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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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Non-invertible symmetries: Half gauging and 't Hooft lines

In (2.27) of https://arxiv.org/abs/2205.05086, when performing a gauge transformation of the background gauge field $B \to B +d \Lambda $, the 't Hooft line $H(\gamma)$ transforms as \begin{equation} ...
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Is work done by a charged particle not gauge invariant?

Work done by a charged point particle with charge $q$ in an external electric field derived from a scalar potential $\phi$ is given by $$W=q \phi.$$ Even if we add a magnetic field the definition ...
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Connection between Noether's theorem for gauge theories and 1-form symmetries?

Applying Noether's theorem to a gauge theory, one can show that the conserved current is generically of the form $$J^\mu=\partial_\nu k^{[\mu\nu]},$$ such that the conserved charge is really ...
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How is Wald deriving this Gauge condition: $\partial^b\, \overline{\gamma}_{ab} = 0$?

R. Wald in Section#4.4 of his book General Relativity derives the EFE in the case of a weak gravitational field by taking the curved spacetime metric $g_{ab}$ to be a "small" perturbation $\...
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What is a gauge transformation? How does it relate to Cauchy intial value problem and second functional derivative of the action?

I am having conceptual problems about 'gauge transformation'. I have well heard that gauge trnasformation is a 'local symmetry' and 'fake symmetry', but I want to understand it more precisely. I am ...
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Relation between the number of curvature functions and dimensions in GR

I am reading Weinberg's Gravitation and Cosmology. On page 10, it reads In $D$ dimensions there will be $D(D+1)/2$ independent metric functions $g_{ij}$, and our freedom to choose the $D$ coordinates ...
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Class of on-shell and gauge equivalent potentials in Chern-Simons theory

Let $(P, M, \pi, G)$ be a principal bundle with three dimensional manifold $M$ and compact, connected, simply-connected, and simple structure group $G$. We define a Lie algebra valued connection $1$ ...
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Why semi-simple and compact Gauge Group in YM Theory? [duplicate]

I'm studying the Yang-Mills theory, with the Action: $$ S=-\frac{1}{2}\int\mathrm{tr}_{\rho}(\mathcal{F}\wedge\star\mathcal{F}) $$ where $\mathcal{F}:=\mathrm{d} \mathcal{A}+\frac{1}{2}[\mathcal{A},\...
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2 votes
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The Abelian versus the non-Abelian commutator of covariant derivatives in field theory

In the case of Abelian symmetry, the covariant derivative is defined as $D_\mu\equiv \partial_\mu + ieA_\mu$, where $e$ is an arbitrary constant and the vector field, $A_\mu$ is a called a gauge field....
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Discrepance between gauge symmetry and Noether's first theorem

In QFT we're interested in the symmetries of our theory (encoded in the invariance of the Lagrangian under symmetries) because they let us study conserved currents of the theory by Noether's theorem. ...
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How are the gauge transformations of $\epsilon(\mu)$ and $A^\mu$ related?

To find a local field description of massless spin-1 particles that is Lorentz invariant, we can identify $\epsilon^\mu_{\pm}(k)$ with $\epsilon^\mu_{\pm}(k)+\alpha(k)k^\mu$. As $A^\mu$ and $\epsilon^\...
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Physical motivation behind gauging a global symmetry

Consider complex scalar field Lagrangian $$\mathcal{L}=(\partial^\mu\psi)^\dagger(\partial_\mu\psi) - m^2\psi^\dagger\psi\tag{1}$$ Which exhibits $U(1)$-invariance, i.e $\psi\mapsto e^{i\alpha}\psi$. ...
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Silly confusion about gauge invariance in supersymmetric Lagrangians - in particular, in the ${\cal N}=1$ superfield formulation of ${\cal N}=4$ SYM

Hoping to resolve a simple confusion I have about supersymmetric gauge theory, one which I ran into while trying to understand the ${\cal N}=1$ superfield formulation of ${\cal N}=4$ supersymmetric ...
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How to derive the gauge invariance of Yang-Mills action with external source?

In the Faddeev-Popov procedure of path integral of $$ Z[J] = \int [DA] e^{iS(A,J)}, \quad S(A,J)= \int d^4x [-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu} + J^{a\mu}A_{a\mu} ] $$ we have used that $S(A,J)$ is ...
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Gauge invariance using equations of motion [duplicate]

I am working with a lagrangian on a homework problem. I expect it to have some gauge invariance. I can show that the Lagranian is invariant under those (gauge) tansformations but I have to use ...
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Why is it valid to only consider linear-order gauge transformation when quantizing non-Abelian gauge theory?

To quantize the non-Abelien gauge theory. We multiply the path integral by: $f[A]=\int \mathcal{D}\pi exp[-i\int d^4 x {1\over \xi}(\partial_\mu D_\mu \pi^a)^2]$ then we can shift the argument in the ...
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Noether's first Vs Noether's Second theorem

I am reading about the first and second Noether's theorem from https://arxiv.org/abs/2112.05289. In the text, there is this piece, which I am not sure I entirely understand. Let us reflect briefly on ...
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In general relativity, is gauge invariance the same as coordinate invariance?

I always understood that gauge invariance of general relativity comes from the fact that the physical observables and states are the same regardless of the coordinates we choose to express them in. I ...
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Parity of a 1d Ising model, and with higher order terms

I don't know if this should be asked here or in a math stack exchange, but I'll try here first. Consider the classical 1d Ising model with periodic boundary condition: \begin{equation} H_2 (\vec{\...
Jun_Gitef17's user avatar
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Gauge redundancy and Gauge fixing

Take any gauge invariant theory, for instance QED. The QED Lagrangian is invariant under $$A_{\mu}(x)\rightarrow A'_{\mu}(x)=A_{\mu}(x)+\partial_{\mu}. \alpha(x)$$ I have chosen a local gauge ...
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Infinitesimal transformation of the Yang-Mills field

I am trying to obtain the infinitesimal transformation for the Yang-Mills field $A_{\mu}$. I want to show that $$ A^{\prime a}_\mu=A_\mu^a-\partial_\mu \theta^a-g_s f^{a b c} \theta^b A_\mu^c $$ For ...
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How to take the second-order gauge covariant derivative in quantum field theory?

I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field. (1) The first way is to write the second order gauge ...
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Gauge symmetries, isometries of spacetime and asymptotic symmetries

I am having a hard time understanding the physical meaning of asymptotic symmetries in General relativity. I think I understand the mathematics, but the meaning eludes me. I'll try to write the things ...
P. C. Spaniel's user avatar
2 votes
1 answer
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Why impose constraints in (Path Integral) Quantization of Proca action?

I was reading the Wikipedia page on Proca Action. To summarize, it is almost like Maxwell action, but with a mass term because of which Proca action does NOT have gauge invariance. From the equation ...
baba26's user avatar
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Does the gauge transformation rule on the gauge fields satisfy the definition of group action?

According to definition group action, it is required that action of a group $G$ on a set $X$ must satisfy the compatibility condition: \begin{equation} g \cdot (h \cdot x) = (gh) \cdot x \text{ for ...
Keith's user avatar
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For these gauge transformations in electromagnetism, $\phi\to \phi-\partial_t \lambda$ and $\vec A\to \vec A+\nabla\lambda$, why do the signs differ?

I was looking at this question on Mathematics S.E, as I would like to know the origin of the signs in the gauge transformations of the scalar and vector potentials components, $\phi$ and $\vec A$, of ...
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Can the derivative of a gauge-invariant quantity be gauge-dependent?

I am wondering whether it is possible for derivatives of a gauge-invariant quantity to be gauge-dependent. Certainly, the converse is true; taking the curl of a gauge-dependent quantity (the vector ...
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1 answer
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Does 2-form curvature $\Omega \in \Omega^2(P,\mathfrak{g})$ represent a physical quantity in gauge theory?

In gauge theory, all measurable physical quantities remain invariant under a gauge transformation. I have always seen that the curvature 2-form $\Omega \in \Omega^2(P,\mathfrak{g})$ associated to a ...
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Expression for the gravitational-wave energy-momentum tensor without choosing a gauge

While studying section 7.6 of Carroll's introduction to general relativity, I encountered difficulties deriving equation 7.165 for the gravitational-wave energy-momentum tensor. Unfortunately, I was ...
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Perturbative expansion and renormalization of non-abelian Yang-Mills theory solely in terms of gauge-invariant quantities?

In standard QFT, each term in the perturbative expansion for a gauge theory is not necessarily gauge-invariant. Only the whole sum of Feynman diagrams is guaranteed so. However, at least for QED, ...
Keith's user avatar
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7 votes
1 answer
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Gauge theories, boundaries and Wilson lines

My understanding of Wilson loops Let's work with classical electromagnetism. The 4-potential $A_\mu$ determines the electric and magnetic fields, which are the physical entities responsible for the ...
P. C. Spaniel's user avatar
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1 answer
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Equivalence of gauge-invariance and physical observable

This is somewhat philosophical than physics. In gauge theories, it is true (more like the first principle) that \begin{equation} \text{ physical observable } \Rightarrow \text{gauge invariant} \end{...
Keith's user avatar
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Why do we use local phase symmetry if we can couple the EM field to a divergence free current without it? [duplicate]

This question is pretty much what I am confused about. However, the answer says that we require local phase symmetry to keep the lagrangian gauge invariant. As far as I can see the argument only works ...
Lukas P's user avatar
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Gauge invariance of simplified weak interaction

I am having difficulties with a homework set. We are given the following lagrangian for a simplified weak interaction between an electron $\psi$, neutrino $\chi$, and a massive (complex) vector-boson $...
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Action for scalar field with externally specified EM fields and gauge

Consider the following Lagrangian for a complex scalar field $\Psi$ along with an electromagnetic environment, $$\mathcal{L} = ([\partial_\mu - i e A_\mu] \Psi)^* [\partial^\mu - i e A^\mu]\Psi - V(\...
evening silver fox's user avatar
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Gauge Invariance in Quantum Mechanics for Charged Particle

In Sakurai's book chapter 2, he has discussed the diffence between canonical momentum and kenitic momentum under gauge transformatiom. The former should be gauge-dependent, and the latter one would be ...
Ting-Kai Hsu's user avatar
5 votes
1 answer
285 views

Why are physical states not eigenstates of BRST charge?

In many texts in quantum field theory or string theory, it is stated that the BRST charge $Q$ must annihilate physical states because the states are required to be BRST invariant. Since $Q$ generates ...
Chang Hexiang's user avatar
13 votes
2 answers
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Trouble reconciling these two views on gauge theory

Very generally speaking, I view gauge theory as asking what local symmetries leave our theory invariant and then seeing the consequences. Thus, taking a look at the Lagrangian for electromagnetism, we ...
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Gauge invariance in QED with just fermion transformations

I've got myself confused about a basic question. If we have a gauge-invariant operator $\mathcal{O}$ whose expectation value is \begin{equation} \left\langle\mathcal{O}\left(x_1, \ldots, x_n\right)\...
Kris's user avatar
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1 answer
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Canonical and kinetic momenta vs gauge dependence

I am struggling a bit to understand the concept of gauge invariance/dependence with canonical momentum. For instance, if we consider a Hamiltonian of a particle in an electromagnetic field described ...
Akhaim's user avatar
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3 votes
0 answers
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Stueckelberg mechanism for interacting QFTs

The Stueckelberg mechanism or "trick" (see e.g. Section 4 of https://arxiv.org/abs/1105.3735) is basically a method to take the massless limit of massive gauge theories in a smooth way, ...
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Why can't we gauge the Lorentz group? (Or can we?)

One of the (many different, somewhat independent) routes to gauge theory is to start from a global symmetry of some kind and "gauge" it, which involves promoting it to a local symmetry and ...
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1 answer
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Gauge transformation with complex parameter in quantum mechanics with minimal substitution

This follows from the question Can an Electromagnetic Gauge Transformation be Imaginary? It's about the Hamiltonian $$H=\frac{(p-A)^2}{2m}$$ in units where $c=e=\hbar=1$. The question regarded a gauge ...
TheQuantumMan's user avatar
1 vote
2 answers
184 views

Are all actions time reparameterization invariant?

Let's concentrate on point particle mechanics on a one dimensional manifold for simplicity. The action is $$S [q,\dot{q}]=\int dt L(q,\dot{q},t).$$ Time reparameterization would involve $t \to t'=f(t)$...
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Can we compute tree-level amplitudes in string theory using the fundamental domain of $SL(2; \mathbb{C})$?

I am not a specialist in string theory. I understand the computation of tree-level string amplitudes (Veneziano or Virasoro-Shapiro), where three variables are fixed using the symmetry $SL(2;\mathbb{C}...
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How to get retarded scalar potential in Coulomb electrodynamics and what's the use?

In Coulomb gauge electrodynamics with potential $(\phi,\vec{A})$ and source $(\rho,\vec{J})$ we obtain the Poisson's equation for the scalar equation and the wave equation with transverse current ...
Sanjana's user avatar
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1 answer
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Deriving the gauge group from the little group

Arguments from the "little group" are used to show that the internal degrees of freedom of a massive particle transform under $SO(3)$, while the internal degrees of freedom of massless ...
Panopticon's user avatar
5 votes
1 answer
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Why is there only one coupling constant in Yang-Mills theory? Why are gluon self-coupling and gluon-matter coupling constants the same?

Is it non-trivial that the coupling constant $g$ in gluon self-interaction terms is the same as the coupling constant $g$ in gluon-fermion interaction term in Yang-Mills theory? Pure Yang-Mills theory ...
TOAA's user avatar
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1 answer
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Gauge invariance of linearized gravity with an arbitrary background spacetime

Consider here a background metric $g_{\mu\nu}$, we impose a perturbation $g_{\mu\nu}+\epsilon h_{\mu\nu}$ with $\epsilon\ll1$. Then we can write down the modified Einstein-Hilbert action with zero ...
Lain's user avatar
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1 answer
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Why did Schwinger [Phys. Rev. 74 (1948) 1439] choose a non-standard form of the Lagrangian density associated with the free electromagnetic field?

This sounds like a science history question, but is not. It is about acceptable forms for the Lagrangian density of electromagnetism. There is also a second question on the distinction between total ...
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