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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Standard Model: Problem with Masses of Elementary Particles?

In his book "Modern Particle Physics", Mark Thomson explains two problems with masses of elementary particles in the SM: (i) If we take the QED Lagrangian $\mathcal L = \bar{\psi}\left( i\...
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Proving that the kinetic energy of the gluon fields $G_{\mu}^{a}$ is $SU(3)$-invariant?

In QCD, we introduce the gluon-fields $G_{\mu}^{a}$ when defining the covariant derivative $D_{\mu} \equiv \partial_{\mu} + ig_{S}G_{\mu}^{a}T^{a}$ to make the free-particle Lagrangian $ \mathcal L_{\...
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Isn't Stress-Energy Tensor of Maxwell field in presence of charges gauge variant?

Versions of this question have been asked on this site before but have not directly addressed by concern. In the $(+---)$ convention the EM lagrangian in the presence of charge sources is $$ \mathcal{...
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Rotational invariance of the the elastic deformation in spherical symmetry

I write the elastic deformation $E$ for an incompressible material in spherical symmetry (Have a look at here Eq.(5.32) ). Since we are incompressible, $\mathbf{det} \,E=1$, so in spherical ...
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Proving that free-particle Lagrangian is not invariant under $SU(3)$ local gauge invariance?

I would like to show that the free-particle Lagrangian $\mathcal L = \bar{\psi}\left( i\gamma^{\mu}\partial_{\mu} - m\right)\psi$ is not invariant under the $SU(3)$ local gauge invariance ...
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$U(1)$ Local Gauge Invariance: What do $q$ and $\alpha(x)$ mean?

When deriving the existence of the photon, we start with the free Lagrangian $\mathcal L_{\text{free}} = \bar{\psi}\left( i\gamma^{\mu}\partial_{\mu}-m\right)\psi$ and require $U(1)$ local gauge ...
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What's the symmetry group $SU(N)/Z_N$?

I'm trying to understand David Tong's notes, specifically the discussion around page 92 where he's arguing that a different symmetry group may the group of QCD, namely $G'=SU(N)/Z_N$ instead of $G=SU(...
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Gauge invariance of loop Diagrams

Say we have a gauge-fixed QED Lagrangian: $$\mathcal{L} = - \frac{1}{4}{F}_{\mu\nu}F^{\mu\nu}+ \frac{1}{2a}\left(\partial_\mu A^\mu\right)^2+\bar\psi_1(i\gamma^\mu D_\mu - m_1)\psi_1.$$ My question ...
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Dirac equation with minimal coupling derivation from Klein-Gordon equation

I am wondering if the form of the Dirac equation given in the case of minimal coupling can be "squared" to give back the corresponding Klein-Gordon equation as in normally done in the field-...
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How local gauge invariance explain charge conservation and electromagnetic force appearance?

Without electromagnetic coupling, the QM charged particle wave function is not invariant under a local gauge transformation — one with a phase that depends on space (space-time): \begin{equation} \psi ...
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When we use Lorenz gauge or Coulomb gauge, the result formula for electric $E$ and magnetic field $B$ is same or different?

Gauge condition can be chosen as you like or not? is the Lorenz gauge is the only one correct? If Coulomb gauge can obtained exactly same results as Lorenz gauge for the electromagnetic fields E and ...
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How to understand gauge fixing condition?

In Peskin & Schröder's book, they wrote that there is no propagator for an Abelian field due to gauge invariance of the action, and then proposed a gauge fixing condition: $$\partial_\mu A^\mu = \...
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Gauge-invariance of the Maxwell Lagrangian in Srednicki's book

When it comes to the demonstration of the gauge-invariance of the Lagrangian of the Maxwell-theory Srednicki's book proceeds as follows: $${\cal L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + J^\mu A_{\mu} \...
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Can a gauge transformation eliminate singularity of gauge potential?

Suppose I have a gauge potential $A$ which goes to infinity at some point $x_0$. Can I use a gauge transformation \begin{equation} A'=U^{-1}AU+U^{-1}dU,~~~U=\exp\{-i\alpha^a(x)T^a\} \end{equation} to ...
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Gauging R-Symmetry

I know that if one gauges the supersymmetry group, you get supergravity. You can then further gauge the R-symmetry and these are the so-called gauged supergravities. But I don't think I've seen anyone ...
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Deriving a general gauge-invariant photon propagator

My understanding is that for a $U(1)$ gauge field $A_\mu$, the most general gauge-invariant kinetic term in the Lagrangian that can be written down which satisfies gauge invariance is something of the ...
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General relativity from the general linear group

I am looking at this answer: https://physics.stackexchange.com/a/225417/747. It states: Let $f\colon U\to V$ be any coordinates transformation on charts of a manifold $U,V\subset\mathcal{M}$ (i. e. a ...
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How does gauge symmetry constrain the dynamics of a field's physical degrees of freedom?

My rough understanding of gauge theory is that some of a field's degrees of freedom (d.o.f.) may turn out to be "non-physical" due to local symmetries. But does gauge symmetry constrain the ...
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Solutions of em potentials in Lorenz's gauge

When we solve the equations for em potentials in Lorenz's gauge: $ \nabla \cdot \textbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0$ we find the solutions: $\phi (\textbf{x}, t) = \frac{1}{...
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Lagrangian density for abelian gauge theories

I'm studying abelian and non-abelian gauge theories, using the geometric approach. I have defined the generalized potential (that is the connection 1-form of the principal bundle $P(M,G)$ where $M$ is ...
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Understanding $\rm SU(3)$ Gauge Invariance Through a Perturbation Theory

Here I am modifying the field theory approach, since I never taken a course on Quantum Field Theory. I am exploring Gauge Invariance in $\rm SU(3)$ by the following approach (which technically is ...
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Fix temporal gauge $A_0=f$ using an appropriate gauge transformation

Consider the Lagrangian \begin{equation} \mathcal{L}= -\frac{1}{4} F_{\mu \nu}F^{\mu \nu} - A_{\mu}J^{\mu} \ \ \ \ \text{ with } \ \ \ \ F_{\mu \nu}=\partial_\mu A_\nu - \partial_{\nu}A_{\mu}. \...
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The relation between gauge symmetry and global internal symmetry

I'm a little confused about the relation between the gauge symmetry and global internal symmetry of a field theory. I'd appreciate any clarification on this. My question can be phrased as the ...
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Gauge invariance and Gauss's law in $U(1)$ E&M theory

I am reading an article: Stable Gapless Bose Liquid Phases without Any Symmetry (and also see Pretko's paper on the same subject). There, the authors wrote that Gauss's laws are associated with ...
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$SO(3)$-invariant Lagrangian and null kinetic term for gauge fields

Let's say we have a Yang-Mills $SO(3)$ theory coupled to a real scalar field $\phi$. Then the Lagrangian can be written as $$ {\cal L} = \frac{1}{2}(D_\mu \phi)^T D_\mu \phi + \mu^2 \phi^T \phi - \...
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Writing the $U(1)$ gauge transformation as coordinate transformation

In quantum mechanics one can "always" write the way an operator acts on a wave function as a coordinate transformation. As an example we can look at unitary representation of the momentum ...
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Galilean invariance of Schrödinger equation [closed]

I'm trying to prove that if $\psi (\mathbf r, t)$ satisfies $$ i\hbar \frac{\partial\psi}{\partial t}(\mathbf r, t) = -\frac{\hbar^2}{2m} \left( \nabla-\frac{iq}{\hbar} \mathbf A \right)^2\psi(\...
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How does a lattice regulator work if we don't want observables to be invariant under “large” gauge transformations?

In quantum field theory (QFT), observables must be invariant under gauge transformations that are continuously connected to the identity, but invariance under "large" gauge transformations ...
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What are the local Lorentz transformations in general relativity?

What is the exact form of local Lorentz transformations (from the point of view of the metric) in a curved spacetime background like in general relativity? It should deviate substantially from ...
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Does introducing a gauge field into the complex scalar field theory Lagrangian change its dynamics?

I've been reading Lancaster & Blundell, and in Chapter 14 they focus on the Lagrangian $$ \mathcal{L}=(\partial^\mu\psi)^\dagger(\partial_\mu\psi) - m^2\psi^\dagger\psi. $$ To impose invariance to ...
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Is there any (locally) conserved charges associated to gauge symmetries?

I'm currently in my second year of master. From what I understand, in QFT, Noether's first theorem implies that for any continuous symmetry (i.e. associated to a $n$-dimensional Lie group $G$, $n\geq ...
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Gauge Invariance and the solutions of the equations

The equations for the potentials in E. M. are $$\nabla^2 \phi + \frac{\partial}{\partial t} \left(\nabla \cdot \textbf{A} \right) = -\frac{\rho}{\epsilon_0} \tag{1}$$ $$\nabla^2 \textbf{A} - \frac{1}{...
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Evolution of the curvature perturbation on super-horizon scales

energy conservation equation(1st order) in density perturbations and on superhorizon scales( $k<<aH $) implies $\delta \dot{\rho } = -3H(\delta p + \delta \rho) + 3 \dot{\Psi} ( \rho + p) $ we ...
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BRST as gauge symmetry or global symmetry or the generalization (e.g. in Peskin and Schroeder 16.4)

In Peskin and Schroeder (PS) Chap 16.4, such as after eq.16.45, in p.518, PS said: "local gauge transformation parameter $\alpha$ is proportional to the ghost field and the anti-commuting ...
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What is the deeper meaning of renormalizability and how is it related to gauge invariance?

I ask myself if the demand of local gauge invariance - say $U(1)$ invariance in free Dirac theory -$$L_D=\bar{\Psi}(i\gamma^\mu\partial_\mu-m)\Psi$$ is enough to define the full Maxwell-Dirac-...
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Why interacting-kinetic terms can't exist? (Can they?)

It seems that the Lagrangian of QED describing electrons and muons cannot include terms like that: $$\overline{\psi_{(e)}}i\not\!\partial\psi_{(\mu)}$$ where $\psi_{(e)}$ and $\psi_{(\mu)} $ are 4-...
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Current conservation in presence of SSB: weak currents

The Standard Model SSB breaks the $SU(2)_W \times U(1)_Y$ to $U(1)_{EM}$. The Noether currents associated to the unbroken group are $J_{a,\mu} = \bar{\psi} \gamma_\mu \frac{\tau_a}{2} P_L \psi $ and $...
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Is it possible to make $c^* c^*$ gauge invariant?

Consider a (fermionic) second-quantized lattice model. We know that we can make $c^*(r')c(r)$ gauge-invariant by applying Peierl substitution, i.e., $$ c^*(r') \exp(iA_{r'r})c(r) $$ so that if $c^*(r)\...
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Does symmetry violation make a approximate conservation law?

I know that Noether's theorem relates conservation laws with symmetries, and I read that to find CPT violation, Lorentz invariance symmetry needs to be broken. This implies that if a symmetry is ...
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Reparametrization invariance of the particle in GR

In the article Antibracket, Antifields and Gauge-Theory Quantization, the relativistic particle in spacetime is studied. Its action is $$\int\text{d}\lambda\,\frac{1}{2}\left(\frac{1}{e}\dot{x}^2-m^2e\...
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Why does the Standard Model predict zero mass for all vector bosons?

This video from 37:33 argues that the Standard Model predicts zero mass for all vector bosons as follows: Gauge bosons must have gauge invariance. For a vector field $A$ define a transformation $\...
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Exercise on gauge invariance and global and local symmetry [duplicate]

The following Lagrangian density describes a charged scalar field. Determine the global U(1) symmetry. Make this global symmetry local, give the Lagrangian density and show that this is Gauge ...
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Gauge invariance for a Lagrangian density with a mass contribution [closed]

Consider the free Maxwell Lagrangian density: \begin{align*} \mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \frac{m^2}{2}A^{\mu}A_{\mu} \end{align*} Where we added a mass contribution $m$. And sign ...
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Ward-Takahashi identity for the 2-point 1PI Green function of photons

I am following Sidney Coleman's lectures of Quantum Field Theory (World Scientific). For the renormalization of QED, he considered the following Lagrangian (Eq 33.54 in the book) \begin{equation} \...
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How do you use gauge symmetry to describe measurement of spin up electron?

I have been reading about gauge symmetry and how it arised from special relativity and the interaction can be understood using economical model as an analogy, I know the grids represent space and ...
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Gauge ivariance and canonical versus kinetic momenta for a charged particle in an EM field

I all, I am struggling to grasp the notion of gauge invariant when talking about an object like the canonical momenta $\frac{\partial L}{\partial \dot{q}_i}$ or kinetic momenta $m\dot{q}_i$. I am very ...
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Cohomology of the Koszul-Tate complex for an irreducible symmetry vanishes in degree $-2$

There must be something really obvious that I am missing here but any help is appreciated. Suppose I have a theory with some action $S$ on some fields $\phi$ such that any function vanishing on-shell ...
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What is the connection between vertex/spin models and gauge theory?

In the usual formulation of lattice gauge theories, one considers gauge variables on the links of a lattice (often hypercubic) taking value in some representation of a gauge group, $U_{ij} \in G$. The ...
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Gauge fixing and instanton calculation

I am reading Cheng&Li's book "Gauge theory of elementary particle physics". In section 16.2, I am confused by some assumptions. Suppose we have a $SU(2)$ gauge theory in $\mathbb{R}^4$ $$...
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Gauge transformation vs field excitation

I think I'm fundamentally misunderstanding something. Say I have a gauged Lagrangian for a complex scalar field $\phi$ with no SSB: $$\begin{equation} \mathcal{L} = (D_{\mu}\phi)(D^{\mu}\phi)^{\dagger}...

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