I Coulomb gauge Lorenz invariant?

Sebas4
Messages
13
Reaction score
2
TL;DR Summary
What is meant by Coulomb gauge not being Lorenz invariant?
Hey,

What is meant by Coulomb gauge not being Lorenz invariant?

The Coulomb gauge is just a constraint on \mathbf{A} and \phi and thus it is independent of inertial frame.

I posted the question in the wrong section. This question is in the context of QFT. The notes says:
A disadvantage of working in Coulomb gauge is that it breaks Lorentz invariance.
 
Last edited:
Physics news on Phys.org
Sebas4 said:
TL;DR Summary: What is meant by Coulomb gauge not being Lorenz invariant?

What is meant by Coulomb gauge not being Lorenz invariant?
It means that if you have some four-potential ##A^\mu=(\phi,\vec A)## where ##A^\mu## satisfies the Coulomb gauge condition in some unprimed inertial frame, then ##A^{\mu'}=\Lambda^{\mu'}_\mu A^\mu## generally will not satisfy the Coulomb gauge condition in the primed inertial frame. It will still be a perfectly valid four-potential in the primed frame, but just not in the Coulomb gauge.
 
Of course you can write the Coulomb gauge in a manifestly covariant way.

The point is that usually you take an arbitrary inertial frame ##\Sigma^{*}## and write the Coulomb-gauge condition in (1+3)-notation as
$$\vec{\nabla}^* \cdot \vec{A}^*=0.$$
You can make this manifestly covariant by introducing the four-vector with components ##U^*=(1,0,0,0)## in this frame.

Then the Coulomb-gauge condition in a general frame reads
$$\partial_{\mu} (A^{\mu}-U^{\mu} U^{\nu} A_{\nu})=0.$$
The point is that you now have introduced a preferred inertial reference frame to define your gauge constraint.

Using this covariant notation, you get a manifestly covariant photon propagator, containing the ##U^{\mu}## of course. Now the important point is that due to gauge-invariance for any physically observable quantities like S-matrix elements for scatterings between photons and electrons+positrons in standard spinor QED the frame-dependent terms, i.e., those containing ##U^{\mu}## cancel thanks to the Ward identities.

The advantage of the Coulomb gauge is that you have a complete gauge fixing and no unphysical degrees of freedom. The disadvantage is this "fictitious breaking of Lorentz symmetry" due to the introduction of an arbitrary reference frame, i.e., the vector ##U^{\mu}##.

You can also use a manifestly covariant gauge like the Landau gauge, demanding ##\partial_{\mu} A^{\mu}=0##, but this fixes the gauge only partially, and you have to deal with unphysical degrees of freedom like longitudinal and timelike photons, which however also cancel using the Gupta-Bleuler formalism. The advantage is that there's no arbitrary preferred frame and the Feynman rules lead to simpler expressions.
 
  • Like
Likes dextercioby and Dale
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
Back
Top