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
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Gauge Invariance of Spin Angular Momentum of EM waves [closed]
How can I prove, the Spin Angular Momentum of Electromagnetic Wave to be not Gauge Invariant?
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0answers
43 views
Constructing gauge symmetry for $SU(2)\times U(1)$
I'm currently reading the book "Classical Theory of Gauge Fields" by Rubakov, therefore I will use his convention in this question. In the following we assume that:
$\phi$ comprises columns ...
3
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4answers
132 views
Does $A_\mu$ transform with the coordinates, or as a vector in a tangent space?
Does the vector potential $A_\mu$ transform when we merely relabel events in space-time (coordinate transformation), or does it transform with the basis vectors of a tangent space in which it lives?
...
2
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1answer
52 views
Is the supercurrent gauge invariant?
If we consider ${\cal N}=1$ renormalizeable chiral gauge theories, specifically discussed in section 27.4 of Weinberg's Quantum Theory of Fields, Supersymmetry book, should the supercurrent be gauge ...
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0answers
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Physical interpretation of Bogolyubov sound with zero momentum in BEC
The point with zero momentum in the spectrum of acoustic phonon corresponds to a translation of the crystal.
Is there an interpretation like this of the Bogolyubov sound in BEC?
I have read that in ...
3
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0answers
95 views
A Naive Question about Gauge Theory
I am suffering from a question I encountered from the lecture notes of gauge theory by David Tong. The problem comes from page 67 on the gauge fixing in back-ground gauge method. In David Tong's ...
1
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1answer
47 views
Dependence of BRST Quantization on the Choice of Gauge-Fixing Function
There is a point which confuses me about BRST procedure. One shows that, if we define physical states as the ones that are annihilated by BRST charge $Q$, the scattering amplitudes don't depend on ...
4
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1answer
69 views
How groups act on fields in QFT?
I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how ...
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0answers
60 views
QCD vs. QED gauge invariance
Trying to understand the difference between QED and QCD gauge invariance treatment I found the following paper: https://arxiv.org/pdf/1101.3425.pdf
I have the following questions:
1) I understand Eq....
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0answers
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Transformation of ADM parameters under diffeomorphisms
I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
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0answers
61 views
Gauge invariance in GR perturbation theory
I have been following this video lecture 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 ...
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0answers
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QCD gauge invariant amplitude?
In order to keep the correct degrees of freedom (which are 2) for massless gauge fields one imposes,
$$p^\mu \epsilon_\mu = 0 \tag1$$
Together with the gauge redundacy/equivalence relation,
$$\...
0
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1answer
41 views
Non-gauge-invariance of EM angular momentum
In 16.7 b) of Zangwill we're asked to show that the expression
$$
{\bf L}=\epsilon_o\int d^3r E_k ({\bf r}\times\nabla)A_k
+ \epsilon_o\int d^3r {\bf E} \times {\bf A}
$$
is not gauge invariant. ...
0
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1answer
37 views
Determine electromagnetic potentials that satisfy Coulomb and Lorenz gauge condition
In our physics lecture, we did the following example of constructing potentials $\vec A$ and $V$ that supposedly satisfy both the coulomb ($\nabla \vec A=0)$ and the lorenz condition $(\nabla \vec A+ \...
2
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3answers
66 views
General Definition of Potential in Circuits with Time-dependent Fields?
Electric potential is a very common observable to measure (especially in electro-engineering fields). It is measured using a multimeter, and thought of as the energy per charge that one charged ...
0
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2answers
79 views
Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$
Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that
$$D_\mu = \partial_\mu - igA_\mu, \tag1$$
you can built the kinetic term for the gauge potential $A_\...
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0answers
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Where is the gauge symmetry in an ideal Bose gas?
It seems in the literature that there is a certain notion of a “macroscopic wavefunction” associated with a Bose-Einstein system (see this PSE answer) which exhibits a global $U(1)$-phase symmetry. ...
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1answer
51 views
Gauge Field Transformation Properties
I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
3
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1answer
72 views
Why do we impose de Donder gauge?
In the field language, a massless particle corresponds to irreducible representations of the Lorentz group. In particular, given a spin-2 massless particle, we can embed the creation and annihilation ...
5
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0answers
41 views
Coordinate Systems vs Reference Frames vs States of Motion [closed]
This is a broad conceptual question. I am really trying to understand symmetry as deeply as possible. Please let that guide your responses.
In particular, I'm looking for help finding clarity on the ...
5
votes
1answer
127 views
QED lagrangian: gauge fixing term
I have a question about the structure of the QED lagrangian, in particular the free photon lagrangian which is contained in it. My premise is: I only know how to exploit canonical quantization in ...
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0answers
69 views
Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)
In Weinberg's QFT textbook, using a gauge transformation $$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_{...
3
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1answer
52 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
172 views
Faddeev-Popov determinant and topology of the worldline
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^...
6
votes
1answer
187 views
Non-local field redefinition and effects on path-integral measure
Consider the partition function
$$
Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
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1answer
54 views
Is the Dirac Lagrangian locally gauge invariant without gauge field $A$?
When it comes to the check of the invariance of the Lagrangian of the Dirac equation under local $U(1)$-transformations I have made the following observation:
$$L = \bar{\psi} (i\gamma^{\mu}\...
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0answers
47 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
votes
2answers
243 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
votes
1answer
71 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}\...
1
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2answers
77 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
$$\...
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0answers
19 views
Why demand local gauge invariance? [duplicate]
Ok, so I see how demanding local gauge invariance leads to gauge fields, and how quantizing those fields then leads to gauge bosons.
But why do we take that step in the first place? What leads us to ...
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1answer
40 views
Algebraic trouble in gauge invariance of Schrodinger equation
I've been trying to prove a component of a proof the gauge invariance of the schrodinger equation. Specifically the part in the first answer here where this is stated:
$$\big(\frac{\nabla}{i}-q(\vec{...
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0answers
41 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 ...
1
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3answers
91 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
vote
1answer
72 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
145 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
41 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 ...
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1answer
51 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:
$$-\frac{1}{2}...
4
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2answers
188 views
Help with understanding the imposition of gauge conditions
Let $s$ be a positive integer and $h_{a_1\dots a_s}$ be a traceless and totally symmetric (real) field which is defined modulo gauge transformations of the form
$$\delta_{\xi}h_{a_1\dots a_s}=\...
0
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2answers
79 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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2answers
136 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 ...
2
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1answer
97 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
$$
J^\mu = i\...
4
votes
2answers
501 views
Does it really make sense to talk about the color of gluons?
It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 ...
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1answer
55 views
Quick question on choosing a gauge (E.g. Lorenz gauge)
I have been quite confused when I read about choosing a gauge. For example we have the gauge transformation
$$
A_\mu\longrightarrow A_{\mu\prime}= A_\mu+\partial_\mu\alpha,
$$
and we can choose any $...
2
votes
1answer
192 views
Proving Gauge invariance of Schrodinger Equation
I am trying to proof explicitly that Schrodinger equation:
$$ i\hbar \partial_t \psi = \big[ -\frac{1}{2m}\big(\frac{\hbar}{i}\nabla-q\vec{A}\big)^2+qV \big]\psi$$
remains the same under the ...
5
votes
1answer
123 views
Is the electroweak $SU(2)$ gauge symmetry an exact symmetry in Standard Model before spontaneous symmetry breaking?
In Standard model, components of a $SU(2)$ doublet (for example $u$ and $d$) have different masses. This means there is no $SU(2)$ symmetry, but I think it is okay because the $SU(2)$ symmetry is ...
1
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3answers
81 views
Why does gauge invariance in electrodynamics mean that there are redundant degrees of freedom? [closed]
It is possible to choose different gauges in electrodynamics. I am familiar with two of them: Coulomb gauge and Lorenz gauge. Let us stick to the Coulomb gauge. It sets $$\nabla\cdot\vec{A}=0.$$ The ...
0
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1answer
46 views
What happens to the gauge covariant derivative if the theory contains multiple fields in different representations?
I'm studying a graduate level course in QFT, and I have a potentially very stupid question that I can't find adressed anywhere (which I guess implies that I'm missing something fundamental). I'm ...
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0answers
94 views
Gauge invariance of pseudo stress-energy tensor of gravitational waves
The pseudo stress-energy tensor of gravitational waves is given by
$$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
88 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}~\...
