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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How to apply the Faddeev-Popov method to a simple integral

Some time ago I was reviewing my knowledge on QFT and I came across the question of Faddeev-Popov ghosts. At the time I was studying thеse matters, I used the book of Faddeev and Slavnov, but the ...
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1answer
988 views

Faddeev-Popov Gauge-Fixing in Electromagnetism

Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. ...
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3answers
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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 ...
13
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2answers
545 views

Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
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Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics

How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics. I need an clear argument that involves the ...
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2answers
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What is a gauge in a gauge theory?

As I study Jackson, I am getting really confused with some of its key definitions. Here is what I am getting confused at. When we substituted the electric field and magnetic field in terms of the ...
11
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1answer
2k views

Gauge fixing and degrees of freedom

Today, my friend (@Will) posed a very intriguing question - Consider a complex scalar field theory with a $U(1)$ gauge field $(A_\mu, \phi, \phi^*)$. The idea of gauge freedom is that two solutions ...
10
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3answers
992 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 ...
10
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2answers
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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 ...
10
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1answer
528 views

Yang Mills Hamiltonian: why do we use the Weyl's temporal gauge?

Do you know why in the quantization of $SU(2)$ Yang-Mills Gauge Theory, it is always chosen the Weyl (temporal) gauge to derive the Hamiltonian? Is it possible to fix another gauge?
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431 views

Why the extra term $\frac{1}{2}(\partial_{\rho}A^{\rho})^2$ in the photon Lagrangian?

In my quantum field theory class we have been told to use this Lagrangian for the photon field $$\mathcal{L}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta} -\frac{1}{2}(\partial_{\rho}A^{\rho})^2.$$ but ...
9
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Why is $p_y$ conserved in the Landau gauge when we know the electron moves in circles?

Considering the cyclotron in $xy$-plane where the magnetic field is $\vec{B}=(0,0,B)^{T}$. In the Landau gauge, we have $\vec{A}=(0,Bx,0)^T$ and we obtain the Hamiltonian $$H=\frac{\hat{p}_x^2}{2m}+\...
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864 views

Why do we use gauges in Maxwell equation?

While solving the Maxwell's equation we often use the Lorenz or Coulomb gauge, but why is that? Are the equations unsolvable if the gauge is not fixed? Or is it just for the simplicity?
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Intuition for gauge parallel transport (Wilson loops)

I'm looking for a geometrical interpretation of the statement that "Wilson loop is a gauge parallel transport". I have seen QFT notes describe U(x,y) as "transporting the gauge transformation", and ...
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1answer
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Clarifications needed on Gauge Fixing and Ghosts [closed]

The first time some kind of gauge fixing appears is during the Gupta-Bleuler procedure, which is used to be able to quantize the photon field: The basic gauge invariant Lagrangian leads to $\Pi_0=0$ ...
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743 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_{\...
8
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3answers
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Gauge fixing choice for the gauge field $A_0$

In many situations, I have seen that the the author makes a gauge choice $A_0=0$, e.g. Manton in his paper on the force between the 't Hooft Polyakov monopole. Please can you provide me a ...
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1answer
548 views

Why should it be allowed to set the einbein to unity?

The Polyakov action for a massive free point particle with worldline $\gamma$ is given by $$ S[\gamma] = \frac{1}{2}\int_\gamma e \biggl(\frac{1}{e^2}\dot{x}^2 - m^2\biggr)\mathrm{d}\tau $$ where $e$...
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2answers
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Are gauge choices in electrodynamics really always possible?

If $B$ is magnetic field and $E$ electric Field, then $$B=\nabla\times A,$$ $$E= -\nabla V+\frac{\partial A}{\partial t}.$$ There is Gauge invariance for the trnasformation $$A'\rightarrow A+\...
8
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1answer
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How do I derive the Lorenz gauge from the continuity equation?

I was reading my old electromagnetics book (Elements of Electromagnetics, by Sadiku, 3rd edition) and after the author explained what the Lorenz gauge is mathematically and why it is useful in ...
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2answers
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History of the names “Feynman-gauge” & “Landau-gauge”. How arised & how settled?

Edit: Use this PO.org question instead. Warning: Students, stay away from antiquities. The aim to learn is to survive. Hi. Today the nomenclatures Feynman gauge and Landau gauge seem established, ...
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2answers
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Showing that Coulomb and Lorenz Gauges are indeed valid Gauge Transformations?

I'm working my way through Griffith's Introduction to Electrodynamics. In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric ...
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2answers
204 views

Coordinate conditions in black hole solutions

The most general metric for a static and spherically symmetric metric is given by: $$ ds^2 = e^{2\gamma(u)}dt^2 - e^{2\alpha(u)}du^2 - e^{2\beta(u)}d\Omega^2 $$ I have the freedom of choosing ...
7
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2answers
668 views

Counting massive degrees of freedom after gauge fixing

Consider the theory of scalar QED with the Lagrangian $$\mathcal{L} = - \frac14 F^{\mu\nu} F_{\mu\nu} + (D^\mu \phi)^* (D_\mu \phi) - m^2 \phi^* \phi \tag{1}$$ where $\phi$ is a complex scalar field ...
7
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1answer
422 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 ...
7
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2answers
388 views

Schwinger-Dyson equations in the Coulomb gauge

Introduction As far as I know (and please correct me if something is wrong!) the usual narrative to deal with perturbation theory in QED with the Coulomb gauge goes as follows: First, the gauge ...
7
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1answer
637 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 ...
6
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2answers
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What is the physical meaning of Lorenz gauge condition? [closed]

What is the physical meaning of Lorenz gauge condition? And what part of the solutions we throw?
6
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2answers
418 views

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} \...
6
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3answers
616 views

Can I call additional conditions on potentials a Gauge choice?

Let's say I have an electromagnetics problem in a spatially varying medium. After I impose Maxwell's equations, the Lorenz gauge choice, boundary conditions, and the Sommerfeld radiation condition, I ...
6
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1answer
719 views

What is the constraint on the Gauge Potential in the Covariant Gauges?

One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term \begin{equation} -\frac{(\partial_\mu A^{\mu})^2}{2\xi} \end{equation} to the Lagrangian. ...
6
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1answer
237 views

What exactly are the sections in gauge theories?

In trying to understand precisely how fiber bundle theory maps to physical models, I came across this quotation: We can think of the elements of the principal bundle as generalized frames for the ...
6
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2answers
712 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}(\...
6
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3answers
815 views

Wess-Zumino Gauge in non-Abelian supersymmetric theory

I've got a question concerning non-Abelian supersymmetric gauge theories. Consider supersymmetric non-Abelian theory realized on chiral superfields $\Phi_i$ in a representation $R$ with matrix ...
6
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4answers
155 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 ...
6
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2answers
684 views

Why can the Lorenz gauge condition always be fullfilled?

Why is the Lorenz gauge condition always possible for classical electromagnetic fields? So far I can only understand the following: If we perform a gauge transformation $A\mapsto A'=A+\mathrm{d}\...
6
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1answer
435 views

Proof that we can always find a gauge transformation such that $A_0=0$?

I'm trying to follow Coleman's proof from his lectures "Aspects of Symmetry" on page 200-201. He proofs it is always possible to work in the temporal gauge for a general Yang-Mills-Higgs theory. I ...
6
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1answer
521 views

QED lagrangian: gauge fixing term

I have a question about the structure of the QED lagrangian, in particular the free photon lagrangian which is contained in it. My premise is: I only know how to exploit canonical quantization in ...
6
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2answers
882 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 \...
6
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1answer
281 views

Is the gauge fixing $\partial_\mu A^\mu + \gamma A_\mu A^\mu=0$ used in the literature and does it have a name?

In an exercise for a course on Gauge Theories, I was asked to derive the action of QED with the method by Faddeev and Popov, using the following gauge-fixing function: $$F(A) = \partial_\mu A^\mu + \...
6
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1answer
281 views

Equations of motion from Polyakov action, before choosing the conformal gauge

My question is the following: It is usual in the standard textbooks to firstly choose a gauge (usually the conformal gauge) and then extract the equations of motion from the Polyakov action by ...
5
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2answers
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What is conformal gauge?

I often see in physics articles on gravity such notion as conformal gauge and Weyl transformation. They use Conformal gauge to change coordinates to transform metrics from arbitrary $$ds^2=g_{\mu \...
5
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2answers
214 views

Confusion on Maxwells equations and Gauge Transformations

I know a little bit about electrodynamics but I don't understand the validity of Gauge Transformations. In particular I am confused on how the theory can be consistent among different gauges, in ...
5
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1answer
230 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
1k views

Equivalency of Gauge Conditions

How is the Lorenz gauge condition $\partial_\mu \overline{h}^{\mu \nu}=0$ equivalent to the harmonic gauge condition $\Box x^\mu=0 $?
5
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3answers
524 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}\...
5
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1answer
269 views

Unitary gauge for non-abelian case

I'm reading Chapter 19 of Mandle and Shaw's Quantum field theory. In the first section it is explained that one can go with a $SU(2)$ followed by a $U(1)$ transformation from $$\begin{bmatrix}\eta_1(x)...
5
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1answer
136 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 ...
5
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2answers
350 views

Trace-reversed EFE and linearized gravity

I have a question about the linearized Einstein Field Equations, and in particular about the Newtonian limit. It goes as follows. If one uses the trace-reversed form of the EFE for the 00-component ...
5
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2answers
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Landau level degeneracy in symmetry gauge, finite system

As we know, Landau level degeneracy in a finite rectangular system is $\Phi/\Phi_0$, where $\Phi=BS$ is the total magnetic flux and $\Phi_0=h/q$ is the flux quanta. This can be easily derived using ...

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