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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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Goldstone Theorem and Goldstone Boson

From "Quantum Field Theory" by Itzykson and Zuber, at the end of the proof of Goldstone Theorem (p. 520) I read "This is a massless state (Goldstone boson) with the same quantum number as $j_0$ ...
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Gribov's phenomenon

In the well known textbook by Itzykson-Zuber "Quantum Field Theory" there is a discussion of the Gribov phenomenon in non-abelian gauge theories (see Section 12-2-1). To my taste, the discussion ...
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Misconceptions in spontaneous symmetry breaking

Spontaneous symmetry breaking occurs when we have a potential like a mexican hat as shown in figure (right) and is unbroken for the potential shape as shown in left figure. Under the Symmetry ...
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Gauge invariance on Yang-Mills Lagrangian

How do I verify the invariance on Yang-Mills' Lagrangian: $$L = -\frac{1}{4} \sum_{a} \left(\partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^bA_\nu^c \right)^2$$ under the transformation:...
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Interpretation of vanishing Noether charge

I was told that Gauge symmetries are redundancies because the Noether charge of a gauge symmetry vanishes, i.e. that there exist no observable quantities that would allow you to distinguish two ...
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Why gauge invariance of the action when equations of motion are satisfied is enough?

This is closely related to this question which in turn has to do with the motivation for gauge invariance. Making a quick recap to make this question self-contained, in his QFT book, chapter 5, ...
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Invariance of Klein-Gordon equation under gauge transformations

I'm sure this is really simple, and I might be right; it's just that I'm not sure. I'm asked to prove that the Klein-Gordon equation it's invariant under global gauge transformations. In Greiner's ...
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Gauge Symmetry & Off-Shell Current Conservation in QED

I’m reading Srednicki - I’m quite confused on bottom page 351 to top page 352 (which I recap below): This is his discussion in a nutshell. The Dirac Field has a conserved current which follows from ...
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Why can't we have a massive photon in the Standard Model of Particle Physics

I've heard that in the Standard Model of Particle Physics you can´t have a massive photon whatever you do but I'm having a few problems showing that. I understand that the trick to make this work is ...
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Connection between gauge invariance and Lorentz invariance

This question is presented in the context of Weinberg's QFT book treatment, in particular considering the electromagnetism chapter. It begins in chapter 5 where Weinberg argues that in order to have ...
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How is possible to see that Maxwell's field have $U\left(1\right)$ symmetry?

As it is well knowing, $U\left(1\right)$ is the group of the unitary matrix of the first order and that this group is connected with rotation operations. Under the complex scalar field perspective $$\...
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What is $U(1)$ symmetry?

I saw there are three intrinsic symmetries in physics,U(1),SU(2) and SU(3).What's the U(1) symmetry talking about?I would appreciate it if you can give me some explaination.
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Spin and Symmetry of SM Particles

I am reading about Quantum Field theory and I cannot understand how do physicists assign spin to particles. Is there a systematic way to assign spin to particles without knowing it first through ...
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Gauge fixing in canonical quantum gravity

In analogy with QFT, the partition function in canonical quantum gravity is defined as a functional integral over the metric tensor (which is now the quantum field), $$ \int \mathcal{D} g \mathcal{D}\...
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Noether current and continuity equation in classical scalar QED

Consider the following scalar QED model \begin{align} S = \int \mathrm{d}^{d+1} x\, \left\{-\left(\mathrm{D}_{\mu}\phi\right)^{\dagger} \left(\mathrm{D}^{\mu}\phi\right) -m^2 \phi^{\dagger}\phi - \...
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Conserved currents in quantum electrodynamics

A general Noether theorem in fields theory says that an infinitesimal symmetry of the action leads to a conserved current $j^\mu$, i.e. $\partial_\mu j^\mu=0$. Below I would like to consider a minor ...
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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 ...
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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? ...
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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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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 ...
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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 ...
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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 ...
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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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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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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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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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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, $$\...
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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. ...
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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+ \...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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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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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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: \...
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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^...
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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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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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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 ...
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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{\...
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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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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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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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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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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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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 ...
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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 ...