ is coupled to a $3$-form gauge field $C_{\alpha\beta\gamma}$ ...
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Gupta-Bleuler Condition

I am studying quantum field theory using David Tong's notes available at http://www.damtp.cam.ac.uk/user/tong/qft.html and I am stuck at page 135 eq. $6.56$ I fail to see how the following equation: $...
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How many degrees of freedom in a massless $2$-form field?

Consider the Kalb-Ramond field $B_{\mu\nu}$ which is basically a massless $2$-form field with the Lagrangian $$ \mathcal L = \frac{1}{2}P_{\alpha\mu\nu}P^{\alpha\mu\nu}\,, $$ where $P_{\alpha\mu\nu} \...
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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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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 ...
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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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What does $\mathcal{N}$ refer to in Gauge theories?

Context: I am a second-year (undergraduate) physics major applying for a summer research position. The investigator is working on Quiver Gauge Theories and in response to my inquiry email he told me ...
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Spinning Particles in Background Gauge Fields

A simple model for a spinning particle is $$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$ with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\...
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What happens to the $U(1)$ factor in the $U(N)$ SYM gauge group of the AdS/CFT correspondence?

I'm learning about the AdS/CFT correspondence. I know that from the open string perspective, the dynamics on a stack of $N$ coincident $D3$-branes is given by a $\mathcal{N} = 4$ Super Yang Mills ...
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Higher form fermionic conserved currents

Higher form conserved currents have already been defined, such as those seen in Klebanov and Polyakov's work in 2002. There, the authors studied the $\text{AdS}_4$/CFT correspondence -- more ...
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What is the dimensionality of each part of a covariant derivative?

In the standard model, we have the following covariant derivative: $$D_\mu = \partial_\mu - ig_sG_\mu^a\lambda_a-igW_\mu^a\frac{\sigma^a}{2}-ig'B_\mu\frac{Y}{2}$$ If we let this work in on e.g. the ...
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Are Maxwell's equations “physical”?

The canonical Maxwell's equations are derivable from the Lagrangian $${\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ by solving the Euler-Lagrange equations. However: The Lagrangian above is ...
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Constrained Hamiltonian systems: spin 1/2 particle

I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action \begin{equation} S=-m\int ...
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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-...
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$\theta$ Term In 2d for $U(1)$ and $SO(2)$

For the $U(1)$ gauge theory in 2d, there can be a theta term $$\frac{\theta}{2\pi}\int_{M} dA$$ where $A$ is the $U(1)$ gauge field and $\theta\sim \theta+2\pi$. However, it is known that $U(1)=SO(2)...
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Is it possible to construct a quiver diagram for electromagnetism?

I have been trying to learn about quiver diagrams and quiver gauge theory for a summer project. All of the lecture notes/papers on the topic give example diagrams that are mathematically simple but ...
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Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)

In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has: $$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
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Non-local field redefinition and effects on path-integral measure

Consider the partition function $$ Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
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If monopoles are excised points in a $U(1)$ bundle, how are they affected by other charges?

We currently understand electromagnetism as a U(1) gauge theory. If you take a point out the space manifold (base space) you can create magnetic monopoles with integral charges by making non-trivial ...