# What is Gauge fixing: Definition and 19 Discussions

In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.
Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration by a particular detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes more realistic; its application to quantum field theory is fraught with complications related to renormalization, especially when the computation is continued to higher orders. Historically, the search for logically consistent and computationally tractable gauge fixing procedures, and efforts to demonstrate their equivalence in the face of a bewildering variety of technical difficulties, has been a major driver of mathematical physics from the late nineteenth century to the present.

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1. ### A Understanding Ghost Fields in QED: Eliminating Unphysical Degrees of Freedom"

I have a question about following statement about ghost fields in found here : It states that introducing some ghost field provides one way to remove the two unphysical degrees of freedom of four component vector potential ##A_{\mu}## usually used to describe the photon field, since physically...
2. ### I 4-Current vector potential transformation under Gauge fixing

I am given an initial vector potential let's say: $$\vec{A} = \begin{pmatrix} g(t,x)\\ 0\\ 0\\ g(t,x)\\ \end{pmatrix}$$ And I would like to know how it will transform under the Lorenz Gauge transformation. I know that the Lorenz Gauge satisfy...
3. ### I Equations of Motion for Massless Particle in Potential

The Lagrangian for a massless particle in a potential, using the ##(-,+,+,+)## metric signature, is $$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - V,$$ where ##\dot{x}^\mu := \frac{dx^\mu}{d\lambda}## is the velocity, ##\lambda## is some worldline parameter, ##e## is the auxiliary einbein and...
4. ### A Solve Conditions to Preserve Bondi Gauge Vector Field

I'm going through the "Advanced Lectures on General Relativity" by G. Compère and got stuck with solving one set of conditions on the subject of asymptotic flatness. Let ##(M,g)## be ##4##-dimensional spacetime and ##(u,r,x^A)## be a chart such that the coordinate expression of ##g## is in Bondi...
5. ### A Gauge breaking and Faddeev-Popov ghost particles

Summary: In QFT, if we add a gauge breaking term to the Lagrangian, do we still need to introduce Faddeev-Popov ghost particles? Ghosts seems to be introduced to maintain gauge invariance. But suppose we have eliminated the gauge invariance, from the start, by explicitly introducing a gauge...
6. ### Gauge invariance in GR perturbation theory

I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the metric. We use an arbitrary small vector field to have the effect of a chart transition map or...
7. ### Mathematical Quantum Field Theory - Gauge Fixing - Comments

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Gauge Fixing Continue reading the Original PF Insights Post.
8. ### I How does Gauge fixing fix anything?

I've looked everywhere and I haven't found an explanation of why is it useful to introduce gauge conditions. I've also searched in this forum, and none of the existing threads I've read answered my question. I apologize if there is and I have failed to find it. My problem is that, as I see it...
9. ### Gauge fixing and gauge transformations

If a theory is gauge invariant and one chooses to fix a particular gauge, having done this is it then possible to make a gauge transformation from this chosen gauge to another gauge, or have we already "spent" the gauge symmetry? Apologies if this is a really basic question, but I've got myself...
10. ### Why does gauge fixing break gauge symmetry?

By fixing a gauge (thus breaking orspending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the...
11. ### Gauge Fixing Term and Equations of Motion

Lets take QED just to simplify. When we are doing Path Integrals and we want to "fix the Gauge": 1 we add in the integral a delta(F) -meaning that we are going to integrate only in one representative of each class of equivalent configuration- 2 We take some factors out because they are constant...
12. ### Gauge Fixing in Yang-Mills gauge theory

I've come across countless sources that gauge fix SU(N) Yang-Mills fields using the typical U(1) gauges (e.g. Lorenz gauge, coulomb gauge, temporal gauge, etc). However, I can't find a single one where they prove that this gauge fixing is valid for all field configurations... I've tried to...
13. ### Why are there several gauge fixing choice for gauge symmetry fields?

Please teach me this: For gauge symmetry fields,only one of any elementary subconfiguration of the whole configuration covers the all physics of the field.So we need to cut off the redundant configuration.It seem to me,in a loose sense,there is only one way to cut off the redundancy(the gauge...
14. ### What does it mean by independent (in gauge fixing of EM field)

What does it mean by "independent"(in gauge fixing of EM field) In my textbook, it gives the Coulomb gauge \phi = 0,\nabla A = 0 and says they will kill two degrees of freedom of the four potential and leave two independent components. I understand \phi = 0 will kill one degree of freedom...
15. ### Gribov (gauge fixing) ambiguities in quantum gravity

Recently I started a new thread regarding Gribov (gauge fixing) ambiguities in quantum field theory, especially in QCD https://www.physicsforums.com/showthread.php?t=429759. Of course every theory incorporating a local gauge symmetry may have this gauge fixing issue - therefore in GR and all...
16. ### Gauge Fixing Term: Physical Effects & Role in Spin Components

Hi... Would you please advise me what does gauge fixing term do (physical point of view) ? Does it eliminate unnecessary spin components from lagrangian for example: Vector particle has two (massless case) or three (massive case) degrees of freedom. Vector itself has four, and a vector...
17. ### Coulomb Gauge Fixing: Adding Gradient & Subtracting Partial Time Derivative

Coulomb gauge fixes gauge by setting div(A)=0. What has it to do with adding a gradient to A and subtract a partial time derivative from V?
18. ### Gauge fixing in 5D vs 4D, and photon polarization

I'm sorry if what I say is not right, or I haven't understood it right, - In 3+1 D we have the photon with spin 1 => it has two polarizations. Our Gauge field A_\mu has 4 components => We have two extra degrees of freedom. => We need to get rid of the extra 2 fixing the gauge. 1. The...
19. ### Gauge fixing and residual symmetries

This question comes from reading Schwarz' string theory book, which is why I put it in this section. But it seems like a general QFT question, so maybe this isn't the right forum for it. Starting with the sigma model action, reparametrization and Weyl invariance allow us to "gauge fix" the...