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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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 ...
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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 ...
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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)\...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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Compactification and gauge choice for instanton solutions

I have several doubts regarding topological solutions in pure YM -- these are related both to less trivial topological misunderstandings as to rudimentary gauge fixing confusions of mine. What is the ...
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Counting number of equations for Rarita-Schwinger field (in Supergravity textbook)

I am reading the book "Supergravity" by Freedman and van Proeyen (2012). On page 96, they are talking about the equation of motion of massless vector-spinor field (the spinor index is ...
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Why the spectrum of square lattice with $\Phi=m \Phi_0$ with $m\in\mathbb{Z}$ is the same as $\Phi=0$?

In solid-state physics, in the study of square lattice in a perpendicular homogenous magnetic field, I have seen that when the flux per cell is an integer multiple of the flux quantum, i.e. $\Phi=m \...
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How is related the exponential map, the covariant derivative and the gauge transformations under Lie groups

I will try to formulate my question the best way possible considering my lack of mathematical formalism. I just want to know, in every aspects, why spinors transforms (under Lorentz transformations) ...
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Flavor changing neutral current with photon

I'm trying to understand why the penguin diagram for photons takes its form. $I$ here is the integral and some CKM elements. The author (Ch1.6.1) says that for $b\rightarrow s\gamma$, we get a factor ...
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Construction of $SU(2)_L$ invariant Lagrangian

I am reading this paper about "Scalar Electroweak Multiplet Dark Matter" written by Wei Chao et al. I am puzzled for their construction of $SU(2)_L$ invariant Lagrangian, Eq.(6). $$ \begin{...
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Gravitino's and vielbein's gauge transformation in Freedman's Supergravity

I'm facing a problem with Ex. 11.7 of Freedman's "Supergravity", specifically about gauge fields and their transformations. At page 218 the authors present a useful list of all structure ...
Fredrigo6's user avatar
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Frames and coordinate transformations in General Relativity

I am learning the frame formalism in differential geometry and I am trying to reconcile this with applications in general relativity, especially in contexts like the tetrad formalism. Consider a ...
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Gauge freedom in general relativity for non-vanishing energy momentum tensor

When studying the linearized Einstein field equations in vaccum $$\partial^{\lambda}\partial_{\nu}\gamma_{\lambda\mu} + \partial^{\lambda}\partial_{\mu}\gamma_{\lambda\nu} - \eta_{\mu\nu}\partial^{\...
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Original motivation for promoting global symmetries to local

Assume we are dealing with a Lagrangian $\mathcal{L}$ for matter field $\psi$ which has a global $G$-symmetry and it's possible to promote this global $G$-symmetry to a local symmetry after the usual ...
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Non-diagonal matrix element in the Wannier functions basis

I'm going through Marzari's paper about maximally localized wannier functions. There is a passage I'm trying to understand but can't seem to get. On the third page, between equation 5 and 6, they have ...
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Differential Forms and Gauge Invariance?

In the Sean Carroll book on GR, page 86 in chapter 2 says $$d(A+d\lambda)=F,$$ where $A$ is vector potential and $\lambda$ is some $0$-form scalar. $F$ is the electromagnetic field strength tensor. I ...
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Gauge-fixing condition invariant under auxiliary gauge transformation

In quantum gravity one usually splits the metric $g= \bar{g}+h$ into a background field $\bar{g}$ and a fluctuation field $h$. In order to obtain a propagator one has to gauge fix the action (e.g. ...
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Section 80 of Dirac's book: Principles of Quantum mechanics

I am looking for informations about one of Dirac's calculation: in his book Principles of Quantum Mechanics (4th edition), he developed inside section 80 a gauge-invariant version of QED in which he ...
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Invariance of gauge-fixing condition in background field method

In the Peskin & Schröder (chapter 16.6) they use the background field method and spilt the gauge field into an background field $A$ and a fluctuation field $\mathcal{A}$. Next they claim that the ...
Silas's user avatar
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Spacetime interval not invariant under gauge transformations

Observables have to be gauge invariant, but clearly the spacetime interval: $$ds = g_{ij}dx^idx^j = (\eta_{ij}+h_{ij})dx^idx^j\tag{1}$$ is not invariant under a transformation: $$ h_{ij} \rightarrow ...
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Gauge covariant derivative for fields in tensor representations with multiple indices

In QFT, for fields transforming under some Gauge group, one defines the covariant derivative as $$ (1)\qquad D_{\mu} \phi = \partial_{\mu}\phi -igA_{\mu}^k \rho(t_k)_{ab}\phi_b $$ If $dim\rho=dim(\...
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Why not design classical electron theory in the vacuum gauge?

The Lienard-Wiechert potentials (published 1898-1900) are used to determine the field strength tensor of a classical (point) electron in arbitrary motion. If we perform a gauge transformation on the ...
Christopher Hayes's user avatar
2 votes
2 answers
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Diffeomorphism invariance for derivative of scalar fields [closed]

In GR, it is well-known that the gravitational stress-energy tensor is a pseudotensor, i.e. it is not gauge-invariant. To make it gauge-invariant one needs to take it under average integral $\langle \...
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Diffeomorphism Invariance of Terms in Lagrangian which use Gauge Fields

A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms ...
Tom's user avatar
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Maxwell's equations in curved spacetime not invariant under metric perturbation gauge transformations

Gauge transformation in general relativity This post states that $$h_{\mu\nu} \rightarrow h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu \tag{1}$$ Is a gauge transformation for a spin-2 ...
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Massive vs. massless relativistic point particle in einbein form: Difference in the gauge structure?

The action for the relativistic point particle with mass $m \geq 0$ in a curved background is given by: \begin{equation} S[X] = \int_{\tau_0}^{\tau_1} d\tau \left[ e(\tau)^{-1} g_{\mu \nu}(X(\tau)) \...
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How can coordinate transformation be gauge transformations?

I'm having great conceptual difficulties with gauge transformations in general relativity. Let me explain my problem by making a comparison to gauge transformations in electrodynamics: In ...
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Physical interpretation of the spin connection field for the Dirac equation in curved spacetime [duplicate]

When dealing with the Dirac equation in curved spacetime one has to replace the partial derivative with the following covariant derivative: ${\partial_{\mu}}-\frac{i}{4}\omega_{\mu}^{\alpha\beta}\...
physics_2015's user avatar
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Gauge invariance or diffeomorphism invariance in GR of observables?

I am confused by the definitions of a gauge transformation, a coordinate transformation and a diffeomorphism. In particular, should observables in GR be fundamentally invariant under gauge ...
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Change in number of gauge symmetries after adding auxiliary fields to the action

As per part (c) of Ex. (3.17) in Ref. 1, the number of gauge symmetries of an action does not change after adding auxiliary fields to it. But we know that a Stueckelberg field is an auxiliary field, ...
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Gravitational waves, gauge transformations and observables

In GR the (perturbed) metric obeys a wave equation in a certian gauge. The same thing happens EM waves, $\Box^2A_{\mu}=0$ only on a certian gauge, but $\Box^2F_{\mu\nu}=0$ without fixing the gauge. I ...
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Are equations of motion invariant under gauge transformations? [duplicate]

We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations? If yes, can you show a mathematical proof (instead ...
vyali's user avatar
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How to derive the gauge transformation of a Lagrangian with auxiliary fields?

Suppose Lagrangian $L_1(y_1,y_2)$ is a functional of fields $y_1$ and $y_2$, and Lagrangian $L_2(y_1,y_2,z_1,z_2)$ is a functional of the fields $y_1,y_2$ and the auxiliary fields $z_1$ and $z_2$. If ...
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Is there an operator to measure pre-SSB boson states?

A common statement is that post-SSB electroweak gauge bosons are linear combinations of pre-SSB gauge bosons. It is also usually stated that pre-SSB bosons can also be thought of as linear ...
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Gravitational wave Riemann tensor in the source gauge discrepancy

In many text books such as Gravitational waves volume 1 by Maggoire and General relativity by Hobson the solution to linearised Einstien field equations can be computed in the far-field using the ...
Flanders's user avatar
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Can we derive that the wavefunction changes as $\Psi'(\vec{r},t)=e^{i(q/\hslash c)\Lambda(\vec{r},t)}\Psi(\vec{r},t)$ under a gauge transformation?

The relevant time-dependent Schrodinger equation, for a spinless charged particle in an EM field, reads $$ i\hslash\frac{\partial \Psi}{\partial t}=\left[\frac{1}{2 m}\left(\vec{p}-\frac{q}{c}\vec{A}\...
Solidification's user avatar
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Faddeev-Popov trick in QED Peskin and Schroeder

On page 297 of Peskin and Schroeder, the book obtains the propogator $$\tag{9.58} \tilde{D}_F^{\mu\nu}(k)=\frac{-i}{k^2+i\epsilon}\bigg(g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2}\bigg).$$ The book then ...
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Symmetry breaking, gauge invariance and superconductivity

I still have some confusions over symmetry breaking in superconductivity. To begin with it’s clear gauge symmetry can’t be spontaneously broken, since it’s not a symmetry to begin with. I want to ...
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Residual gauge freedom and complete residual gauge fixing in lorenz gauge

What I understand after reading all answers from physics stack exchange related to residual gauge freedom and complete residual gauge fixing are as follows; The gauge transformation is: $A'_{\mu}$=$A_{...
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