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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1answer
521 views

Why is Berry connection a connection?

The Berry connection, following the derivation of the Berry phase for a non degenerate system, is $\mathcal{A}_{k}(\lambda) = i \langle n|\frac{\partial}{\partial \lambda^{k}}|n\rangle$ This result ...
2
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2answers
69 views

Violation of Derrick's theorem for finite energy, time independent solutions?

How are vortices the finite energy time independent solutions for 2+1 dimensions abelian higgs model? Doesn't it violate derricks theorem that there are no finite energy time independent solutions in ...
2
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1answer
122 views

How to calculate a real (vector) superfield in Wess-Zumino gauge

In Wess, Bagger's "SUSY and SUGRA", Eq. (6.6), they write down the real superfield in WZ-gauge as $$ V = -\theta\sigma^\mu \overline\theta v_\mu +\text{i}\theta\theta\overline\theta\overline\lambda - ...
5
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1answer
212 views

Locality in the coulomb gauge of classical electrodynamics

In the coulomb gauge, the equations that describe the dynamics of $\Phi$ and $\vec{A}$ simplify to: $$ \Delta \Phi = - \frac{\rho}{\epsilon_0} \\ \Delta \vec{A} - \frac{\partial_t^2}{c^2} \vec{A} = - \...
2
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2answers
327 views

Equivalence between Lorenz gauge and continuity equation

I want to show that the Lorenz gauge condition$$ \nabla\cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial\Phi}{\partial t}~~=~~0 \,,$$where $\mathbf{A}$ and $\Phi$ are the vector and scalar potential of ...
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2answers
279 views

Coulomb gauge in special relativity (for QFT)

I don't totally understand the procedure of Coulomb Gauge that we do in special relativity. Here is what I understood. We have: $$ F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$$ It ...
0
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1answer
471 views

Gauge fixing with vector potential: Coulomb gauge

There is something I would like to clarify with gauge fixing. In E.M, we can introduce the potential vector. As $div(\vec{B})=0$ we know that we can write $\vec{B}=\vec{curl}(\vec{A})$. But as $\...
1
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1answer
74 views

In simple words, why does a Lorenz Gauge does not have any physical effects?

I'm studying vector calculus via Arfken & Weber's "Mathematical Methods for Physicists", and, in page 40, he is deriving the electromagnetic wave equation. During the demonstration he states that ...
1
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1answer
66 views

Counting degrees of freedom without fixing the gauge?

In electrodynamics, the current-current interaction in the momentum space is described by $$p^2 A_\mu J^\mu = J_\mu J^\mu \, ,$$ where $J$ denotes an arbitrary external current. Since photon-...
2
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2answers
433 views

Gauge theory and eliminating unphysical degrees of freedom

In free space we can express Maxwell's equations as \begin{align} \varepsilon^{abcd}\partial_bF_{cd}=0 ~~\text{ and }~~ \partial_aF^{ab}=0 \tag{1} \end{align} where $F^{ab}=-F^{ba}$. The most general ...
4
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2answers
174 views

TQFT- Adding a $Q$-exact term which is equal to the action itself

It is known that Witten-type topological quantum field theories (TQFT) are invariant when $Q$-exact terms are added to the classical action, where $Q$ is the BRST charge. But for these theories, the ...
5
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1answer
198 views

Can we choose other than Gaussian integral for Faddeev-Popov gauge fixing?

for $U(1)$ field $A_\mu$ and its longitudinal gauge component $\partial_\mu \alpha(x)$, Faddeev-Popov gauge fixing written in Peskin (eq.9.56) is: $$ N(\xi)\int \mathcal{D}\omega\hspace{0.1cm}\text{...
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1answer
111 views

Choice of gauge for vector potential in Schroedinger Equation

When we learn about the Schroedinger equation of a particle in a magnetic field, we are told $ \displaystyle \frac{(p-eA/c)^2}{2m}\Psi = E\Psi $ The momentum operator is $-i\hbar \nabla$. The ...
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1answer
221 views

Aharonov-Bohm effect: a particle on a ring vs. a particle confined to the x-axis with periodic boundary conditions

Is there a fundamental difference between the Aharonov-Bohm effect in the two cases below? (a) A particle on a ring (polar coordinates) with a magnetic flux through the ring. (b) A particle confined ...
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4answers
492 views

The definition of the Lorenz gauge condition

The inner product of two vectors in space-time is: $$(x_1, y_1, z_1, t_1) \cdot (x_2, y_2, z_2, t_2) = x_1 x_2 + y_1 y_2 + z_1 z_2 - t_1 t_2$$ So $$(\frac{\partial }{\partial x}, \frac{\partial }{\...
6
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2answers
792 views

Question about physical degree of freedom in Maxwell Theory: Why Coulomb gauge can fix all redundant degree of freedom

Given $4$-potential $A^\mu(x)=(\phi(x),\mathbf{A}(x))$, the vacuum Maxwell equations: $$\nabla^2\phi+\frac{\partial}{\partial t}(\nabla\cdot \mathbf{A} )=0$$ $$\nabla^2 \mathbf{A} -\frac{\partial^2 \...
4
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1answer
319 views

Gauge-fixing term for the Hilbert action

In Yang-Mills theories, it is possible to fix the gauge directly in the action, via a gauge fixing term of the form (for EM for instance) $$S_{EM} = \int_D d^nx [-\frac{1}{4}F^{\mu\nu} F_{\mu\nu} - \...
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1answer
96 views

Solution to Schrödinger's equation for an electron gas in presence of an electric field by Gauge transform

Schrödinger's equation for an electron gas in presence of an electric field, which is given by: $$ \left( - \frac{\hbar^2}{2m} \mathbf{\nabla}^2 - q \mathbf{E} \cdot \mathbf{r} \right) \psi(\mathbf{r}...
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0answers
254 views

Ghost Fields in Abelian and Non-Abelian gauge theories

I have some questions about ghost fields in QED and in a non Abelian gauge theory: Does the fact that ghosts and photons are decoupled depend on the choice of the gauge-fixing function? In the Lorenz ...
2
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1answer
196 views

Ensuring Lorenz Gauge condition in Green Function solution

In the Lorenz Gauge, we can write Maxwell's equations as: $$\tag1 \Box A^\beta=\mu_0j^\beta.$$ We then go on to solve this by treating each component $A^\beta$ as an independent solution of the ...
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1answer
248 views

Delta function conversion into gauge-fixing Lagrangian in the path integral

So, I am at the moment working on gauge-fixing a path integral. The procedure involves adding a delta function $\delta g$ to the path integral (together with the faddeev-popov determinant, but that is ...
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3answers
2k views

Radiation gauge and choice of the gauge function

In electrodynamics, the scar potential $\phi$ and the vector potential $\textbf{A}$ satisfy the equations $$\frac{\partial}{\partial t}(\boldsymbol{\nabla}\cdot\textbf{A})+\nabla^2\phi=-\frac{\rho}{\...
2
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0answers
91 views

Axial gauge in lattice gauge theories with matter fields

I am studying the lattice gauge theory coupled with matter (Higgs) fields in the 1979 Fradkin-Shenker paper (https://inspirehep.net/record/132906). Taking the matter fields $\sigma$ and the link ...
3
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1answer
269 views

Quantization of a spin-1 field canonical commutator

This is a question regarding the spin-1 massive field commutator $[A_i(\mathbf{x},t),\Pi_j(\mathbf{y},t)]$, where $\Pi$ is the conjugate field and $A^\mu$ is the four-potential. My result was, $$[A_i(\...
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2answers
324 views

Can the Coulomb gauge and the Lorenz gauge be satisfied simultaneously?

The Coulomb Gauge: $\nabla \cdot A=0\\$ The Lorenz Gauge: $ \nabla \cdot A= { \mu }_{ 0 }{ \epsilon }_{ 0 }\frac { \partial V }{ \partial t }$ Can both of these gauges be satisfied for some ...
6
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2answers
658 views

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}(\...
4
votes
1answer
293 views

How to prove that Feynman propagators of $U(1)$ spin-$1$ field are equivalent in Coulomb gauge and $R_\zeta$ gauge?

How to prove that Feynman propagator are equivalent in Coulomb gauge and $R_\zeta$ gauge? (Be more specific, they are same when they contract with external current) In $R_\zeta$ gauge, the propagator ...
5
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1answer
216 views

Gauge transformation and large gauge transformation

Recently, Strominger posted his lecture notes on the infrared structure of gravity and gauge theory 1703.05448. In section 2.5, the equation (2.5.16) takes the following form $$e^2\partial_zN=A_z^{(0)...
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2answers
713 views

Gravitational waves in the Lorenz gauge?

The linearized Einstein field equations in the Lorenz gauge (with $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$ and $\bar h_{\mu \nu}=h_{\mu \nu}-\frac{\eta_{\mu \nu}}{2}h$) are given by: $$ \Box \bar h_{\...
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0answers
146 views

Polarization of gauge boson and gauge choices

Consider the following facts: For a particle with momentum $k$, the two transverse polarization vectors $\epsilon({\bf k}, \lambda_{1})$ and $\epsilon({\bf k}, \lambda_{1})$ are purely spatial and ...
2
votes
3answers
223 views

Gauge fixing for spin 1 field

Consider the Lagrangian for a spin 1 massless field with a gauge fixing term: \begin{equation} \mathcal{L}=-\frac{1}{4}F_{\mu\nu}^2-\frac{1}{2\xi}\left(\partial_\mu A^\mu\right)^2-J_\mu A^\mu, \end{...
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1answer
56 views

Gauge symmetries related to kinetic matrix determinant

Maybe it is a trivial question, but in my books I can not find an explanation why the kinetic matrix determinant gives us information about the gauge symmetries of the theory. The operator I'm ...
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1answer
93 views

How do gauge transformation imply gauge conditions?

In classical EM I understand the electric and magnetic fields are invariant under the potential transformations $\varphi\to\varphi - \partial_t\chi$ and $\mathbf{A}\to\mathbf{A} + \nabla\chi$. From ...
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0answers
180 views

Lorenz gauge and residual gauge freedom

I have this 4-potential: $A^\mu= a^\mu e^{ik^\alpha x_\alpha}$ Working in the Lorenz gauge, it is known that we can still make a transformation of this kind $$ A'^\mu = A^\mu + \partial ^\mu \theta \...
2
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0answers
36 views

Relevance of Gauge Transformations in Physical Interpretations of a System

In the simple example of a stationary electric field (and some other quantum mechanical examples) it is shown in the papers https://arxiv.org/pdf/physics/0506203.pdf https://arxiv.org/pdf/1302.1212....
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0answers
50 views

Proving that the 4-potential respects Lorenz gauge

We solved the maxwell equations $$ \Box \, A_{ret}^\mu =\frac{4 \pi}{c} J^\mu \tag{1}$$ using the Green function method finding $$ A^\mu = \frac{4 \pi}{c} \int d^4x' G_{ret} (x-x') J^\mu(x') \tag{2}$$...
2
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1answer
363 views

Remaining Gauge Freedom in Lorenz gauge

If one fixes the gauge in Electrodynamics to fulfill the Lorenz gauge $\partial_\mu A^\mu=0$, then the gauge scalar field $\chi$ has to fulfill (eq 3 page 5): \begin{equation} \partial^\mu\partial_\mu ...
9
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3answers
906 views

Can we do path integrals in gauge theories without fixing a gauge?

I am aware that when quantizing gauge theories with a path integral, one needs to add a gauge fixing term to avoid over-counting gauge related field configurations. From an aesthetic perspective, I ...
4
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0answers
419 views

Gauge invariance and the unitarity

I want to discuss the relation between the unitarity and the gauge invariance. Suppose we have for simplicity an abelian gauge theory (say, EM theory). We want to quantize it in terms of 4-potential $...
0
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1answer
199 views

Magnetic field, Vector potential

I have a question about the vector potential $\textbf A$ of a magentic field $\textbf B$, we know that $$\nabla\times \textbf A=\textbf B$$ and that $$\nabla ^2 \textbf A = -\mu _0 \textbf j. $$ ...
10
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2answers
1k views

A question on gauge fixing

As I understand it, a physical theory that has a gauge symmetry is simply one that has redundant degrees of freedom in its description, and as such, is invariant under a continuous group of (in ...
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1answer
148 views

Energy Density in Coulomb Gauge [closed]

I want to show that the energy density $$\mathcal{H} = \frac{1}{8 \pi}(\vec{E}^2 + \vec{B}^2)$$ of the EM field can be written as the following in the Coulomb gauge: $$\mathcal{H} = \frac{1}{8\pi}(\...
0
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1answer
435 views

How use Lorentz Gauge in Einstein tensor?

i have the Einstein tensor in terms h of first order $$ \begin{eqnarray} \label {eq: wf5o} G_{\beta \mu}& = & -\frac{1}{2}\left[ \bar h_{\beta \mu},_{\lambda}^{\;\;,\lambda}+\eta_{\beta\mu}\...
0
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1answer
662 views

Discontinuity of the derivative of vector potential

On p.251 of Introduction to Electrodynamics by Griffiths, there is the formula for the discontinuity of the derivative of vector potential across a surface: $$ \frac{\partial \textbf{A}_{\text{above}}}...
3
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1answer
254 views

Laplacian of Lorenz gauge magnetic potential

My textbook, Gettys's Physics (Italian language edition), says that the Lorenz gauge choice uses the magnetic vector potential $$\mathbf{A}(\mathbf{x},t):=\frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\...
5
votes
3answers
483 views

How fundamental is the transversality condition in QED?

This question is probably answered somewhere in textbooks, but I haven't got there yet, sorry for my ignorance in advance. There is a famous transversality condition in E&M and QED $$\vec{k}\...
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1answer
76 views

Different Hamiltonians because of different choice of magnetic vector potential

I'm talking Quantum Mechanics here. Let's say that we have a constant magnetic field $\mathbf{B} = (0,0,B)$. Then I can pick two different vector potentials: $\mathbf{A} = (\ -\tfrac{1}{2}By ,\ \...
2
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1answer
114 views

PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2', '...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix ...
1
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1answer
613 views

Residual Gauge Freedom [closed]

How are we still left with one Residual Gauge Freedom in the choice of Electromagnetic Potential after having already exploited the Gauge Freedom once. As is mentioned in Halzen and Martin Section 6.9....
7
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1answer
387 views

For a constant magnetic field, is there a gauge with both canonical momenta conserved?

To describe a constant magnetic field $\mathbf B=(0,0,B)$ (ignoring the motion along the $z$ dimension) within hamiltonian (or quantum) mechanics, one needs to choose a gauge. One common gauge is the ...