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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Series expansion of unitary operators in terms of other operators
I am reading lecture notes on local gauge invariance, part of Prof. Ethan Neil's course on Quantum Mechanics at the University of Colorado.
There, he writes about introducing a so-called comparator $U(...
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Extending Wigner's Classification with Gauge Symmetry
In Wigner's Classification, as far as I understand, one uses unitary irreps. of the Poincare group for treating an elementary particle, since then mass $m$ and helicity $h$ emerge naturally as ...
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Conserved charge at null infinity associated with Large gauge transformation
I am reading Strominger's lecture notes "Lectures on the infrared structure of gravity and gauge theory" (https://arxiv.org/abs/1703.05448). At some point, following (I guess) the authors of ...
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Is my expression of the Noether current $J^\mu$ for a local $\rm U(1)$ symmetry correct? If not, what's wrong?
The Lagrangian of electrodynamics reads $$\mathcal{L}=i\bar\psi\gamma^\mu D_\mu\psi-m\bar\psi\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$
where $D_\mu=\partial_\mu+iqA_\mu$. It is unchanged under the set of ...
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Is Galilean boost actually a gauge transformation?
In elementary physics, it is well-known that the Newton's law
$$\vec{F}=m\vec{a}$$
is invariant under Galilean transformations. However, Galilean relativity is not introduced in details in ordinary ...
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Functional integral for unconstrained superfields
Context
In this paper by Srivastava (also in his book "Supersymmetry, Superfields and Supergravity"), he proposes the functional integral for a chiral superfield $\Phi$. In order to work ...
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Is gauge symmetry necessary for charge conservation?
The common view is that gauge symmetry is necessary for conservation of charge(s) in Yang-Mills theory. But one thing I have never been able to get out of my head is, if there isn't any other possible ...
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The constraint commute with Hamiltonian in Gauge theory
When canonical quantizing gauge theory, we find that the canonical momentum corresponding to $A_0$ vanish since the Lagrangian contains no $\dot{A_0}$ . Thus we need to choose a gauge, for example, $...
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Wave equation for lightcone coordinate $X^-$
A quick question from Polchinski volume.1 : He claims in p.20 that the worldsheet lightcone coordinates $X^\pm$ also (i.e. in addition to the transverse coordinates $X^i$) satisfy the wave-equation. ...
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Do internal symmetries always leave the Lagrangian strictly invariant?
In order for the action to be invariant under a transformation, the Lagrangian can change by a total derivative. However, for internal symmetries (where the fields transform but not the coordinates), ...
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MTW Box 11.2: Any measure of geodesic deviation must be independent of stretchout. What? And why?
From Misner, Thorne and Wheeler, Box 11.2 (facsimile below):
Test geodesic same as before, except for uniform stretch-out in scale of affine parameter. Any measure of departure of the actual course ...
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Is physical energy is invariant under reflection?
I am wondering whether the energy landscape of a physical system (in particular a molecular conformation) can be considered invariant under reflection of the 3D space. My understanding is that some ...
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Proof of unitarity of gauge-invariant S-matrix in Peskin and Schroeder
I'm reading chapter 9.4 "Quantization of the electromagnetic field" of Peskin's and Schroeder's book.
When proving the unitarity of the gauge-invariant S-matrix, a trick is used.
$$
SS^\...
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Why insisting global invariance should hold locally? [duplicate]
In QED, when the Dirac Lagrangian is found to be not invariant under a local phase transformation,
$\psi$ $\longrightarrow$ $\psi'$ = $e^{i\theta(x)} \psi$
one tries to force it to get the desired ...
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Why does symmetry of the system depends on the gauge for particle in magnetic field?
Consider a particle in two dimensions with an external magnetic field in the $z$-direction. The vector potential can be chosen to be $$\mathbf{A}=-By\ \hat{x}$$ so that the Hamiltonian given by
$$H=\...
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Intuition/Motivation behind necessity of Spontaneous Symmetry Breaking to generate massive gauge bosons
In field theory textbooks, it is shown that while any gauge invariant Lagrangian must involve massless gauge fields, to obtain massive gauge bosons, we must postulate the existence of a Higgs scalar ...
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Can we say that, in some sense, the gauge group of gravity is the group of diffeomorphisms or coordinate changes? [duplicate]
In General Relativity Theory, there is a great freedom in the choice of space-time coordinates. As long as two coordinate systems can be related by a diffeomorphism, it seems that they both serve to ...
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What does it mean for a gauge field to have no gauge force?
The electromagnetic gauge field is $A + d\theta$, where $\theta \colon \mathbb{R}^n \to \mathbb{R}$ comes from a gauge function, $e^{i\theta(\vec x)} $. Let's set $A=0$. The curvature form is $0$ ...
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Laughlin gauge argument, integer quantum hall and periodic boundary conditions
In every treatment I have seen of the Laughlin gauge argument, it is suggested that as a flux quantum, $\Phi_0 = h/e$, threads through the cylinder or ribbon, that one unit of charge is pumped from ...
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Momentum Conserving Delta Function and Soft Gravitons
I am reading a paper called "Testing subleading multiple soft graviton theorem for
CHY prescription" written by Subhroneel Chakrabarti, Sitender Pratap Kashyap, Biswajit Sahoo, Ashoke Sena ...
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In lay terms, what are the real world consequence of the gauge invariances/symmetries upon which the Standard Model is built?
We learn that the SM is based on gauge invariance. Gauge invariance in turn is a consequence of symmetries (as I understand it) - meaning that a gauge theory having a symmetry is what makes it a gauge ...
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Is gauge invariance necessary to have Ward identity hold for off-shell amplitudes?
In this other SE post: Is it really proper to say Ward identity is a consequence of gauge invariance? it is shown that the on-shell Ward identity is a consequence of global $U(1)$ symmetry for QED. ...
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Gauge invariance of scalar QED
Let's consider a complex $\phi$ coupling minimally to $U(1)$ gauge field:
$$
\mathcal{L} = - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + (D_\mu\phi)^*(D^\mu\phi) - m^2 \vert\phi\vert^2 + \dots
$$
For now, I ...
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Confusion with gauge symmetry and spin
Suppose we have an electron with some arbitrary spin. This means that a 360 degree rotation in space will cause a phase shift of 180 degrees. However, the electron description (Dirac Equation) is ...
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Does a local spacetime symmetry lead to gravity?
In the case of charge a global $U(1)$ symmetry leads to the conservation of charge, however upgrading the global symmetry to a local symmetry leads to the electromagnetic potential field $A^\mu$ such ...
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Demanding $U(1)$ symmetry on Dirac Equation
For some personal studies I tried to derive the $U(1)$ invariant Dirac Equation as follows (I’ll be assuming natural units):
$$i\gamma^{\mu}\partial_{\mu}\psi-m\psi=0.$$
Then applying a $U(1)$ ...
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Noether identities and the relativistic point particle
I am trying to better understand Noether identities, i.e. relations between equations of motion in the presence of gauge symmetries for the example of the relativistic point particle.
Formally, a ...
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Can we use a magnetic vector potential in the case of time varying $E$-fields?
Most discussions of the magnetic vector potential defined through $\mathbf{B}=\nabla\times\mathbf{A}$ are only for working with static electric fields (for example, Griffiths:
If we instead require ...
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Expand an infinitesimal Wilson loop
I have a question about expanding an infinitesimal Wilson loop operator to get the field tensor $F_{\mu \nu}$ in chapter 3 of Fradkin's notes Classical Symmetries and Conservation Laws. For a ...
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Transforming potentials to the appropriate gauge
In my script I had the following example of gauge transformation (when $\rho=0$, $\vec j(\vec r,t)=0)$:
\begin{gather}
\phi(\vec r)= -\frac{\partial f(\vec r,t)}{\partial t}, \\
\vec A(\vec r,t)= -\...
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Transforming the potentials that satisfy Lorenz & Coulomb gauge to potentials that satisfy only Lorenz gauge
If $\vec E(\vec r,t)=\vec E_0sin(\vec k \vec r- \omega t)$ and also that $\rho(\vec r,t)=0$ and $\vec j(\vec r,t)=0$
I was asked to find $\vec A(\vec r,t)$ and $\phi (\vec r,t)$ which satisfy both the ...
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Lorenz Gauge different definitions
For the lorenz gauge we can either write:
$$\nabla \vec A(\vec r,t)+\frac{1}{c^2}\frac{\partial \phi(\vec r,t)}{\partial t}=0$$
If we also consider the following invariant transformations:
$$\vec A(\...
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Coulomb Gauge misunderstanding
If we have $\vec A(\vec r,t)$ and $\phi (\vec r,t)$ and we make the following gauge transformations:
$$\vec A(\vec r,t)'= \vec A(\vec r,t) + \nabla f(\vec r,t)$$
$$\phi(\vec r,t)'=\phi(\vec r,t) - \...
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Harmonic coordinates in General Relativity
I was going through chapter 10 of Wald's book on GR. The part I am reading concerns the harmonic gauge:
\begin{equation}
H^\mu\equiv \Box x^\mu=0.
\end{equation}
In eq. (10.2.34) he gives the Ricci ...
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"One-parameter" gauge transformation
In my advanced classical physics course, it was stated that the electromagnetic field strength tensor $F_{\mu\nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu}$ is invariant under "one-...
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In Schrodinger Equation Local $U(1)$ Gauge Invariance, How is Laplacian Simplified?
So I'm trying to derive the conditions necessary for local $U(1)$ gauge invariance in the Schrodinger equation, and I don't understand how Laplacian of the wavefunction is simplified the way it is. So ...
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Gauge independence of the gravitational-wave frequency and gauge dependence of the binary separation?
In the seminal paper by Cutler & Flanagan (1994), which uses multi-timescale analysis to derive waveforms in the post-Newtonian approximation without spin effects, they state that,
"In Eq. (...
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Is Weyl gauge together with Coulomb gauge a possible choice?
I am working in the Weyl gauge with an action in euclidean signature (so the Lagrangian is a Hamiltonian):
\begin{equation}
S=-\frac{1}{96\pi^2}\int_{\mathbb{R}^4} d^4 x\,\text{Tr}\left( E_i \cdot ...
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Intuition for Hilbert space of a quantized gauge theory
In the standard explanation, the physical Hilbert space of a quantized gauge theory (such as QCD) is given by the cohomology of the BRST charge acting on some larger, unphysical Hilbert space.
More ...
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Seeing gauge-covariance of the wavefunction for a uniform magnetic field
Consider the Hamiltonian
$$H=\frac1{2m}(\mathbf p-q\mathbf A)^2+q\phi$$
and let $\psi$ be a solution to the Schrödinger equation
$$i\hbar\frac{\partial\psi}{\partial t}=H\psi$$
Then if we gauge ...
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Can gauge fields be avoided by not specifying a coordinate system?
I've been studying quantum field theory, and recently my focus has been on the complex scalar field $\phi(x_\mu)\in\mathbb{C}$ because it seems to be the simplest field that can couple to an ...
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Can local supersymmetry be characterized entirely in terms of observables?
Global symmetries can be defined through their effect on observables. In contrast, quantum theories are often constructed with the help of symmetries that leave observables invariant, like the ...
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The residual gauge symmetry of Yang-Mills theory after Wick rotation
I am a bit puzzled by a statement in this question here. In particular, the claim that the residual gauge symmetry in Yang-Mills theory disappears upon Wick rotation to the Euclidean theory.
For ...
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Gauge invariance / charge conservation on string ending on brane
Consider a fundamental string charged under the Kalb-Ramond 2-form $B$ and ending on a D5-brane. Charge conservation implies the existence of a 1-form gauge field $A$ with field strength $F$ living on ...
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Transformation of field strength tensor in non-abelian gauge theory
The field strength tensor is defined as
$$F_{\mu\nu}^a=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu +g f^{abc} A_\mu^b A_\nu^c$$
where $f^{abc}$ are the antisymmetric structure constants and $A_\mu^a$ ...
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Peierls substitution, periodic boundary condition, gauge invariance
I am considering a tight-binding model in a magnetic field, and studying the Peierls substitution
$$t_{jk} \to t_{jk} e^{i\frac{q}{\hbar}∫_j^k \vec{A} \cdot d \vec{r}}$$
In some papers, such as this ...
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Residual symmetry of Polyakov action in general backgrounds
In Becker & Becker, Schwarz book String Theory and M-Theory, page $40$ is stated that after choose the conformal gauge $h_{ab} = \eta_{ab}$ in the Polyakov action with background field $G_{\mu \nu}...
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About the use of static gauge in string theory
In the paragraph before equation (2.1.44) of GSW 1st Volume, 25th anniversary edition, it's said that after fix the choice of conformal gauge $h_{\alpha \beta} = \eta_{\alpha \beta}$, there still a ...
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BRST variation of $\delta_{\alpha}F^A$ in $S_3$ in BRST section of Polchinski
The Faddeev-Popov action reads
$$S_3=b_Ac^{\alpha}\delta_{\alpha}F^A(\phi).\tag{4.2.5}$$
I want to find the BRST variation of the gauge variation of $F^A$ in $S_3$ i.e. $$b_Ac^{\alpha}\color{red}{\...
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Angular momentum of vacuum solution in Einstein gravity
In Strominger's "Lecture Notes on Infrared Structure of Gravity", page 38, he mentioned about how part of this whole mess about "vacuum degeneracy" (classically, i.e. in the sense ...
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