All Questions
91
questions
4
votes
1
answer
72
views
Can we impose Coulomb gauge without using temporal gauge in source-free Maxwell electrodynamics?
Coulomb gauge is $$\vec{\nabla} \cdot A=0$$ Now, from expression for electric field in terms of potentials $\vec{E}=-\vec{\nabla} \phi-\frac{\partial \vec{A}}{\partial t}$ and Gauss Law $\vec{\nabla} \...
- 43
0
votes
0
answers
61
views
Degree of freedom - Lorentz transfomation reduces it? [duplicate]
I am having a real difficult to counting degree of freedom. In fact, I notice that sometimes I am confused about what exactly we count as DoF, and what we do not count.
See, for example, the ...
- 980
1
vote
0
answers
51
views
Gauge redundancy and Gauge fixing
Take any gauge invariant theory, for instance QED. The QED Lagrangian is invariant under
$$A_{\mu}(x)\rightarrow A'_{\mu}(x)=A_{\mu}(x)+\partial_{\mu}. \alpha(x)$$
I have chosen a local gauge ...
- 3,992
1
vote
1
answer
77
views
Magnetic vector potential in 1+1 spacetime dimensions
In the theory of electromagnetism in 1+1 spacetime dimensions (one temporal and one spatial coordinate), one can define the 2-potential vector (analogous to the 4-potential vector in 3+1 spacetime ...
0
votes
1
answer
99
views
Can we express the electrodynamic potentials $V$, $\mathbf{A}$ in terms of the electrodynamic fields $\mathbf{E}$, $\mathbf{B}$?
In Griffiths' Introduction to Electrodynamics problem 10.25, I am asked to draw a "triangle diagram" illustrating the relationship between (1) the sources $\rho$, $\mathbf{J}$, (2) the ...
- 687
1
vote
0
answers
50
views
Bibliography for the Quantization of the free electromagnetic field with the Lorenz gauge
Recently I have been studying QFT and when I arrived at the Gauge theory I learned that one can quantize the electromagnetic field with the Coulomb gauge and the Lorenz gauge.
Regarding the Coulomb, I ...
13
votes
2
answers
2k
views
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 ...
- 3,350
3
votes
3
answers
235
views
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 ...
- 1,515
3
votes
2
answers
573
views
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
0
answers
107
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
1
answer
181
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}\...
- 3,992
2
votes
0
answers
56
views
$R_\xi$ 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
1
vote
0
answers
159
views
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.
$$ \int \mathcal{D} A e^{i S[A]}=\operatorname{det}\...
- 1,421
2
votes
2
answers
265
views
Does Coulomb gauge imply constant density?
Say we have
$$\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
2
votes
0
answers
273
views
Coulomb gauge choice: Does $A_0=0$ imply that we also need to choose $\nabla \cdot \vec{A} =0$ from the EOM of $A_0$?
How to justify the Coulomb gauge fixing condition choice with
$$
A_0=0, \quad \nabla \cdot \vec{A} =0?
$$
Below in the text image, I find a text explaining that imposing $A_0=0$ is always possible ...
- 593
1
vote
0
answers
267
views
How to find Weyl/temporal gauge fixing condition?
Transformations that leave the field invariant:
$$\vec{A}' = \vec{A} + \nabla f$$
$$\phi' = \phi -\frac{\partial f}{\partial t}$$
I would like to solve for the weyl gauge, aka a gauge that leaves
$$\...
- 6,636
1
vote
1
answer
199
views
What is 't Hooft-Veltman gauge? What are the interactions in SM in 't Hooft-Veltman gauge?
What is 't Hooft-Veltman gauge? I can't really find any suitable answer online. If we introduce this gauge in SM, then what becomes interactions?
- 23
0
votes
0
answers
64
views
Steps in Quantizing Electromagnetic Field for the Gauge Condition $A_0=0$
While reading section 9.3 of QFT An Integrated Approach by Fradkin, it is shown (see equations $(9.49)$ and $(9.54)$ of the book)
$$B_{j}(\boldsymbol{x})^{2}=\boldsymbol{p}^{2} A_{j}^{T}(\boldsymbol{p}...
- 381
-1
votes
2
answers
667
views
Coulomb gauge with $\rho = 0$ implies Lorenz gauge?
Maxwell equations take the form:
$$\nabla^2 \phi + \frac{\partial}{\partial t} \nabla \cdot \vec{A}= - \frac{\rho}{\epsilon_0}\qquad (\nabla^2 \vec{A} - \mu_0\epsilon_0\frac{\partial^2 \vec{A}}{\...
- 6,636
0
votes
0
answers
65
views
The choice of gauge seems has contradiction
Suppose I have a quantum object, inside it the electric field distribution is $\vec{E}(\vec{r})$, with this field we can obtain the scalar potential $\phi(\vec{r})$, a charged particle in this object ...
- 6,095
1
vote
2
answers
203
views
Existence of the Coulomb gauge
In reading about the Coulomb gauge, my mind seems to have painted itself into a corner. For, lets assume that Maxwells equations for the physics of the problem are solved by the magnetic vector ...
- 446
2
votes
1
answer
543
views
Do the retarded potentials satisfy the Lorenz Gauge condition?
Every source I have ever seen derives the retarded and advanced potentials by finding the Green's functions of the inhomogeneous Lorenz gauge conditions, and I have always thought that any linear ...
- 269
1
vote
1
answer
110
views
Transforming the potentials that satisfy Lorenz & Coulomb gauge to potentials that satisfy only Lorenz gauge
If $\vec E(\vec r,t)=\vec E_0sin(\vec k \vec r- \omega t)$ and also that $\rho(\vec r,t)=0$ and $\vec j(\vec r,t)=0$
I was asked to find $\vec A(\vec r,t)$ and $\phi (\vec r,t)$ which satisfy both the ...
- 1,398
0
votes
1
answer
72
views
Lorenz Gauge different definitions
For the lorenz gauge we can either write:
$$\nabla \vec A(\vec r,t)+\frac{1}{c^2}\frac{\partial \phi(\vec r,t)}{\partial t}=0$$
If we also consider the following invariant transformations:
$$\vec A(\...
- 1,398
1
vote
2
answers
286
views
Coulomb Gauge misunderstanding
If we have $\vec A(\vec r,t)$ and $\phi (\vec r,t)$ and we make the following gauge transformations:
$$\vec A(\vec r,t)'= \vec A(\vec r,t) + \nabla f(\vec r,t)$$
$$\phi(\vec r,t)'=\phi(\vec r,t) - \...
- 1,398
2
votes
2
answers
207
views
"One-parameter" gauge transformation
In my advanced classical physics course, it was stated that the electromagnetic field strength tensor $F_{\mu\nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu}$ is invariant under "one-...
- 234
1
vote
0
answers
31
views
Gauge fixing terminology (math terms) [duplicate]
In the majority of the sources I've read regarding gauge fixing, the authors sometimes use (IMHO) a vague terminology. Let's take the case of the magnetic vector potential $\vec{A}$ defined as
$$ \vec{...
- 149
0
votes
1
answer
140
views
Gauge invariant Green's function for a point particle
This question is a follow up to the question (Gauge invariant Green's function for electrodynamics).
It is not possible to generally solve the eqution
\begin{equation}
\square A^{\mu}-\partial^{\...
- 498
2
votes
2
answers
214
views
Gauge invariant Green's function for electrodynamics
Varying the electromagnetic action
\begin{equation}
S=-m c \int d s\left(\dot{z}^{2}\right)^{\frac{1}{2}}-\frac{e}{c} \int d s A_{\mu} \dot{z}^{\mu}-\frac{1}{16 \pi c} \int d^{4} x F_{\mu \nu} F^{\mu \...
- 498
5
votes
1
answer
940
views
Gauge symmetry of massive vector field
Consider a real massive vector field with lagrangian density
$$\begin{align}\mathcal{L}&=-\frac{1}{4}(\partial_\mu A_\nu-\partial_\nu A_\mu)(\partial^\mu A^\nu-\partial^\nu A^\mu)+\frac{1}{2}m^2 A^...
- 2,284
1
vote
1
answer
425
views
Gauge fixing in the classical $U(1)$ gauge theory
My question concerns the gauge fixing in classical v.s. quantum $U(1)$ gauge theory. I will ask about the gauging fixing in quantum $U(1)$ gauge theory in a separated Phys-SE post.
For the classical $...
- 4,336
1
vote
1
answer
318
views
Quantisation of gauge field in temporal gauge
Whenever we use temporal gauge and quantise gauge field we implement Gauss law. I have seen some papers but the point is not cleared to me that why we implement Gauss law there. Please explain this if ...
- 43
1
vote
1
answer
203
views
When we use Lorenz gauge or Coulomb gauge, the result formula for electric $E$ and magnetic field $B$ is same or different?
Gauge condition can be chosen as you like or not?
is the Lorenz gauge is the only one correct? If Coulomb gauge can obtained exactly same results as Lorenz gauge for the electromagnetic fields E and ...
- 149
1
vote
1
answer
180
views
Photon Path Integral and Lorenz Gauge
I am reading Srednicki's QFT book (http://web.physics.ucsb.edu/~mark/qft.html). In chapter 57, specifically in page 343, the book stated that there's a problem with the path integral because the ...
- 673
1
vote
1
answer
153
views
Gauge transformation of the gauge-fixing term in the QED action
In the classroom my teacher stated that the Gauge-fixing term in the action
$$\frac{1}{2\alpha}\int d^4x (\partial_\mu A^\mu(x))^2$$
transforms under $A_\mu(x) \rightarrow A_\mu(x)+\partial_\mu \...
- 99
5
votes
1
answer
1k
views
$R_\xi$ gauges and the EM-field
$R_\xi$-gauges are said to be a generalization of the Lorenz gauge. I dont quite get why we add the term
$$
\mathcal L_{GF} = - \frac{(\partial_\mu A ^\mu)^2}{2\xi}\tag{1}
$$
to the Lagrangian. If i ...
- 2,196
2
votes
2
answers
504
views
Gauge invariance for classical fields
I recently did some exercises in classical field theory and tried to think deeply about the gauge symmetry of the free electromagnetic field described by the Lagrangian
$$
\mathcal L = -\frac 1 4 F^{\...
- 2,196
0
votes
1
answer
186
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 ...
- 1,427
1
vote
1
answer
249
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 ...
- 26.8k
7
votes
1
answer
744
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 ...
- 2,296
1
vote
1
answer
113
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 ...
- 311
1
vote
2
answers
1k
views
How does gauge-fixing really work?
Leaving technical issues like Gribov copies and residual gauge freedom aside, how do gauge fixing conditions like the Coulomb condition
\begin{equation}
\partial_i A_i =0
\end{equation}
or the axial ...
- 10.1k
6
votes
1
answer
1k
views
Why does Coulomb gauge condition $\partial_i A_i =0$ pick exactly one configuration from each gauge equivalence class?
There are infinitely many configurations of a vector field $A_\mu$ that describe the same physical situation. This is a result of our gauge freedom
$$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + \...
- 10.1k
5
votes
4
answers
1k
views
Why does Lorenz gauge condition $\partial_\mu A^\mu =0$ pick exactly one configuration from each gauge equivalence class?
For a vector field $A_\mu$, there are infinitely many configurations that describe the same physical situation. This is a result of our gauge freedom
$$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + ...
- 10.1k
2
votes
1
answer
143
views
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: ...
- 538
1
vote
0
answers
173
views
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 ...
- 195
2
votes
1
answer
319
views
Landau levels in symmetric gauge
On Shankar’s Quantum Many body page 394 it says for one electron in a magnetic field, ignoring spin,
$$H_0=\frac{(\bf{p}+e\mathbf{A})^2}{2m}$$
$$e\mathbf{A}=-\frac{\hbar}{2l^2}\hat{z}\times \mathbf{...
- 301
1
vote
0
answers
149
views
Why is the Coulomb Gauge enough to fix extra degrees of freedom?
In classical electrodynamics, we have after the Coulomb gauge is applied:
$$ \Delta U = -\frac{\rho}{\epsilon_0} $$
$$ \Box \vec{A} = \mu_0 \vec{j}-\frac{1}{c^2} \vec{\nabla} \frac{\partial U}{\...
- 1,450
0
votes
1
answer
133
views
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 ...
2
votes
1
answer
636
views
Why Coulomb gauge is a possible gauge choice?
In classical field theory we can get, that adding gradient of some scalar field to magnetic vector potential does not change the physics at all. So, we have such a symmetry:
$\boldsymbol{A}\...
- 331