All Questions
91
questions
4
votes
1
answer
73
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} \...
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 ...
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 ...
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 ...
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
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 ...
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 ...
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}\...
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)
$\...
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.
$$ \int \mathcal{D} A e^{i S[A]}=\operatorname{det}\...
2
votes
2
answers
265
views
Does Coulomb gauge imply constant density?
Say we have
$$\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\;.$$
Thus,
$$\...
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 ...
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
$$\...
1
vote
1
answer
199
views
What is 't Hooft-Veltman gauge? What are the interactions in SM in 't Hooft-Veltman gauge?
What is 't Hooft-Veltman gauge? I can't really find any suitable answer online. If we introduce this gauge in SM, then what becomes interactions?
0
votes
0
answers
64
views
Steps in Quantizing Electromagnetic Field for the Gauge Condition $A_0=0$
While reading section 9.3 of QFT An Integrated Approach by Fradkin, it is shown (see equations $(9.49)$ and $(9.54)$ of the book)
$$B_{j}(\boldsymbol{x})^{2}=\boldsymbol{p}^{2} A_{j}^{T}(\boldsymbol{p}...
-1
votes
2
answers
667
views
Coulomb gauge with $\rho = 0$ implies Lorenz gauge?
Maxwell equations take the form:
$$\nabla^2 \phi + \frac{\partial}{\partial t} \nabla \cdot \vec{A}= - \frac{\rho}{\epsilon_0}\qquad (\nabla^2 \vec{A} - \mu_0\epsilon_0\frac{\partial^2 \vec{A}}{\...
0
votes
0
answers
65
views
The choice of gauge seems has contradiction
Suppose I have a quantum object, inside it the electric field distribution is $\vec{E}(\vec{r})$, with this field we can obtain the scalar potential $\phi(\vec{r})$, a charged particle in this object ...
1
vote
2
answers
203
views
Existence of the Coulomb gauge
In reading about the Coulomb gauge, my mind seems to have painted itself into a corner. For, lets assume that Maxwells equations for the physics of the problem are solved by the magnetic vector ...
2
votes
1
answer
543
views
Do the retarded potentials satisfy the Lorenz Gauge condition?
Every source I have ever seen derives the retarded and advanced potentials by finding the Green's functions of the inhomogeneous Lorenz gauge conditions, and I have always thought that any linear ...
1
vote
1
answer
110
views
Transforming the potentials that satisfy Lorenz & Coulomb gauge to potentials that satisfy only Lorenz gauge
If $\vec E(\vec r,t)=\vec E_0sin(\vec k \vec r- \omega t)$ and also that $\rho(\vec r,t)=0$ and $\vec j(\vec r,t)=0$
I was asked to find $\vec A(\vec r,t)$ and $\phi (\vec r,t)$ which satisfy both the ...
0
votes
1
answer
72
views
Lorenz Gauge different definitions
For the lorenz gauge we can either write:
$$\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
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:
$$\vec A(\vec r,t)'= \vec A(\vec r,t) + \nabla f(\vec r,t)$$
$$\phi(\vec r,t)'=\phi(\vec r,t) - \...
2
votes
2
answers
207
views
"One-parameter" gauge transformation
In my advanced classical physics course, it was stated that the electromagnetic field strength tensor $F_{\mu\nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu}$ is invariant under "one-...
1
vote
0
answers
31
views
Gauge fixing terminology (math terms) [duplicate]
In the majority of the sources I've read regarding gauge fixing, the authors sometimes use (IMHO) a vague terminology. Let's take the case of the magnetic vector potential $\vec{A}$ defined as
$$ \vec{...
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^{\...
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 \...
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^...