Questions tagged [gauge]

Use this tag to discuss gauge-fixing conditions, as in the phrase 'choosing a gauge', such as, e.g. the Lorenz gauge, Coulomb gauge, Feynman gauge, Landau gauge, axial gauge, temporal gauge, light cone gauge, etc.

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11 views

Integral of gauge field bulk to boundary propagator in AdS

I'm studying from the book "Introduction to the AdS/CFT Correspondance" by Horatiu Nastase. In page 190, he defines the gauge field bulk-to-boundary propagator in Euclidean $AdS_{d+1}$ given ...
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71 views

Why does the Lorenz gauge condition impose a relationship between $\vec A$ and $V$?

background I read in the book Introduction to Electrodynamics by D. J, Griffiths the process of solving the four Maxwell equations in the most general form: Firstly, the four equations were simplified ...
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57 views

Faddeev-Popov Gauge Fixing Procedure

I want to know that does $F^{a}[A_{\mu}] = 0$ condition used in Faddeev-Popov Quantization has unique solution $A_{\mu}$ or is it $F^{a}[A^{\theta}_{\mu}] = 0$ should have unique $\theta$ as solution ...
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91 views

Quantising Yang-Mills, analogy to Gauss' law

In David Tong's lecture notes on Gauge Theory, in the section on 'Quantising the Colour Degree of Freedom', the following action is discussed, $$ S_{w}=\int d \tau i w^{\dagger} \frac{d w}{d t}+\...
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62 views

Ward Identity and Gupta-Bleuler condition

Reading David Tong notes on QFT, he mentions about Gupta-Bleuler condition $$\partial^{\mu}A_{\mu}^{+}|\Psi\rangle=0\tag{6.54},$$ which makes sure that matrix elements vanish,$$\langle \Psi|\partial_{\...
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127 views

Transverse traceless gauge in linearized GR

I'm reading about gravitational waves and I'm wondering how we know we can always go to the transverse and traceless gauge? Going to the de Donder gauge is fine, I can follow that, but then showing ...
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1answer
72 views

Hamiltonian for charge particle in electromagnetic field in dipole approximation

Why can the Hamiltonian $$ H = \frac{1}{2m}\left(\vec{p}-q\vec{A}\right)^2+q\Phi + V $$ be transformed to $$ H = \frac{1}{2m}\vec{p}^2 - q \vec r \vec E + V $$ in the dipole approximation, in which ...
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136 views

Is special relativity falsified by the Aharonov Bohm effect?

The Lorenz gauge is the only Lorentz invariant electrodynamic gauge. If the vector potential has physical meaning, as in the Aharonov–Bohm effect, then the gauge condition can not be arbitrarily ...
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24 views

Singularities in gauge-fixing conditions and topological defects

I am studying the 't Hooft's paper "Topology-of-the-gauge-condition-and-new-confinement" https://doi.org/10.1016/0550-3213(81)90442-9 and there are several points which I would like to ...
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53 views

Examples of Chern number calculations where more than two $U(1)$ gauge of wavefunction has been used

While computing the Chern number of electronic wave functions \begin{align} \left|\psi\right\rangle = \begin{pmatrix}\cos\left(\frac{\theta}{2}\right) \\ \sin\left(\frac{\theta}{2}\right)e^{i \phi} \...
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52 views

How does the gauge-fixing term break gauge invariance?

Given the gauge-fixing term $\mathcal{L}_\mathrm{gf}=-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$ and the gauge transformation $A_\mu\mapsto A_\mu+\partial_\mu\alpha$, how does the term break gauge ...
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60 views

Existence of $(\tilde{V},\mathbf{\tilde{A}})$ for each $(V,\mathbf{A})$ that gives rise to the same EM-fields

The electric field $\mathbf{E}$ and the magnetic induction $\mathbf{B}$ can be parameterized in terms of potentials $V$ and $\mathbf{A}$: $$ \mathbf{E}=-\nabla V-\frac{\partial \mathbf{A}}{\partial t},...
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63 views

Is there a physical mechanism explaining the link between Gauge-Bosons and local Gauge-Invariance?

Imposing local gauge-invariance naturally couples e.g. a charged fermion-field to the electromagnetic field. To my understanding local gauge-invariance is imposed because a gauge in one system should ...
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59 views

Uniqeness of Coulomb gauge

Say I have some magnetic vector potential $A$ which is not in Coulomb gauge, meaning $\nabla \cdot A \neq 0$. I can set it to Coulomb gauge by adding some scalar potential function $\nabla \phi$ (and ...
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49 views

Why doesn't choosing an arbitary divergence of $\vec A$ change electric and magnetic field?

I am currently reading the 2nd volume of Feynman Lectures and I am stuck in the part where he solves the Maxwell's equation to find the potentials and wave equations. I can't understand the thing he ...
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88 views

Question about Faddeev-Popov gauge-fixing in Schwartz textbook

I am trying to understand equation (25.91) from Schwartz's Quantum Field Theory textbook. The goal is to gauge-fix the path integral for Quantum chromodynamics using the Faddeev-Popov trick. Briefly, ...
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50 views

Regarding Gauge fixing in General relativity

I have been reading about gauge fixing in general relativity but I have failed to understand what does choosing a particular gauge really mean? Is it just a choice of a coordinate system where a ...
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85 views

Gauge fixing conditions in general relativity

Is there a limit to gauge fixing conditions we can impose in gravity ? I have seen two gauge fixing conditions. The DeDonder gauge $\partial_\mu g^{\mu\nu}$ and then in 3+1 formalism the gauge fixing ...
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124 views

What is the physical interpretation of the Lorenz gauge condition?

If we integrate both sides of the Lorenz gauge condition, $\nabla \cdot \mathbf{A} = -\frac{1}{c^2}\frac{\partial \phi}{\partial t}$, over a small volume (free of charges for simplicity), we get: $$ \...
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67 views

Eliminating residual gauge in BRST quantization of Yang-Mills theory

I would like to know if there is a procedure to completely fix a gauge, which I believe we must do in order to make sense of the path integral? In chapter 74 Sredniki introduces the Lagrangian $$ \...
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True Hamiltonian in Geometrodynamics

If I set $N=1$ and $N^a=0$ in the Einstein-Hilbert action $$S[N,N^a,q_{ab}]=\int\sqrt{\mathrm{det}(g)}\,R\,\mathrm d^4x \,\text{,}$$ expressed in terms of ADM variables, then $N$ and $N^a$ are no ...
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67 views

Setting $N=1$ and $N^a=0$ in the Einstein-Hilbert action

In the ADM formalism of general relativity, one obtains a $3+1$ split of spacetime by setting $$\mathrm d s^2=(-N^2+N_a N^a) \,\mathrm d t^2 + 2N_a\,\mathrm d t\,\mathrm d x^a + q_{ab} \,\mathrm d x^a\...
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60 views

Gauge transformation of the gauge-fixing term in the QED action

In the classroom my teacher stated that the Gauge-fixing term in the action $$\frac{1}{2\alpha}\int d^4x (\partial_\mu A^\mu(x))^2$$ transforms under $A_\mu(x) \rightarrow A_\mu(x)+\partial_\mu \...
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45 views

$R_\xi$ gauges and the EM-field

$R_\xi$-gauges are said to be a generalization of the Lorenz gauge. I dont quite get why we add the term $$ \mathcal L_{GF} = - \frac{(\partial_\mu A ^\mu)^2}{2\xi} $$ to the Lagrangian. If i ...
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101 views

Gauge invariance for classical fields

I recently did some exercises in classical field theory and tried to think deeply about the gauge symmetry of the free electromagnetic field described by the Lagrangian $$ \mathcal L = -\frac 1 4 F^{\...
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47 views

Gauge invariance and the degrees of freedom [duplicate]

I am currently starting to learn some QFT. And I am very confused by the fact that the freedom to choose a gauge for a theory controls the physical degrees of freedom. Please correct me if i am wrong ...
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40 views

What are all the gauge symmetries & derivatives of the QED lagrangian?

I find that the gauge symmetries of the lagrangian are a topic that gets obfuscated quite a bit. I'm trying to understand the big picture of this in QED. My understanding is that: Gauge derives its ...
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23 views

Relation between polarization vector sums and propagators and the use of ghosts

It is commonly argued that, according to the BRST treatment, ghosts cannot appear as initial or final states, and that the only states that can be used as initial or final states are physical gauge ...
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32 views

What are the main issues with gauge theories of gravity? [duplicate]

From what I understand, the obvious problems with a gravity being a posed as a gauge theory of the Lorentz group are the residual tetrad fields having no obvious gauge theoretic interpretation, as ...
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1answer
90 views

Where does the Goldstone couplings go in the unitary gauge?

Imagine that you have some model with an enlarged scalar potential, such that there is, for instance, a quartic coupling $\kappa$ between the Higgs charged component and three other scalars, which do ...
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25 views

Difference in the Lagrangian when working with different gauges

like, I have a Standard model Lagrangian, then what will be the differences in interaction terms and in interaction vertices if we work with the Feynman gauge and unitary gauge?
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44 views

Understanding the gauge condition in light-cone gauge

When going over Cambridge String Theory Notes (page 38) I came across the following gauge conditions: $$X^+ (t, \sigma) = x^+(t)\hspace{5mm}, \hspace{5mm} P_−(t, \sigma) = p_−(t) \tag{1} $$ which is ...
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41 views

Lagrangian density for a free EM field

I wanted to derive the Lagrangian density of a free EM field $\mathcal{L}$ in the Lorenz gauge as: $$ \mathcal{L} =- \frac{1}{4} F_{\mu \nu} F^{\mu \nu} = \frac{1}{2} A_{\mu} \partial^2 (g^{\mu \nu}- ...
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78 views

For the free electromagnetic field, is it possible make single gauge transformation to achieve $\phi={\bf \nabla}\cdot{\bf A}=0$?

For any electromagnetic field, it is easy to impose the Coulomb gauge condition ${\bf\nabla}\cdot{\bf A}=0$. To start with, if ${\bf \nabla}\cdot{\bf A}_{\rm old}\neq 0$, the trick is to make a gauge ...
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32 views

Why retardation showed up for Lorentz gauge but not coulomb gauge?

There's two common gauge used for EM, Lorentz gauge and Coulomb gauge However, if you look at the solution of those gauge, the retardation only showed up for Lorentz gauge, but not for coulomb gauge. ...
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48 views

Consistency condition for Yang-Mills on a Torus

So I was recently studying 't Hooft's paper on self-dual solutions to Yang-Mills on $\mathbb{T}^4$. So the basic idea is that you consider a box with periodic boundary conditions and then you impose ...
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50 views

Is Lagrange multiplier an arbitrary gauge?

A related post might be found here: Is the gauge transform field in electromagnetism a Lagrange multiplier? Consider the case of Lagrange multiplier $L=f(x)+\lambda g(x)$ where $g(x)=0$ was a ...
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85 views

Path integral formulation of an Abelian Field Theory, unclear identity

TL;DR: How exactly does one come to this identity $$\int\mathcal{D}G(A^\alpha)\delta(G(A^\alpha)) = \int\mathcal{D}\alpha(x)\delta(G(A^\alpha)) \mathrm{det}\left(\frac{\delta G(A^\alpha)}{\delta\...
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100 views

Landau levels in rotationally invariant gauge

I try to find wave-function of electron in external constant magnetic field in gauge $$A=\frac{B}{2}(-y,x,0).$$ I substitute anzats, $\psi=e^{-i\omega t}e^{ip_zz}F(x,y)$. Then, I rewrite equation in ...
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50 views

Coulomb gauge in magnetostatics should give divergence-free vector potential

Say we're dealing with magnetostatics ($\vec{\nabla} \cdot \vec{j} = 0 $). If we define $\vec{A}$ to satisfy $\vec{B} = \vec{\nabla} \times \vec{A}$, and we take the assumption that $\vec{\nabla} \...
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208 views

What does adding a gauge fixing term $-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$ really mean?

Given any electric and magnetic field (or $F_{\mu\nu}$), there is always some freedom in defining what $A_\mu(x)$ should be. In fact, there are infinite choices for $A_\mu(x)$. This is because for an ...
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48 views

Independence of $S$-matrix in QED of the gauge choice

The Feynman rules in QED often use different expressions for the free photon propagator (e.g. Feynman gauge, Landau gauge, and others). Is there a textbook on the subject which explicitly checks ...
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44 views

Gauge transformations of potentials with a step function

In have a question about the following exercise. I am given the following expressions for the scalar and vector potential, $$\vec{A}(\vec{r},t) = \frac{\sigma}{\epsilon_0}\left[\frac{x}{c}\hat{x} + tU(...
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86 views

When one says that Yang Mills theory is a non-abelian field theory, what specific gauges and gauge transformations are implied in this statement?

What specific gauges and gauge transformations are implied when one states that the order of such gauge groups are vital? Can this please be explained as simple as possible (:
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147 views

Gauge transformation issues of Proca Lagrangian

It seems to be trivial to claim that the mass term in Proca Lagrangian is NOT "gauge invariant". The claim show up in Wikipedia or section 6.3 in Greiner's book "Field Quantization". But I'm puzzled ...
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52 views

Is comoving coordinate a gauge choice?

Consider the Roberston Walker metric in 4D, $$ds^2 = -dt^2 + a^2(t) \left( \frac{dr^2}{1-kr^2} +r^2 d\Omega^2_2 \right)$$ Now, if we consider the collapse of a spherically symmetric fluid, spherically ...
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27 views

What is the way to express Yang-Mills symmetry groups without gauges?

Given a Yang-Mills theory such as $SU(3)$ which has 8 gluons. After we gauge-fix this theory, it no longer has $SU(3)$ guage symmetry. Yet, we still use the group constants and the 8 types of gluons ...
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182 views

How to show the that Lorenz gauge is true given that the scalar and the magnetic vector potentials are not unique?

So I understand that we select the divergence of A (magnetic vector potential) to be: $$\frac{1}{c^2}\dot{\phi} + \nabla\cdot\vec{A} = 0.$$ The Lorenz gauge (1). because of the symmetries in ...
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205 views

Why is the gauge-fixing condition squared in the QED Lagrangian?

Consider the free Maxwell Lagrangian: $$L= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}. $$ As we know, the gauge symmetry $A_{\mu} \rightarrow A_{\mu}+\partial_\mu \lambda$ must be fixed when quantizing the ...
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50 views

Quantisation of gauge theory with minimal coupling

I have a question on the quantization of the gauge theory with minimal coupling term. What I understand is that if one is given an action $$ S=-\int d^4 x \frac{1}{4}F^2 \tag1 $$ Since this action has ...

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