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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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longitudinal and transverse components in higher dimensions

I am familiar with the Helmholz decomposition of a vector field in three dimensions: $$\vec{V}=\vec{\nabla}\wedge\vec{A}+\vec{\nabla}\phi$$ But I am interested to show that something similar can be ...
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Reference for proof of renormalizability

I have been trying to truly understand the renormalizability of quantum (i.e., without anomalies) gauge theories (after which I will focus on the case with spontaneous symmetry breaking). The problem ...
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What's the meaning of “inequivalent quantizations”?

The notion "inequivalent quantizations" is regularly used when topological terms are discussed. From what I've gathered so far, "inequivalent quantizations" means that there are different quantum ...
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What happens to large gauge transformations in gauges different from the temporal gauge?

There are already several questions regarding the meaning and definition of large gauge transformations. Discussions of large gauge transformations typically only happen in the context of the ...
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How to check for invariance in Lagrangian after gauge transformation?

If I have the Lagrangian density: $$\mathcal{L}=\left(\partial_{\mu} \phi^{*}\right)\left(\partial^{\mu} \phi\right)-m^{2} \phi^{*} \phi$$ How can I show it is invariant under the following gauge ...
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Soft charges and infrared physics in standard gauge and gravitational physics [closed]

Recently, we have learned that there are an infinite number of possible soft charges in black hole physics. It turns that the commonly assumption of that infrared physics is completely independent of ...
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In what sense $Z_\mu^0$ is orthogonal to $A_\mu$?

I am reading Standard model. Please explain in what sense the $Z$-boson $$Z_\mu^0=(g^2+g^{\prime 2})^{-1/2}(g A^3_\mu-g^\prime B_\mu)$$ is an orthogonal linear combination of the photon $$A_\mu=(g^2+...
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What is a gauge (for someone who has not studied gauge theory)? [duplicate]

I am taking a Quantum Mechanics II course and we were studying the relativistic corrections to the hydrogen atoms in perturbation theory. I was looking at the assignment, and a question is as follows: ...
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Explicit counting of gauge field degrees of freedom

Consider a connection on a principal $U(1)$-bundle $A_\mu$ over the flat base manifold $M_4$. The action of the theory is described in terms of the curvatures of such connection coupled to some source ...
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How do interaction terms appear in the Lagrangian?

How does forcing the Lagrangian to be invariant under $U(1)$ group give rise to the electromagnetic interaction term?
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Computation of the Faddeev-Popov determinant

I am studying the Faddeev-Popov method from Pokorski's Gauge Theory book, and I am puzzled by what happens in the step below. He is writing the group element $g = 1 - i T_a\ \Theta^a(x)$ in a ...
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Fields and gauge transformations vanishing at infinity

I find that, in field theory, it is very often assumed that the fields (classical) vanish at infinity. The same assumption is also applied to gauge transformations, for example, when saying that the ...
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Landau levels in symmetric gauge

On Shankar’s Quantum Many body page 394 it says for one electron in a magnetic field, ignoring spin, $$H_0=\frac{(\bf{p}+e\mathbf{A})^2}{2m}$$ $$e\mathbf{A}=-\frac{\hbar}{2l^2}\hat{z}\times \mathbf{...
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Can we do one-loop integrals in the unitary gauge?

$\hspace{5cm}$ Imagine we want ot compute one of the diagrams for the self-energy of the quark $u$, with external momentum $p$. Inside the loop, we would have a $W^+$ and a $d$-quark propagator, with ...
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Is it possible to have a complex gauge field?

Beyond the obvious fact that the particles in the standard model described by gauge fields do not have an anti-particle pair, is there a reason why a complex gauge field is typically not considered? ...
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Why is the Coulomb Gauge enough to fix extra degrees of freedom?

In classical electrodynamics, we have after the Coulomb gauge is applied: $$ \Delta U = -\frac{\rho}{\epsilon_0} $$ $$ \Box \vec{A} = \mu_0 \vec{j}-\frac{1}{c^2} \vec{\nabla} \frac{\partial U}{\...
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Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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Inconsistency between $d_A = d + A \wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative. $d_A A=F$ --- @...
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A question on supersymmetry variation of the Wilson loop in $\mathcal{N}=4$ SYM

The Wilson loop in $\mathcal{N}=4$ SYM is $$W=\frac{1}{N}tr P \exp \int ds (i A_\mu(x) \dot{x}^\mu+\Phi_i(x)\theta^i|\dot{x}|).\tag{2.3}$$ In order to check whether this operator is supersymmetric I ...
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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 ...
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Why is the relation $M_W=M_Z\cos\theta_W$ true only at tree-level?

In Glashow-Weinberg-Salam electroweak theory, the relation $$M_W=M_Z\cos\theta_W\tag{1}$$ is said to be remain true only at the tree-level; it receives corrections from the loop diagrams. See here. ...
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Particle on a circle with magnetic flux$.$

I am trying to understand the model studied in 1905.09315 §2, to wit, a $0+1$ dimensional theory with target space $\mathbb S^1$ with non-trivial magnetic flux: $$ \mathcal L=\frac12m\dot q^2-\frac{i}{...
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Particles and associated fields

When a particle is associated with a field. 1) It is said that the excitation of the field produces the particle, 2) it is also said that when the field is quantized, the quanta of the field is ...
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Reference request for Gribov ambiguity

I was hoping to find a reference (book or article) with a good introduction to the Gribov Ambiguity in non-abelian gauge theories. I’ve looked through QFT books by Schwartz and Srednicki, Rubakov’s ...
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Gribov's phenomenon

In the well known textbook by Itzykson-Zuber "Quantum Field Theory" there is a discussion of the Gribov phenomenon in non-abelian gauge theories (see Section 12-2-1). To my taste, the discussion ...
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Polar representation of complex scalar field in spontaneous symmetry breaking

In Rubakov's "Classical Theory of Gauge Fields", in the chapter on Higgs mechanism, he mentions that you can switch to the polar form of the complex scalar field only for non-zero vacuum value of the ...
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Gauge invariance on Yang-Mills Lagrangian

How do I verify the invariance on Yang-Mills' Lagrangian: $$L = -\frac{1}{4} \sum_{a} \left(\partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^bA_\nu^c \right)^2$$ under the transformation:...
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Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source $$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$ My question is that what are the most simple form we can boil down this ...
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Classical Yang-Mills equation of motion with both electric and magnetic sources?

We know the classical Maxwell equation of motion (eom) with both electric and magnetic source can be written as: (1) Explicit form or more schematically as: (2) Differential form $$ d * F = * J_e $$...
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$U(N)$ & $SU(N)$ : What's the conceptual difference in Gauge Theory?

I know the mathematical difference that one means $ absolutevalue(det) = 1$ and one means det = 1 (rotation) and that ones the subgroup of the other and so on. But: has a local/gauged $SU(3)$ ...
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Mistake or Rewriting of Yang-Mills in Nakahara

I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form. $$ D * F =0 $$ and Bianchi identity $$ D F=0 $$ where $F= dA + A \wedge A$ and $D=d + [...
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Connection between gauge invariance and Lorentz invariance

This question is presented in the context of Weinberg's QFT book treatment, in particular considering the electromagnetism chapter. It begins in chapter 5 where Weinberg argues that in order to have ...
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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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Why aren't gravitons spin 1?

Expressing the metric as $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$, assuming $h_{mu \nu} \ll 1$ we can write the Einstein Hilbert action to leading order in $h_{\mu \nu}$ and quantize the ...
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Why does a triangle anomaly appear in a gauge theory?

I have read that when we construct a theory with abelian gauge symmetry there will appear some anomalies when we do the quantum correction to the theory. In 4D space such anomalies are explained by ...
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Relating the Yang-Mills field-strength to the Maxwell tensor in $SU(2)$ gauge theory

I'm studying topological monopoles in a $SU(2)$ Yang-Mills theory with spontaneous symmetry breaking, through the book "Topological Solitons", by Manton and Sutcliffe. In section 8.2, the authors ...
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What is $U(1)$ symmetry?

I saw there are three intrinsic symmetries in physics,U(1),SU(2) and SU(3).What's the U(1) symmetry talking about?I would appreciate it if you can give me some explaination.
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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 ...
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1answer
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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{...
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If the gauge symmetry is not broken by spontaneous symmetry breaking, what symmetry is broken?

In this post, the answer by buzhidao showed that the $U(1)$ gauge symmetry is not broken by spontanous symmetry broken and Higgs mechanism. What role does "spontaneously symmetry breaking" ...
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A Question about Wess-Zumino Gauge in Non-Abelian Gauge Theory

I have a question about the Wess-Zumino gauge in non-Abelian supersymmetric gauge theory. I am following BUSSTEPP Lectures on Supersymmetry. The spinortial field strength is defined as $$W_{\alpha}(...
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1answer
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Z and $\gamma$ bosons as mixtures of W and B: Part II

When it is said that the photon is ["a mixture of W and B"][1] ($B$ being a gauge field associated with the $U(1)$ hypercharge) I have one question on this: Why there isn't a boson directly ...
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Gauge fixing in canonical quantum gravity

In analogy with QFT, the partition function in canonical quantum gravity is defined as a functional integral over the metric tensor (which is now the quantum field), $$ \int \mathcal{D} g \mathcal{D}\...
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Z and $\gamma$ bosons as mixtures of W and B: Part I

When it is said that the photon is "a mixture of W and B" ($B$ being a gauge field associated with the $U(1)$ hypercharge) I have a question on this: When speaking of "mixtures", this is meant as ...
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Motivation for Weinberg angle in electroweak gauge interaction?

Suppose I have the following lagrangian If we only focus on the neutral current in the lagrangian: Where $L$ is defined as: And $Y_L$,$Y_{R}^{\nu}$,$Y_{R}^{e}$, are the hypercharge values of the ...
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Why do we require gauge symmetries to commute?

My question arises after reading the 87th page of Elementary particle physics in a nutshell by Tully: which is also given by the following link: https://books.google.se/books?id=vLy2YlkXZuEC&pg=...
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Why does the divergence of a QFT's coupling constant under RG flow trivialize the theory if it occurs in the UV but not in the IR?

When you first learn quantum field theory, at some point you calculate the beta function (to leading order) for a renormalizable coupling constant of some theory like $\varphi^4$ theory, Yukawa theory,...
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How do I show that the Lorenz gauge is consistent?

I have been asked to show that the Lorenz gauge condition, written as $$\nabla_T \bullet \vec{A} + \dfrac{1}{c^2}\dfrac{\partial}{\partial t}\Phi = 0$$ is mathematically consistent with the vector ...
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Noether current and continuity equation in classical scalar QED

Consider the following scalar QED model \begin{align} S = \int \mathrm{d}^{d+1} x\, \left\{-\left(\mathrm{D}_{\mu}\phi\right)^{\dagger} \left(\mathrm{D}^{\mu}\phi\right) -m^2 \phi^{\dagger}\phi - \...
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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 $SO(N)$ ...