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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Vectors in AdS/CFT: scaling dimension and near boundary behaviour
I'm trying to understand how the near boundary expansion of a field in AdS$_{d+1}$ is related to the conformal dimension of the corresponding operator in the dual CFT$_d$.
I use coordinates in which ...
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Gauge invariance of a lagrangian density
Considering the following Lagrangian density:
$$\mathcal{L} = -A_{\mu\nu}A^{\mu\nu}$$
with $A_{\mu\nu}$ a generic field.
I have to check if the action is invariant under:
$$A_{\mu\nu} \rightarrow A_{\...
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Why are these loop diagrams involving Goldstone bosons cancelling one another in the zero-momentum limit?
I have been working with one-loop corrections to certain processes. Let's say, corrections to the $Z\bar{f}f$ vertex. Of, course I need to consider the following diagrams:
Here $\phi^{\pm}$ is the ...
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Is the Dirac action invariant under $U(1)$ local gauge transformations?
I have usually found in books/lectures that the Dirac theory, given by
$$S=\int d^4x\bar\psi(i\gamma^\mu\partial_\mu-m)\psi, $$
is invariant under $U(1)$ global transformations (which is evident) but ...
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What is the meaning of gauge theory and Yang-Mills theory? [duplicate]
I would appreciate it if you guys would help me to understand the idea behind these two concepts: Gauge field and Yang-Mills theory.
What I think I understand is: Suppose we have a Lagrangian that ...
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GW gauge choice shortcut
When solving the linearised Einstein field equations for gravitational waves, we define the trace reversed metric $\bar{h}_{\mu\nu}$ for which we write the coordinate transformations $x^\mu \...
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What is the gauge symmetry of wave equations in curved spacetime?
The lagrangian of electroweak interaction is invariant under $SU(2)\times U(1)$ gauge transformations, QCD lagrangian is invariant under $SU(3)$ gauge transformations. There're lagrangians of ...
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Can residual gauge symmetries have compact support?
I have been reading this review about asymptotic symmetries, and one definition that is used is apparently due to Penrose:
$$
G = \frac{\mbox{gauge symmetries preserving boundary conditions}}{\mbox{...
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Applying the Optical Theorem in Non-Abelian Gauge Theories
I am reading P&S (Peskin's and Schroeder's book on QFT), Chapter 16.3 entitled Ghosts and Unitarity. The authors employ the optical theorem to calculate the imaginary part of a $f\bar{f}\...
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How (if) is the gauge group for gravity incorporated in the Calabi-Yau manifold of string theory?
The Calabi-Yau manifold is a stable complex 3D (or real 6D) manifold on the geometry of which information about strings can be stored as fibre bundles of tensors, more or less like the electromagnetic ...
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Transverse and longitudinal component of a photon propagator
I'm studying the QFT with Peskin and Schroeder's book. There is a point which is mentioned several times in the book and I don't quiet understand which is the transverse and longitudinal component of ...
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Gauge-invariant vertex structure for $h\to\gamma\gamma$ via fermion loop
I am struggling (a bit) with the following diagram for scalar Higgs to two photons.
$h\to\gamma\gamma$" />
If I put $q_\mu$ on-shell (or at the very least if I put both $q_\mu$ and $q'_\nu$ on-shell),
...
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Can gauge theory be explained completely non-relativistically?
I have been reading in the book: "Particles, fields and forces - a conceptual guide to quantum field theory and the standard model". The specific chapter I have been reading you can find ...
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Local gauge transformations and Noether Current
I am trying to derive the conserved Noether current, corresponding to a local gauge transformation in the theory of charged matter, coupled to the electromagnetic field: $$\mathcal{L}=-\frac{1}{4}F_{\...
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Why are photons massless? [duplicate]
I am trying to understand, from the QFT perspective, why photons are massless. Let's consider the following Lagrangian for a massive U(1) gauge boson
$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+m^2A^...
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On the Ward Identity in QED
I am reading P&S, particularly Chapter 5.5. The authors are trying to derive an expression for the Ward identity (not formally, but still). They claim that the amplitude describing a photon ...
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How to find Weyl/temporal gauge fixing condition?
Transformations that leave the field invariant:
$$\vec{A}' = \vec{A} + \nabla f$$
$$\phi' = \phi -\frac{\partial f}{\partial t}$$
I would like to solve for the weyl gauge, aka a gauge that leaves
$$\...
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Gauge fixing and explicit view of arbitrary scalar function [duplicate]
With the introduction of the vector $ \mathbf {A} $ and scalar $ \varphi $
of the potentials of the electromagnetic field, an ambiguity arises that does not create any problems of a fundamental nature,...
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Vasiliev equations Cartan gauge invariance - arXiv:1208.3880
I am going through https://arxiv.org/abs/1208.3880 and I am somehow struggling to show that the Vasiliev field equation (2.24)
\begin{equation}
d_{x}W + W *W = 0
\end{equation}
is invariant under the ...
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Are there "physical fields" in non-abelian gauge theories?
In QED the field strength $F_{\mu\nu}$ is gauge-invariant. This is reasonable since its components are physical fields $\vec{E}$, $\vec{B}$, so it doesn't matter in which gauge you express it.
However,...
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Why is it that in gauge theories the assumption "all fields decay sufficiently rapidly at infinity" not justified anymore?
I read that in gauge theories the assumption that "all fields decay sufficiently rapidly at infinity" is not justified anymore and therefore, one needs to consider boundary terms that ...
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Does Pauli-Villars regularization protect gauge invariance in QED?
I have read this answer, where the statement that Pauli-Villars protect gauge invariance at least for QED is talking about. There @ved stated that, the modified propagator
\begin{equation}
\frac{1}{k^{...
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Can we make a massive non-abelian gauge field renormalizable by gauge fixing without Higgs mechanism?
There have been a lot of similar questions about this topic on this website, such as Gauge invariance is just a redundancy. Why is massive abelian gauge field renormalizable but massive non-abelian ...
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Explicit derivation that the Faddeev-Popov functional determinant is gauge invariant
I am trying to show that the Faddeev-Popov functional determinant used in the quantisation of non-Abelian gauge theory is indeed gauge invariant. As shown in my previous question when we follow the ...
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Are there any theories of the beginning or the end of the universes where all fundamental symmetries would be violated?
There are theories trying to explain the origin of the universe (such as inflationary models) and also theories that try to predict how will the universe end (like the "big freeze", the &...
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Showing that the integration measure is preserved under gauge transformation in the non-Abelian case
I am trying to show that the integration measure we use in the Fadeev-Popov method of quantisation of non-Abelian gauge theory is invariant under a gauge transformation.
I am using Peskin & ...
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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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