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Questions tagged [gauge]

Use this tag to discuss gauge-fixing conditions, as in the phrase 'choosing a gauge', such as, e.g. the Lorenz gauge, Coulomb gauge, Feynman gauge, Landau gauge, axial gauge, temporal gauge, light cone gauge, etc.

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3
votes
1answer
333 views

Why does reparameterisation invariance lead to gauge-fixing?

In Becker, Becker and Schwarz, the point particle action is given in terms of an auxiliary field $e(\tau)$ as: \begin{align} \tilde{S}_0 = \frac{1}{2}\int \,d\tau \left(e^{-1}\dot{X}^2 - m^2e\right) \...
9
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1answer
3k views

Clarifications needed on Gauge Fixing and Ghosts [closed]

The first time some kind of gauge fixing appears is during the Gupta-Bleuler procedure, which is used to be able to quantize the photon field: The basic gauge invariant Lagrangian leads to $\Pi_0=0$ ...
8
votes
2answers
1k views

Are gauge choices in electrodynamics really always possible?

If $B$ is magnetic field and $E$ electric Field, then $$B=\nabla\times A,$$ $$E= -\nabla V+\frac{\partial A}{\partial t}.$$ There is Gauge invariance for the trnasformation $$A'\rightarrow A+\...
11
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2answers
3k views

What is a gauge in a gauge theory?

As I study Jackson, I am getting really confused with some of its key definitions. Here is what I am getting confused at. When we substituted the electric field and magnetic field in terms of the ...
12
votes
2answers
490 views

Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
6
votes
1answer
666 views

What is the constraint on the Gauge Potential in the Covariant Gauges?

One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term \begin{equation} -\frac{(\partial_\mu A^{\mu})^2}{2\xi} \end{equation} to the Lagrangian. ...
7
votes
2answers
1k views

Showing that Coulomb and Lorenz Gauges are indeed valid Gauge Transformations?

I'm working my way through Griffith's Introduction to Electrodynamics. In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric ...
27
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0answers
2k views

How to apply the Faddeev-Popov method to a simple integral

Some time ago I was reviewing my knowledge on QFT and I came across the question of Faddeev-Popov ghosts. At the time I was studying thеse matters, I used the book of Faddeev and Slavnov, but the ...
6
votes
3answers
764 views

Wess-Zumino Gauge in non-Abelian supersymmetric theory

I've got a question concerning non-Abelian supersymmetric gauge theories. Consider supersymmetric non-Abelian theory realized on chiral superfields $\Phi_i$ in a representation $R$ with matrix ...
6
votes
2answers
323 views

How many degrees of freedom in a massless $2$-form field?

Consider the Kalb-Ramond field $B_{\mu\nu}$ which is basically a massless $2$-form field with the Lagrangian $$ \mathcal L = \frac{1}{2}P_{\alpha\mu\nu}P^{\alpha\mu\nu}\,, $$ where $P_{\alpha\mu\nu} \...
18
votes
1answer
916 views

Faddeev-Popov Gauge-Fixing in Electromagnetism

Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. ...
12
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2answers
2k views

Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics

How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics. I need an clear argument that involves the ...
9
votes
2answers
752 views

Why is $p_y$ conserved in the Landau gauge when we know the electron moves in circles?

Considering the cyclotron in $xy$-plane where the magnetic field is $\vec{B}=(0,0,B)^{T}$. In the Landau gauge, we have $\vec{A}=(0,Bx,0)^T$ and we obtain the Hamiltonian $$H=\frac{\hat{p}_x^2}{2m}+\...
9
votes
3answers
904 views

Can we do path integrals in gauge theories without fixing a gauge?

I am aware that when quantizing gauge theories with a path integral, one needs to add a gauge fixing term to avoid over-counting gauge related field configurations. From an aesthetic perspective, I ...
7
votes
1answer
385 views

For a constant magnetic field, is there a gauge with both canonical momenta conserved?

To describe a constant magnetic field $\mathbf B=(0,0,B)$ (ignoring the motion along the $z$ dimension) within hamiltonian (or quantum) mechanics, one needs to choose a gauge. One common gauge is the ...
5
votes
2answers
296 views

Trace-reversed EFE and linearized gravity

I have a question about the linearized Einstein Field Equations, and in particular about the Newtonian limit. It goes as follows. If one uses the trace-reversed form of the EFE for the 00-component ...
2
votes
4answers
135 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) + ...
1
vote
1answer
176 views

Do we fix divergence of the vector potential $A$, because $\nabla \cdot \nabla \psi \ne 0$?

Because $\nabla \times \nabla \psi = 0$, we can transform the vector potential $A \longmapsto A + \nabla \psi$, without changing the magnetic field. Is the reason we specify $\nabla \cdot A$ in the ...
8
votes
3answers
3k views

Gauge fixing choice for the gauge field $A_0$

In many situations, I have seen that the the author makes a gauge choice $A_0=0$, e.g. Manton in his paper on the force between the 't Hooft Polyakov monopole. Please can you provide me a ...
8
votes
1answer
6k views

How do I derive the Lorenz gauge from the continuity equation?

I was reading my old electromagnetics book (Elements of Electromagnetics, by Sadiku, 3rd edition) and after the author explained what the Lorenz gauge is mathematically and why it is useful in ...
2
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1answer
1k views

Gauge theory in classical electromagnetism

I understand gauge theory as the theory of continuous transformation group which keeps Lagrangian (or dynamics) invariant. So some integral invariants could be found. In terms of classical ...
2
votes
1answer
524 views

Why is the electromagnetic four-potential $A_{\mu}$ not an observable?

Why within classical field-theory the electromagnetic four-potential (usually $A_{\mu}$) not an observable? In classical mechanics we don't have problems with energy measurements and in quantum ...
6
votes
2answers
771 views

Question about physical degree of freedom in Maxwell Theory: Why Coulomb gauge can fix all redundant degree of freedom

Given $4$-potential $A^\mu(x)=(\phi(x),\mathbf{A}(x))$, the vacuum Maxwell equations: $$\nabla^2\phi+\frac{\partial}{\partial t}(\nabla\cdot \mathbf{A} )=0$$ $$\nabla^2 \mathbf{A} -\frac{\partial^2 \...
3
votes
1answer
12k views

Physical difference between Lorenz and Coulomb gauge-fixing conditions

Lorenz and Coulomb gauge-fixing conditions. What is physical difference between these two gauge-fixing conditions? Mathematical expression are clear but how to we choose one of these means what they ...
13
votes
3answers
1k views

Are Maxwell's equations “physical”?

The canonical Maxwell's equations are derivable from the Lagrangian $${\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ by solving the Euler-Lagrange equations. However: The Lagrangian above is ...
11
votes
1answer
2k views

Gauge fixing and degrees of freedom

Today, my friend (@Will) posed a very intriguing question - Consider a complex scalar field theory with a $U(1)$ gauge field $(A_\mu, \phi, \phi^*)$. The idea of gauge freedom is that two solutions ...
9
votes
2answers
402 views

Why the extra term $\frac{1}{2}(\partial_{\rho}A^{\rho})^2$ in the photon Lagrangian?

In my quantum field theory class we have been told to use this Lagrangian for the photon field $$\mathcal{L}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta} -\frac{1}{2}(\partial_{\rho}A^{\rho})^2.$$ but ...
4
votes
0answers
416 views

Gauge invariance and the unitarity

I want to discuss the relation between the unitarity and the gauge invariance. Suppose we have for simplicity an abelian gauge theory (say, EM theory). We want to quantize it in terms of 4-potential $...
10
votes
1answer
511 views

Yang Mills Hamiltonian: why do we use the Weyl's temporal gauge?

Do you know why in the quantization of $SU(2)$ Yang-Mills Gauge Theory, it is always chosen the Weyl (temporal) gauge to derive the Hamiltonian? Is it possible to fix another gauge?
8
votes
1answer
537 views

Why should it be allowed to set the einbein to unity?

The Polyakov action for a massive free point particle with worldline $\gamma$ is given by $$ S[\gamma] = \frac{1}{2}\int_\gamma e \biggl(\frac{1}{e^2}\dot{x}^2 - m^2\biggr)\mathrm{d}\tau $$ where $e$...
4
votes
1answer
388 views

Why do we have to choose a gauge to quantize a gauge theory?

Why do we have to choose a gauge to quantize a gauge theory? This was an exam question but I couldn't answer it.
7
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2answers
583 views

Counting massive degrees of freedom after gauge fixing

Consider the theory of scalar QED with the Lagrangian $$\mathcal{L} = - \frac14 F^{\mu\nu} F_{\mu\nu} + (D^\mu \phi)^* (D_\mu \phi) - m^2 \phi^* \phi \tag{1}$$ where $\phi$ is a complex scalar field ...
4
votes
2answers
3k views

Landau level degeneracy in symmetry gauge, finite system

As we know, Landau level degeneracy in a finite rectangular system is $\Phi/\Phi_0$, where $\Phi=BS$ is the total magnetic flux and $\Phi_0=h/q$ is the flux quanta. This can be easily derived using ...
0
votes
1answer
107 views

Properties of Fluids-theoritical confusion

I am having a very basic confusion on how we calculate the height of atmosphere when we assume that the density does not change with altitude(density remains 1.29 kg/m$^3$). I want to know why we say ...
4
votes
1answer
136 views

Why can we pick the divergence of the vector potential? [duplicate]

I'm aware that the vector and scalar potential in E&M can be modified using a function $\lambda(t)$ in the following way: $$\mathbf{A}' = \mathbf{A} + \nabla\lambda,\;\; \textrm{ and } \;\;\Phi' =...
3
votes
2answers
468 views

The location of an object is gauge dependent. Therefore, it's not measurable?

The location of an object $x$ depends on how we choose our coordinate system. If we move the zero point, $x$ also changes. However, since we have translational invariance, we can always do such shifts ...
2
votes
2answers
418 views

Gauge theory and eliminating unphysical degrees of freedom

In free space we can express Maxwell's equations as \begin{align} \varepsilon^{abcd}\partial_bF_{cd}=0 ~~\text{ and }~~ \partial_aF^{ab}=0 \tag{1} \end{align} where $F^{ab}=-F^{ba}$. The most general ...
2
votes
1answer
51 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) + \...
0
votes
4answers
475 views

The definition of the Lorenz gauge condition

The inner product of two vectors in space-time is: $$(x_1, y_1, z_1, t_1) \cdot (x_2, y_2, z_2, t_2) = x_1 x_2 + y_1 y_2 + z_1 z_2 - t_1 t_2$$ So $$(\frac{\partial }{\partial x}, \frac{\partial }{\...
4
votes
2answers
764 views

Lorenz gauge fixing

Is it always possible to define function $\psi$ satisfying the Lorenz gauge equation $$ \partial_{\mu}\partial^{\mu} \psi + \partial_{\mu}A^{\mu} = 0? $$
2
votes
1answer
113 views

PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2', '...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix ...
2
votes
1answer
191 views

Ensuring Lorenz Gauge condition in Green Function solution

In the Lorenz Gauge, we can write Maxwell's equations as: $$\tag1 \Box A^\beta=\mu_0j^\beta.$$ We then go on to solve this by treating each component $A^\beta$ as an independent solution of the ...
0
votes
2answers
87 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 ...
0
votes
1answer
452 views

Determining the Lorenz gauge condition

I'm having a bit of trouble understanding how the gauge condition is found. Consider the potentials $V$ and $\vec A$ and $V'$ and $\vec A'$ so that $$\vec E = -\vec\nabla V -\frac{\partial\vec A}{\...
0
votes
1answer
809 views

General relativity: gauge fixing

In his lectures professor Hamber said that the metric tensor is not unique, just like the 4 vector potential is not unique for a unique field in electrodynamics. Since the metric tensor is symmetric, ...
0
votes
1answer
300 views

EM vector potential

We can write the electromagnetic field tensor as $$\begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & ...