Questions tagged [gauge-invariance]
Invariance of a physical system (its action) under a continuous group of local transformations underlain by a global symmetry whose group parameters fixed in space-time have now been extended to vary in space-time instead. Use for buildup of the invariance, fixing the gauge, and accounting for the corresponding changes in the functional measure of the system.
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1answer
60 views
Why do we impose de Donder gauge?
In the field language, a massless particle corresponds to irreducible representations of the Lorentz group. In particular, given a spin-2 massless particle, we can embed the creation and annihilation ...
5
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0answers
39 views
Coordinate Systems vs Reference Frames vs States of Motion [closed]
This is a broad conceptual question. I am really trying to understand symmetry as deeply as possible. Please let that guide your responses.
In particular, I'm looking for help finding clarity on the ...
5
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1answer
67 views
QED lagrangian: gauge fixing term
I have a question about the structure of the QED lagrangian, in particular the free photon lagrangian which is contained in it. My premise is: I only know how to exploit canonical quantization in ...
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65 views
Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)
In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has:
$$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
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1answer
48 views
Why is the projective symmetry group a group?
I am reading the paper from X. Wen about quantum orders and symmetric spin liquids. It can be found here: https://arxiv.org/abs/cond-mat/0107071
The Hamiltonian he is writing about looks like this:
\...
3
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1answer
109 views
Faddeev-Popov determinant and topology of the wordline
I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral
\begin{equation}
Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^...
4
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1answer
141 views
Non-local field redefinition and effects on path-integral measure
Consider the partition function
$$
Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
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1answer
37 views
Is the Dirac Lagrangian locally gauge invariant without gauge field $A$?
When it comes to the check of the invariance of the Lagrangian of the Dirac equation under local $U(1)$-transformations I have made the following observation:
$$L = \bar{\psi} (i\gamma^{\mu}\...
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0answers
40 views
A puzzle about Green's function and S-matrix
From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but ...
5
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2answers
217 views
Comparison of covariant form of Maxwell equations with Einstein's GR
We know, the the vector form of Maxwell equations
\begin{align}
\vec\nabla\cdot\vec{E} &= 4\pi\rho \label{Diff I}\\
\vec\nabla\times\vec{B} &= \dfrac{4\pi}{c} \vec{j}+\dfrac{1}{c}\dfrac{\...
2
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1answer
60 views
Why Coulomb gauge is a possible gauge choice?
In classical field theory we can get, that adding gradient of some scalar field to magnetic vector potential does not change the physics at all. So, we have such a symmetry:
$\boldsymbol{A}\...
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2answers
53 views
Global $U(1)$ transformation properties of gauge fields
What are the Global gauge transformations of gauge bosons in Standard Model?
To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as
$$\...
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0answers
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Why demand local gauge invariance? [duplicate]
Ok, so I see how demanding local gauge invariance leads to gauge fields, and how quantizing those fields then leads to gauge bosons.
But why do we take that step in the first place? What leads us to ...
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1answer
33 views
Algebraic trouble in gauge invariance of Schrodinger equation
I've been trying to prove a component of a proof the gauge invariance of the schrodinger equation. Specifically the part in the first answer here where this is stated:
$$\big(\frac{\nabla}{i}-q(\vec{...
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0answers
34 views
How does the photon arise from the $U(1)$ symmetry in QED?
I'm trying to study a bit of QED and I really don't get where the photon does arise.
I know that we want our field theory to be gauge invariant under a $U(1)$ phase transformation, because that is ...
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3answers
86 views
Is $F^{\mu\nu}F_{\mu\nu}$ equivalent to $A^{\mu}\nabla^{\alpha}\nabla_{\alpha}A_{\mu}$ for $U(1)$ gauge field lagrangian?
The two seem to yield the same equation of motion is why I asked. Where of course the standard notation for exterior forms applies $dA=F$.
We all know how the field strength tensor plays into the ...
1
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1answer
48 views
Why does gauge invariance HAVE to correspond to an observable?(Or is it the other way round)
Under the line integral of the geometrical Berry phase, a close-loop integral is gauge invariant as if we were to perform a gauge transformation of the initial state, with the end point of the path in ...
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2answers
131 views
Can the mass of a hydrogen atom be calculated in a gauge-invariant way?
Please excuse the lengthy question. It involves an interesting controversy which has arisen in discussions on this site:
Energy/mass of Quantum Vacuum
Relative potential energy vs Absolute potential ...
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1answer
39 views
Can fermions and bosons exist in the same representation?
Some people have made theories where they claim fermions and bosons exist in the same representation for example $E_8$. I can't see how this is possible.
But say for example it is. This would imply ...
0
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1answer
40 views
Why can we write lagrangian for gauge theory without the traces?
I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true:
$$-\frac{1}{2}...
4
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2answers
184 views
Help with understanding the imposition of gauge conditions
Let $s$ be a positive integer and $h_{a_1\dots a_s}$ be a traceless and totally symmetric (real) field which is defined modulo gauge transformations of the form
$$\delta_{\xi}h_{a_1\dots a_s}=\...
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2answers
54 views
Why does Maxwell's equations $\partial_{\mu} F^{\mu \nu} = 0$ have 3 independent components (DOF) in $D = 4$?
And how can we generalize this to the statement that it has $D-1$ independent components in dimension $D$?
I know that $F_{\mu \nu}$ has six independent components (because of antisymmetry), how do ...
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0answers
83 views
Parity Anomaly and Gauge Invariance
In Fermionic Path Integral and Topological Phases, Witten shows that in $2+1$ dimensions, the free massless Dirac fermion suffers from parity anomaly. To be specific, he shows that it is impossible to ...
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1answer
77 views
Conserved current in scalar QED
Consider a theory of a free massless complex scalar $\phi$ which undergoes global $U(1)$ transformations. The conserved current associated to this symmetry is the usual scalar current
$$
J^\mu = i\...
4
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2answers
498 views
Does it really make sense to talk about the color of gluons?
It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 ...
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1answer
52 views
Quick question on choosing a gauge (E.g. Lorenz gauge)
I have been quite confused when I read about choosing a gauge. For example we have the gauge transformation
$$
A_\mu\longrightarrow A_{\mu\prime}= A_\mu+\partial_\mu\alpha,
$$
and we can choose any $...
2
votes
1answer
119 views
Proving Gauge invariance of Schrodinger Equation
I am trying to proof explicitly that Schrodinger equation:
$$ i\hbar \partial_t \psi = \big[ -\frac{1}{2m}\big(\frac{\hbar}{i}\nabla-q\vec{A}\big)^2+qV \big]\psi$$
remains the same under the ...
5
votes
1answer
107 views
Is the electroweak $SU(2)$ gauge symmetry an exact symmetry in Standard Model before spontaneous symmetry breaking?
In Standard model, components of a $SU(2)$ doublet (for example $u$ and $d$) have different masses. This means there is no $SU(2)$ symmetry, but I think it is okay because the $SU(2)$ symmetry is ...
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3answers
75 views
Why does gauge invariance in electrodynamics mean that there are redundant degrees of freedom? [closed]
It is possible to choose different gauges in electrodynamics. I am familiar with two of them: Coulomb gauge and Lorenz gauge. Let us stick to the Coulomb gauge. It sets $$\nabla\cdot\vec{A}=0.$$ The ...
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1answer
39 views
What happens to the gauge covariant derivative if the theory contains multiple fields in different representations?
I'm studying a graduate level course in QFT, and I have a potentially very stupid question that I can't find adressed anywhere (which I guess implies that I'm missing something fundamental). I'm ...
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0answers
66 views
Gauge invariance of pseudo stress-energy tensor of gravitational waves
The pseudo stress-energy tensor of gravitational waves is given by
$$T_{\mu\nu}^{(\mathrm{G}\mathrm{W})} =
\frac{1}{32\pi}\left\langle \partial_{\mu} \bar{h}_{\alpha\beta} \partial_{\nu}\bar{h}^{\...
1
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1answer
78 views
Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?
Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
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0answers
37 views
Lapse and shift inside or outside the Poisson bracket?
For general relativity in the 3+1 ADM formulation, one has $H=\int dx [N{\cal H}+N^a{\cal H}_a]$ with $N$ and $N^a$ the lapse and shift which are undetermined Lagrange multipliers. The dynamical ...
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0answers
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Conserved currents in Yang-Mills theory: gluon current vs. quark current
In Yang-Mills theory there are two currents we can construct. There is the well-known quark current related to the global $SU(3)_C$ symmetry,
$$j{}^{\mu\,A}_\text{quark} = \overline{\psi}{}^i \gamma{}^...
7
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1answer
113 views
Counting degrees of freedom in the Higgs mechanism for different gauges
I am wondering how to count the degrees of freedom (dof) for a massive gauge field in different gauges. I've been reading some other answers, but haven't found a solution yet.
I am looking at the ...
3
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1answer
62 views
Gauge transformations with varying phase give us conservation of the charge density. Hence charged particles cannot move?
I stumbled upon the following paragraph in Quark confinement and Topology of gauge theories by Polyakov
"Gauge invariance with constant phase $\Psi \to e^{i \alpha}$ lead to conservation of the
...
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1answer
45 views
Why do we require that functions which parametrize gauge transformation are smooth?
A local $U(1)$ transformation is given by
\begin{equation}
f(x) = e^{i\epsilon(x)} \qquad \text{with} \qquad \epsilon(x) \in C^\infty \, .
\end{equation}
Why do we require that the functions in ...
0
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1answer
34 views
Gauge covariance of the magnetic momentum operator
In the book about Schrödinger Operators by Cycon et al. there is a step in their calculations I don’t understand. When I pick two vector potentials $A_1$ and $A_2$ such that their curl (i.e. the ...
3
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1answer
89 views
What happens to the global $U(1)$ symmetry in alternative formulations of Quantum Mechanics?
The global $U(1)$ symmetry in Quantum Mechanics corresponds to the freedom to shift the phase of the wave function
$$ \Psi \to e^{i\varphi} \Psi \, $$
and can be used to understand the conservation ...
3
votes
2answers
83 views
The meaning of gauge-fixing in covariant quantization of the electromagnetic field
I am having trouble wrapping my head around the idea behind the covariant quantization for the electromagnetic field that is usually done in textbooks (I'm currently following Mandl & Shaw and ...
3
votes
2answers
438 views
The location of an object is gauge dependent. Therefore, it's not measurable?
The location of an object $x$ depends on how we choose our coordinate system. If we move the zero point, $x$ also changes. However, since we have translational invariance, we can always do such shifts ...
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2answers
165 views
Gauge Invariance in Electrodynamics
I am studying Electrodynamics and I have been introduced to the concept of Gauge Invariance.
This was introduced by noting that $E$ and $B$ amount to 6 six degrees of freedom and the Maxwell ...
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0answers
35 views
Are there more general gauge transformations than simple phase shifts?
Usually, in the context of a $U(1)$ gauge theory, we only consider gauge transformations of the form
\begin{equation}
\Psi(x) \to \mathrm{e}^{i\epsilon(x)} \Psi(x) \, .
\end{equation}
Are there ...
2
votes
1answer
109 views
Momentum eigenfunction for non vanishing vector potential
I'm reading a section in a textbook about the Aharonov-Bohm effect, which claims that the eigenfunction (outside the solenoid, so $\operatorname{curl} A=0$) for the canonical momentum operator $$\pi = ...
3
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1answer
37 views
Why are transformations that only change something within a finite region redundancies?
I'm trying to build some intuition for a very particular definition of the notions global and local gauge symmetries. The definition goes as follows and appears, for example, in "Quantum Field Theory -...
3
votes
2answers
82 views
What's the physical meaning of the gauge invariant quantity $\partial_\mu \varphi(x) - A_\mu(x)$?
A famous locally gauge invariant quantity is
$$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , $$
which is interpreted as the measurable electric and magnetic field strengths.
Now, ...
2
votes
1answer
43 views
Are the nonphysical degrees of freedom in Yang-Mills theory analogous to the worldsheet metric in the Polyakov formalism?
The Polyakov string action on a flat background (in the Euclidean signature)
$$S_{P}[X,\gamma]\propto\int_{\Sigma}\mathrm{d}^2\sigma\,\sqrt{\text{det}\gamma}\,\gamma^{ab}\delta_{\mu\nu}\partial_{a}X^{...
2
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4answers
92 views
Is fixing the gauge the same thing as performing a Lorentz transformation?
Let's say I have a moving charged particle, with constant velocity.
Its electric field is given by (generally):
$$ \mathbf{E} = -\nabla\phi - \frac{\partial \mathbf{A}}{\partial t}. $$
If I perform ...
3
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0answers
71 views
Confusion Between Associated and Principle-G-Bundles
I realize there have been similar questions on stack before, but none of them have answered what I'm after.
-My question is really whether I can import wholesale everything from the principle bundle ...
4
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5answers
222 views
In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?
In Carroll's Appendix B, he says
You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with ...
