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.
411
questions
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0
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40
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Possible mistake at Nakahara: $F_{\mu\nu}$ vanishing for a particular $A_\mu$
In equation $(1.293)$ of Nakahara's Geometry, Topology and Physics, he says that the Yang-Mills field tensor $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+g[A_\mu,A_\nu]$ for a gauge field
$$A_\mu=...
- 2,217
4
votes
0
answers
77
views
What is meant by Coulomb gauge not being Lorenz invariant? [duplicate]
This question is in the context of QFT. The notes says:
A disadvantage of working in Coulomb gauge is that it breaks Lorentz invariance.
What is meant by Coulomb gauge not being Lorenz invariant?
The ...
- 217
0
votes
0
answers
30
views
Is any choice of electric and vector potentials valid, as long as the correct gauge transformation exists?
For example, in general one would not choose a gauge with $\phi \neq 0$ and $\vec{A}=0$ because that simply gives you the electrostatic case where $\vec{B}=0$. However one can transform this into $\...
- 1,585
3
votes
0
answers
115
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Find smooth gauge numerically in the presence of symmetries
Let's assume a gapped two-band model in $2$D characterized by a Bloch-Hamiltonian $h(k_x,k_y)$ in the presence of reflection symmetry along the $x$ direction represented on the orbital degrees of ...
- 369
13
votes
2
answers
1k
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Trouble reconciling these two views on gauge theory
Very generally speaking, I view gauge theory as asking what local symmetries leave our theory invariant and then seeing the consequences. Thus, taking a look at the Lagrangian for electromagnetism, we ...
- 2,478
3
votes
3
answers
212
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When we solve the Maxwell equations for $(\phi,{\bf A})$ in a gauge, will the solution $(\phi,{\bf A})$ automatically obey the gauge condition?
As the title of the question suggest; how you could determine if a gauge fixing is a condition or a requirement. Let me explain.
Imagine you are working with Maxwell's Equations. By the definition of ...
- 1,315
0
votes
0
answers
40
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2D Quantum Hamonic oscillator in magnetic field with a shiftted position
Background
Consider a hole in a 2D parabolic potential in a magnetic field which is generate by the following gauge:
$$
\vec{A} = \left( - \frac{B_z y}{2}, \frac{B_z x}{2},0\right)
$$
Our quantum ...
1
vote
0
answers
47
views
Faddeev-Popov Method for Gauge Fixing in CFT (Light-ray Operators)
I was attempting to go through the paper by Petr Kravchuk and David Simmons-Duffin: https://arxiv.org/abs/1805.00098 where I encountered the following
Just below E.4, it is mentioned that for the ...
- 41
0
votes
0
answers
58
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Is there an error in this Wikipedia article: "Landau quantization"?
I am trying to do a homework, so I had to consult this Wikipedia article Landau Quantization where it is mentioned that for the symmetric gauge $\vec{\textbf{A}} = \frac{1}{2}(-\textbf{B}y, \textbf{B}...
1
vote
0
answers
52
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Gauge Fixing in Derivation of Lorentzian OPE Inversion Formula in 2D CFT
I have been looking through the following article: https://arxiv.org/abs/1711.03816 and wish to understand the derivation from scratch. The definition of conformal partial waves and the object of ...
- 41
1
vote
1
answer
86
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Why does $\mathbf{A}(x) = \frac{1}{2}(\mathbf{B}(x) \times \mathbf{x})$ work?
In my textbook, the identity for a possible vector potential
$$\mathbf{A}(x) = \frac{1}{2}(\mathbf{B}(x) \times \mathbf{x})$$ is used. How is this valid? If I compute the curl, I get
$$\mathbf{B}(x) \...
- 1,045
0
votes
0
answers
33
views
How to get retarded scalar potential in Coulomb electrodynamics and what's the use?
In Coulomb gauge electrodynamics with potential $(\phi,\vec{A})$ and source $(\rho,\vec{J})$ we obtain the Poisson's equation for the scalar equation and the wave equation with transverse current ...
- 91
4
votes
1
answer
108
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Why is $Z_3= Z_\xi$ in a non-abelian gauge theory?
In my lecture notes for a course on QFT it is said that, also in non-abelian gauge theories, the identity $Z_3 = Z_\xi$ holds, where those renormalization parameters belong respectively to the ...
- 220
1
vote
0
answers
22
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Choosing the Gauge in Randall–Sundrum model
Given the $5D$ conformally flat metric in Randall–Sundrum model
$ds^2=e^{-2A(y)}\eta_{MN}dx^Mdx^N$, where $x^5=y$
The Einstein tensor is given by
$G_{MN}=\frac{1}{2M^3}(\Lambda g_{MN}+T_{\mu\nu}\delta^...
- 39
3
votes
2
answers
544
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Quantum Theory of Radiation Enrico Fermi 1932
I was reading Fermi's review on Dirac's "Quantum Theory of Radiation", which he published in 1932. I was unable to know why he expressed electric field as the following:
I understand that ...
2
votes
2
answers
90
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Propagator and Ward identity in the $R_\xi$ gauge
The full gauge propagator in the $R_\xi$ gauge is
$$D_{\mu\nu} = \frac{i}{k^2+i\epsilon}\left(-g_{\mu\nu}+\frac{1-\xi}{k^2}k_\mu k_\nu\right).\tag{1}$$
Now if we take $\xi=0$, we get the Lorenz gauge, ...
3
votes
0
answers
44
views
Why is there always a $TT$-gauge coordinate system comoving with a test particle?
In section 35.5 of Gravitation by Misner, Thorne & Wheeler, when discussing the action of a gravitational wave on two test particles A and B, there is this statement:
My question is why, to the ...
- 56
2
votes
1
answer
171
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Gauge-fixing condition invariant under auxiliary gauge transformation
In quantum gravity one usually splits the metric $g= \bar{g}+h$ into a background field $\bar{g}$ and a fluctuation field $h$. In order to obtain a propagator one has to gauge fix the action (e.g. ...
- 502
3
votes
1
answer
201
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Temporal Gauge with periodic boundary conditions
In Yang-Mills theory with periodic boundary conditions in time, is the temporale gauge, i.e. $A_0 = 0$, well defined?
Periodic boundary conditions would be
$$A_\mu(T_2,x) = A_\mu(T_1,x).$$
Naively I ...
- 2,183
2
votes
1
answer
68
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Invariance of gauge-fixing condition in background field method
In the Peskin & Schröder (chapter 16.6) they use the background field method and spilt the gauge field into an background field $A$ and a fluctuation field $\mathcal{A}$. Next they claim that
the ...
- 502
5
votes
0
answers
80
views
Validity of Lorenz gauge in non-Abelian gauge theory
I understand that this is a long shot, especially because it's such a niche question but: has it been mathematically proven that (under sufficient smoothness conditions, etc.) any field configuration ...
- 557
4
votes
1
answer
139
views
Where is the Yennie gauge useful in Gupta-Bleuer formalism (or QED in general)?
Consider the Lagrangian of the Gupta-Bleuer formalism given by:
$$\mathcal{L}
=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}
-\frac{1}{2\xi}(\partial A)^2.$$
I understand the necessity of the gauge fixing term: ...
- 2,136
2
votes
0
answers
88
views
Proving that the path integral formulation of scalar QED theory is independent of the choice of the gauge-fixing parameter $\xi$
I am considering the following scalar QED lagrangian:
$$L = −\frac{1}{4}F_{\mu\nu}^2 + |D_{\mu\varphi}|^2 − m^2|\varphi|^2− \frac{1}{2\xi}(\partial_\mu A^\mu)^2.$$
Where I want to show that the ...
- 41
1
vote
0
answers
25
views
Proving that the Faddeev-Popov path integral is independent of the gauge choice? [duplicate]
I know that the Faddeev-Popov path integral is gauge invariant. But how does one show that
\begin{equation}
I = \int \mathcal{D}\mathcal{A}_\mu \bigg|\frac{\delta\mathcal{G}}{\delta{\omega}}\bigg|\...
- 673
1
vote
1
answer
130
views
Gauge choice and observable quantities
Assume that I have the usual $U(1)$ gauge field $A_{\mu}$. We know that observable quantities are invariant under global transformations of the form $A_{\mu}\rightarrow A_{\mu}'=A_{\mu}+\partial_{\mu}\...
- 3,799
2
votes
0
answers
45
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$R_\xi$ gauge and degrees of freedom counting
In the standard classical Maxwell theory, we use the following arguments to claim that there are only two propagating degrees of freedom
$A_\mu$ has 4 components
$A_0$ is non-dynamical (-1)
$\...
- 669
0
votes
1
answer
117
views
Gauge invariance to solve Dirac equation in external EM field
Consider a Dirac equation in external EM field $A_\mu$
$(i\gamma^\mu\partial_\mu-m)\psi=q\gamma^\mu A_\mu\psi$
Consider a solution without EM field $\psi_0$. Let us do the gauge transformation $\psi_0\...
- 925
3
votes
1
answer
116
views
Why can we always find $\vec A$ such that it satisfies Coulomb (or Lorenz) gauge and Maxwell's equations?
I have a short question about the Coulomb potential.
Let $\vec{E}$ and $\vec{B}$ be the electric field and magnetic field respectively.
The electric field and magnetic fields are described by the ...
- 217
1
vote
1
answer
160
views
4-Vector Potential, transformation under Lorenz Gauge [closed]
I am given an initial vector potential let's say:
\begin{equation}
\vec{A} = \begin{pmatrix}
g(t,x)\\
f(t,y)\\
0\\
g(t,x)\\
\end{pmatrix}
\end{equation}
- 21
1
vote
0
answers
32
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Holomorphic sections that lead to the original parameterization of the gauged STU model
The original action, given in 9903214, comes from the $\mathcal{N} = 2$ abelian truncation of the maximal $\mathcal{N} = 8$, $SO(8)$ gauged supergravity that can be obtained from $S^
7$
reduction of ...
0
votes
0
answers
31
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Why are gauge theories so significant? [duplicate]
As a disclaimer, the only gauge theory I know so far is from electromagnetism.
I have read that gauge theory is a big part of modern physics and it heavily underlies much of the standard model. ...
- 2,478
0
votes
0
answers
49
views
Gauge fixing in the magnetic induction equation
The magnetic induction equation for a fluid with zero resistivity may be written as
\begin{equation}
\frac{\partial \mathbf{B}}{\partial t} + \nabla \times (\mathbf{B} \times \mathbf{u}) = 0
\end{...
1
vote
1
answer
117
views
Feynman-Propagator of the gauge-fixed electromagnetic field
I want to find the Feynman propagator for the so called $R_\zeta$ gauge fixed electromagnetic field. The lagrangian density is given by:
\begin{align}
L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\...
- 423
0
votes
3
answers
73
views
Problem with the vacuum energy upon quantization of the EM field in Lorenz gauge
There seems to be an issue arising concerning the vacuum energy of the EM field after quantizing in Lorenz gauge.
As usual, we consider an extra gauge-fixing term in the Lagrangian of the form $-\frac{...
- 115
0
votes
1
answer
111
views
Coulomb gauge covariance
Why do we say Coulomb gauge is not covariant, whereas Lorenz gauge is? What's the ultimate reason why Coulomb gauge cannot be covariant?
- 21
2
votes
0
answers
142
views
Gauge freedom in (3+1) general relativity
In the ADM formalism of general relativity, in which one rewrites Einstein's field equations in terms of evolution and constraint eqautions of the $3$-metric and the extrinsic curvature, after fixing ...
- 834
2
votes
1
answer
258
views
How to get $\mathbb{Z}_2$ gauge symmetry from $\mathrm{U}(1)$ gauge theory?
I'm reading this paragraph from Tong's gauge theory lecture note.
This is the previous phrase of my question:
This is my understanding of getting $\mathbb{Z}_N$ gauge symmetry from $\mathrm{U}(1)$ ...
- 81
1
vote
0
answers
108
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QED gauge invariance
On P & S page.297, in the second paragraph from bottom, the book discussed gauge invariance of Faddeev-Popov procedure, following
a QED example. Where the photon propagator is:
$$ \widetilde{D}_F^{...
- 1,391
0
votes
1
answer
157
views
Unitarity gauge in Higgs mechanism in P&S's QFT
To my understanding, after spontaneous symmetry breaking, if we parametrized Higgs field:
$$\\ \phi(x)= \frac{1}{\sqrt{2}} \left(\begin{array}{c} 0 \\ v+h(x) \end{array}\right)e^{i\pi(x)/v}, $$
we ...
- 1,391
1
vote
0
answers
127
views
Understanding the Bound State Aharonov Bohm Effect
The bound state Arahanov Bohm Effect in most textbooks is addressed by explicitly solving the time-independent S.E. in order to solve for the energies of the system and show that they explicitly ...
- 41
5
votes
1
answer
291
views
QED scattering cross sections are independent of the gauge-fixing terms
In QED, the Lagrangian with gauge-fixing terms is $$\tag{7.2}L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\xi(\partial^\sigma A_\sigma)^2 $$ (See Greiner field quantization), from which we can obtain ...
- 730
1
vote
0
answers
85
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Lorentz invariance of Quantum Electrodynamics [closed]
In chapter 5 of Weinberg QFT (Vol.$1$), in section 5.9, Weinberg demonstrated through an explicit calculation that there cannot be a massless vector field with helicities $\pm 1$. In fact, Weinberg ...
- 730
1
vote
0
answers
125
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Peskin and Schroeder's QFT eq.(9.56)
On Peskin and Schroeder's QFT book, page 296, the book give the functional integral formula after inserting Faddeev and Popov's trick of identity.
$$ \int \mathcal{D} A e^{i S[A]}=\operatorname{det}\...
- 1,391
3
votes
1
answer
267
views
Can we choose the Coulomb gauge if we're in a gauge where the gradient of the scalar potential is zero?
If we start in the gauge
\begin{align*}
\textbf{E}=-\nabla\phi-\frac{\partial\textbf{A}}{\partial t},
\end{align*}
\begin{align*}
\textbf{B}=\nabla\times\textbf{A}
\end{align*}
We can express ...
- 849
1
vote
1
answer
64
views
Normalization of a four-velocity when constructing the metric from the EM tensor
I've been following the MIT General Relativity Course. In lecture (14) the concept of linearized gravity is introduced. This is then demonstrated by showing how we can recieve the Newtonian limit from ...
- 283
6
votes
1
answer
233
views
Path integral quantization of the EM field in Peskin and schroeder
I'm studying path integral quantization of the electromagnetic field using Peskin and Schroeder secdtion 9.4. We want to compute the functional integral
$$\tag{9.50} \int \mathcal{D}A\,e^{iS[A]}.$$
We ...
- 730
1
vote
1
answer
55
views
Computation of functional determinant of Lorenz gauge
In the Peskin and Schroeder's book P295, there is a derivation I don't quiet understand.
In Lorenz gauge we have $$G(A^\alpha)=\partial^\mu A_\mu+(1/e)\partial^2 \alpha.$$ Then it says that we could ...
- 497
0
votes
0
answers
62
views
Unitarity gauge
Weinberg in Chapter 21, section 21.1, QFT 2, says that, in unitarity gauge $\tilde{\phi}_n(x) = \gamma^{-1}_{nm}(x)\phi_m(x)$,
we do not have degrees of freedom with negative probability, like time-...
- 423
2
votes
1
answer
1k
views
"Lorentz gauge" or "Lorenz gauge"?
In electrodynamics there is a gauge condition named after Ludvig Lorenz: $$\partial^\mu A_\mu = 0.$$
In general relativity we also have a gauge condition defined as follow:
$$\partial_\mu \gamma^{\mu\...
- 35
2
votes
2
answers
239
views
Does Coulomb gauge imply constant density?
Say we have
$$\Box A = J$$
and
$$\nabla \cdot A = 0\;.$$
Then
$$0 = \Box (\nabla \cdot A) = \nabla \cdot J\;.$$
But,
$$\nabla \cdot J - \partial_t \rho = 0\;.$$
So
$$ \partial_t \rho = 0\;.$$
Thus,
$$\...
- 184