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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 $$ \vec{B} = \nabla \times \vec{A} \tag{1} $$ The magnetic vector potential $\vec{A}$ is not unique because when one applies this transformation $$ \vec{A} \longmapsto \vec{A}+\nabla\psi \tag{2} $$ computing $\nabla \times \vec{A}$ will yield the same $\vec{B}$ as (1). A typical gauge fixing condition (at least I think that's how it is called) is the Coulomb gauge $$ \nabla\cdot\vec{A}=0 \tag{3} $$ which serves to simplify many calculations.

What is somewhat clear to me: the transformation (2) is called a gauge transformation and the equation (3) is called a gauge fixing condition. What is not clear:

  • Which is the gauge? Is it the particular choice for $\psi$ or the one for $\vec{A}$ in (2)? If the second statement is true, then what would we call the particular choice for $\psi$?
  • To make things even more complicated the equation (3) is sometimes called "the Coulomb gauge", although I suspect this might be a terminology abuse.

Could someone please clarify this for me?

Qmechanic
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Iam
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    We say some quantity or expression is "in the X gauge" if we have applied the "X gauge fixing condition" to it. I don't know a standard name for what $\psi$ is. – Connor Behan Dec 12 '21 at 19:46
  • Possible duplicate: https://physics.stackexchange.com/q/146585/2451 – Qmechanic Dec 12 '21 at 19:49
  • The gauge is the scalar and vector potential together. It is together because QFT is a relativistic theory. Thus a gauge transformation changes them in a coherent manner whilst a gauge fixing sets them to particular values or a range of values. In fact, in the geometric description of gauge theory, the gauge is described as a connection form on a bundle. – Mozibur Ullah Dec 12 '21 at 20:35
  • The Coulomb gauge is a specific choice of gauge fixing. It's basically a case of terminological elision rather than of abuse. But of course the latter term is often used to mean elision isca form of terminological abuse. – Mozibur Ullah Dec 12 '21 at 20:37

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