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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2answers
87 views

Local symmetry restoration via a gauge field

In the book Quantum Field Theory for the Gifted Amateur, the author stated that, having a field that transforms locally via $\psi(x) \rightarrow \psi(x)e^{i \alpha(x)}$ will destroy local symmetry -...
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Gauge invariance doesn't actually force usual choice of covariant derivative?

As we all know, a gauge invariant theory is of the form $$ \mathcal{L} = \bar{\psi} \gamma^\mu \left( i\partial_\mu + A_\mu^a T^a\right) \psi.$$ The multiplet $\psi$ and gauge field $A_\mu = A^a_\mu ...
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Review: If true, what makes the vacuum of a local ${\rm U(1)}$ gauge theory unique?

Long back, I posted a question with title "Is the vacuum of a local U(1) gauge theory unique?", which, as the title suggests asked whether the vacuum of a "spontaneously broken" the gauge theory is ...
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Why gauge transformations does not/cannot change a quantum state?

Sometimes it is argued that the gauge symmetry is not a symmetry of the quantum field theory. The Xiao-Gang-Wen argument is as follows. Gauge symmetry is a redundancy in our description of the ...
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Normalization of the Chern-Simons action in the Dijkgraaf-Witten paper

I am trying to understand the seminal paper "Topological gauge theories and group cohomology" by Dijkgraaf and Witten. They consider an oriented three-manifold $M$, compact Lie group $G$ and a $G$-...
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What causes here an apparent violation of Elitzur's theorem?

Elitzur's theorem [Ref. Andreas Wipf, Statistical approach to quantum field theory] states that A local gauge symmetry cannot break spontaneously. The expectation value of any gauge non-invariant ...
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Does general covariance really imply that there is no conserved stress energy tensor in gravity?

There are many questions on this website which ask similar questions to this, but none I have found have asked this exact question. First, a bit of history on the origin of Noether's theorem. David ...
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1answer
66 views

Why does this amplitude not vanish by the Ward identity?

Consider the process $e^-\rightarrow e^-\gamma$ depicted in the following Feynman diagram. The spin-averaged amplitude with linearly polarised photons is $$\overline{|M|^2}=8\pi\alpha\left(-g^{\mu\nu}...
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Why does gauge invariance have physical consequences?

My understanding is that gauge invariance occurs when the description of a physical field as a mathematical field (i.e., function whose domain is space-time) contains a redundancy: there are ...
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Definition of gauge freedom in electromagnetism and general relativity

The freedom we have in choosing the vector potential $\vec{A}$ in E&M is referred to as the gauge freedom, whereas in general relativity (GR), we refer to the freedom to choose any coordinate ...
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Gauge symmetries not from promotion of global symmetries

The most intuitive example of a gauge symmetry is such where you take a theory that has some global symmetry, and ask what needs to be done for this symmetry to be local. This procedure involves the ...
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Field Strength and Source terms

This question is related to my recent unanswered question, but it was too complicated so please let me make this new question at first. First, I consider a field strength which is expressed as \...
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Field strength invariance on curved manifold

Promoting a global gauge symmetry $\Psi\rightarrow\Psi^\prime=U\Psi$ to a local symmetry requires the definition of a gauge covariant derivative as ${D_\mu}^\prime\Psi^\prime=UD_\mu\Psi\Rightarrow{D_\...
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Local Symmetry or gauge transformation of second kind in QED

While ensuring the gauge invariance of the lagrangian $$\mathcal{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}+\bar{\psi}(i \not \partial-m) \psi-e \bar{\psi} \gamma^{\mu} A_{\mu} \psi$$ we consider the ...
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Vacuum expectation value (VEV) of a Gauge theory - Spontaneous Symmetry Breaking (SSB) - Higgs Mechanism

I am dealing with a sort of scalar QED with a term of SSB \begin{equation} \mathcal{L}=\left|D_{\mu} \phi\right|^{2}-\frac{1}{4}\left(F_{\mu \nu}\right)^{2}-V\left(\phi^{*} \phi\right) \end{equation} ...
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How can I prove that the Noether charge represents actually conservation of electric charge?

I have a question about Noether's theorem for global gauge invariance of a complex scalar field. Starting from \begin{equation} \mathscr{L} = \partial_{\mu}\Phi \partial^{\mu}\Phi^{*} + \frac{m^2c^2}{...
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Linearized gravity and local Lorentz symmetry

Action for linearized gravity is well-known, see for example David Tong: Lectures on General Relativity: $\mathbf{The\;Fierz-Pauli\;Action}$ The linearised equations of motion can be derived ...
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Is $F_{\mu\nu}F^{\mu\nu}$ the only possible gauge-invariant Lagrangian for the electromagnetic field?

Maxwell's equations can be derived from a Lagrangian formulation using the Lagrangian term (modulo some constants) $$\mathcal L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^\mu.$$ Focusing on the free ...
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When one says that Yang Mills theory is a non-abelian field theory, what specific gauges and gauge transformations are implied in this statement?

What specific gauges and gauge transformations are implied when one states that the order of such gauge groups are vital? Can this please be explained as simple as possible (:
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Gauge transformation issues of Proca Lagrangian

It seems to be trivial to claim that the mass term in Proca Lagrangian is NOT "gauge invariant". The claim show up in Wikipedia or section 6.3 in Greiner's book "Field Quantization". But I'm puzzled ...
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Path integral for particle with spin and Dirac propagator

I am currently trying to compute a path integral for fermion particle using the action, provided in chapter 9 of Polyakov "Gauge fields and strings", and show that it yields Dirac propagator in the ...
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What is a 'potential' in the context of physics and gauge symmetry?

I heard someone say that the definition of a gauge transformation is any formal, systematic transformation of the potentials that leaves the fields invariant. What is the definition of a potential in ...
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1answer
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Applications and limitations of the Hamilton-Jacobi formalism

It was my understanding that the Hamiltonian formalism was inadequate to describe systems that are invariant under time reparametrization or that have gauge symmetries. However, I see in Classical ...
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1answer
124 views

Spacetime translation is not a gauge invariant symmetry in Maxwell's theory

Consider the Action for Electromagnetism, \begin{align} I=-\frac{1}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu}, \end{align} Suppose we perform a space-time translation, $x^{\mu} \to x^{\mu} + \epsilon ^{\mu}...
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1answer
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Usage of the word “gauge” in these contexts

This is probably a trivial question to someone with more knowledge than myself. I have met the word "gauge" now in the context of the gauge transformations that change the scalar and vector potentials ...
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Higgs mechanism and gauge symmetry

In the case of massive bosons of weak interaction we have Higgs mechanism to save the symmetry. I do not see how with Higgs the symmetry is preserved? Because the mass term is what gives us problems ...
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Whcih is more fundmental of Ampere's law or Biot Savart Law?

There's couple of related question before. See Reference: Ampère's law vs Biot Savart law What is the difference between Biot-Savart law and Ampere's law? Is Biot-Savart law obtained ...
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63 views

General gauge algebra identity

In https://arxiv.org/abs/1011.1145 the following rather general gauge algebra identity (2.4) is used $$\delta_{gct}(\xi) B_\mu^{\>\>A} + \xi^\lambda R_{\mu\lambda}^{\quad\! A} -\sum_{\{C\}}\...
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125 views

Why is gauge invariance so important?

Quantizing the electromagnetic field (without ghosts or gauge fixing terms) using path integrals is not possible because the propagator is not well defined. Textbooks such as P&S or Ashok Das say ...
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How to prove the covariance of the covariant derivative for adjoint representation?

Consider a triplet $\Phi$ transforming under the adjoint representation of the $SU(2)$ group. The covariant derivative is $$ (D_{\mu}\Phi)_{a} = \partial_{\mu}\Phi_{a} - g\epsilon_{abc}A_{\mu}^{(b)}\...
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Topological number and gauge invariance

In QCD or other non-abelian gauge theories, we come across infinitely many vacua that are gauge equivalent but have different topological numbers. We then say that the instanton solution is tunnelling ...
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1answer
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Infinite conservation laws applying Noether's theorem to the local $U(1)$ symmetry of QED

The Lagrangian of QED is \begin{equation} \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\big(i\not{D}-m\big)\psi \end{equation} where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$...
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Why do we require that the gauge condition $\alpha(x)$ falls off at infinity?

Let's say we are working in QED. The lagrangian is \begin{equation} \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\big(i\not{D}-m\big)\psi \end{equation} where $F_{\mu\nu}=\partial_{\mu}A_{\...
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Conserved currents in the Standard Model

As we know, the gauge bosons $W^{\pm}, Z$ and $\gamma$ couple to currents in the Standard Model. Now my question is are these currents conserved? I mean, there is no doubt about the electromagnetic ...
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Gauge invariance for a Feynman diagram/set of diagrams in the Standard Model

As the question suggests, I am interested if there is a "rule of the thumb" (or some similar "trick") that allows one to tell which set of subgraphs in the Standard Model are gauge invariant. To be ...
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1answer
62 views

Showing that the $A$-$j$ coupling in classical electromagnetism is gauge invariant

I am attempting an early exercise from Altland's Condensed Matter Field Theory. The electromagnetic field's action is given as: $$S[A]=\int d^4x(c_1F_{\mu\nu}F^{\mu\nu}+c_2A_\mu j^\mu),$$ and I wish ...
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Why isn't too restrictive to consider a connection of the form $-iA$ in the magnetic Ginzburg-Landau model?

I'm facing this equation as a mathematician, so the problem could easily be my lack of knowledge in physics. In the magnetic Ginzburg-Landau model the energy has the form $$ E(A,\phi)=\int_{\mathbb{R}...
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1answer
79 views

Local phase invariance of our lagrangian

So in lectures we have just started looking at classical field theory. We were introduced to symmetries and we were told in general our lagrangian wont be invariance wrt to local phase changes but if ...
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What is the way to express Yang-Mills symmetry groups without gauges?

Given a Yang-Mills theory such as $SU(3)$ which has 8 gluons. After we gauge-fix this theory, it no longer has $SU(3)$ guage symmetry. Yet, we still use the group constants and the 8 types of gluons ...
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Help with Invariance of Euler-Lagrange equations under a gauge transformation?

I'm presented with the following question: From the Lagrangian for a relativistic particle in an electromagnetic field: $$L=-\frac{m}{\gamma} - eV + e\vec A \cdot \vec v$$ show that the gauge ...
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1answer
68 views

Gauge invariance of the path integral measure

When we have an abelian group the gauge transformation of gauge field is given by $$ A^\mu \rightarrow A^\mu + \partial^\mu \alpha \equiv A^{\prime \mu} $$ Here it's easy to see that the path ...
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Is there a connection between spontaneous symmetry breaking and massless photons?

I haven't studied a lot about these topics to put it that way. But I wonder if there is a connection between spontaneous symmetry breaking and the fact that photons are massless? The spontaneous ...
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2answers
67 views

What happens to the symmetry after gauge fixing?

Given a theory with gauge symmetry. After gauge fixing where does the symmetry go? Does the gauge symmetry turn into a global symmetry? For example there is a way to quantize fields theory with BRST ...
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2answers
102 views

Is this a gauge symmetry?

Imagine a hypothetical action: $$S=\int \left(\frac{\partial}{\partial t}\phi(x,t)\right)^2 d^3x dt$$ Now we have a symmetry of the action: $$\phi(x,t)\rightarrow \phi(x,t)+\chi(x).$$ At time $t$, $\...
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137 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 ...
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1answer
75 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) + \...
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4answers
192 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) + ...
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Does gauge invariance of scalars/fermions in the adjoint representation induce the existence of Wilson loop and (then) covariant derivative?

First, for the known case of $U(N)$ gauge invariance we have scalars (it works for fermions too) transforming as (fundamental representation) $$ \phi(x)\to V(x)\phi(x), \ \ V(x)\in U(N) $$ So then we ...
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1answer
78 views

Gauge-independence of the “$n$-particle” probability current

Problem Show for the non-relativistic quantum mechanical problem of $n$ electrons in a static homogenous magnetic field $\bf B$ and ignoring spin that the probability current density is gauge ...
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1answer
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Bosonization and gauge symmetry

The bosonization map relates the fermionic current $\bar{\psi}\gamma\psi$ to the bosonic current $\partial\phi$, and also the components of $\psi$ to $e^{i\sqrt{\pi}\left(\phi\pm\bar\phi\right)}$. ...

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