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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London current density gauge
As far as i know, the origin of the Meissner effect can be explained with the fact that the supercurrent is proportional to the current density $\vec{A}$, i.e.
$$\vec{j(\vec{r})}=-\frac{1}{\mu \,\...
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Gauge invariance of magnetic vector potential
If we go by transformation for magnetic vector potential $\mathbf{A} \rightarrow\tilde{\mathbf{A}}=\mathbf{A}-\nabla\psi$, as well as $\varphi \rightarrow \tilde{\varphi}=\varphi+\frac{\partial\psi}{\...
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Interaction for QED with charged, scalar particles
Let $\mathcal{L}$ be the Lagrangian for usual QED with scalar, charged particles (with photons and electrons as well):
$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\left(i\gamma^{\mu}\...
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Local Lorentz invariance of General Relativity
Consider the gravitational action as an integral over a differential 4-form $$ S = \int_{\mathcal{M}} \star F_{ab}\wedge e^a \wedge e^b$$ where $\star F_{ab} = \epsilon_{abcd} F^{cd}$ and $F$ is the ...
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1answer
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Gauge invariant Green's function for a point particle
This question is a follow up to the question (Gauge invariant Green's function for electrodynamics).
It is not possible to generally solve the eqution
\begin{equation}
\square A^{\mu}-\partial^{\...
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Why do we talk about $U(1)$-principal bundles rather than torus bundles?
I'm trying to understand gauge theory in terms of differential geometry, and one thing I'm confused about is why we talk about $U(1)$-principal bundles when the gauge potential $A$ usually has ...
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Does an electromagnetic gauge transform induce a $U(1)$ transform on the field?
For the free complex scalar Lagrangian,
$$\mathscr{L}=\partial_\mu \phi\partial^\mu\phi^{\dagger}-m^2 \phi \phi^{\dagger} $$
if we want it to be invariant under a transformation of the form $\phi\...
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Ward Identity and Proca Fields
I'm following the book Quantum Field Theory and the Standard Model by Schwartz and I came to the rigorous non-perturbative proof of the Ward identity with path integrals via the Schwinger-Dyson ...
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1answer
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A question about charge in the gauge covariant derivatives of the $U(1)$ Higgs model
I'm studying the $U(1)$ Higgs model, with the lagrangian:
\begin{gather}
\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\left(D_\mu\phi\right)\left(D^\mu\phi\right)^\dagger-\mu^2\left(\phi\phi^\dagger\...
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Quantum gauge transformation definition
I have always thought that a gauge transformation of a quantum Hamiltonian $H(\Psi,\Psi^{\dagger},A)$ ($A$ is the vector potential and $\Psi$ a matter field) is given by:
$$\Psi(r) \rightarrow \Psi(r) ...
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What is the connection between the symmetry group, number of dimensions and the particles predicted by a quantum field theory?
I did an undergraduate degree in physics a number of years ago and have a basic knowledge of quantum mechanics, quantum field theory, relativity, Noether's theorem, etc.
Recently, I have been reading ...
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Trying to understand the conformal gauge “derivation” in Polyakov action symmetries
In section 2.3 of the book "Basic Concepts of String Theory" by Blumenhagen, Lüst, Theisen, 3 symmetries of Polyakov action are discussed: Poincarè invariance, diffeomorphism invariance and ...
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How to prove the amplitude of $q\overline{q}$-$g\overline{g}$ with unphysical backward polarization is zero in Yang-Mills theory
In Peskin&Schroeder 16.1, we calculate the tree diagram level scattering ampitude of $q\overline{q}$-$g\overline{g}$. There are three terms:
$$i\mathcal{M}^{\mu\nu}_{1,2}=(ig)^2\overline{v}(p_+)\...
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1answer
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How is the Lie algebra connection form a potential energy field?
I'm pretty familiar with the math of principle bundles etc, but I have no intuition for physics at all, so I'm curious how some facts fit together.
When a potential is undetermined up to a gauge, we ...
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Linearised diffeomorphisms on arbitrary gravitational background Part 2
This question is a follow on from my recent post here, in the sense that I will use the notation introduced there. In that post, I considered infinitesimal diffeomorphisms of a metric $g_{\mu\nu}$ ...
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Gauge invariance prohibits the existence of other universes?
I'm only a student, so most likely misunderstood. The Wikipedia article on Relational quantum mechanics says:
"The universe is the sum total of everything in existence with any possibility of ...
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1answer
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Why do we have residual symmetry when we already used all symmetry in gauge fixing of the worldsheet metric?
In Becker, Becker and Schwarz' book about string theory, the following symmetries are listed for the $\sigma$-model of the string:
Poincaré transformations
Reparametrizations $\sigma^{\alpha}\...
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1answer
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Gauge invariance for the electric field
We know that $$\boldsymbol{E}=-\nabla V-\frac{\partial\boldsymbol{A}}{\partial t}$$
$$\boldsymbol{B}=\nabla\times\boldsymbol{A}$$
But I see that the following changes do not change these fields:
$$\...
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2answers
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How to interprete gauge symmetry in following way?
From Wikipedia:
In here, $G$-principal bundle $P$ on spacetime $\Sigma$ is given. Gauge transformation $\phi$ can be written in three ways, and it seems to use the third way. More specifically,
\...
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1answer
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Kobe 1980 AJP “Derivation of Maxwell’s equations from the gauge invariance of classical mechanics”. Confusion between field and source locations
In his paper (Kobe 1980 AJP “Derivation of Maxwell’s equations from the gauge invariance of classical mechanics”) Kobe writes down a new Lagrangian, Eqn (2.10) using the definitions of charge and ...
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Is massless QED more “natural” than massive QED?
My understanding is that massive and massless QED share some key physical features including (see this PSE post and 8-4 of Ref. 1):
renormalizability
charge conservation
The key differences of ...
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With a local anomaly, is the determinant of the Dirac operator still a section of a complex line bundle?
In the literature about anomalies in quantum field theory, the determinant of the Dirac operator plays an important role. The Dirac operator may depend on some background data, and the subject of ...
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1answer
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The integer eigenvalues for a generator of $SU(N)$
I am studying gauge theory by V.P. Nair's QFT textbook.
He explains in p. 455 that all components of all fields which are $\mathbb{Z}_N$ invariant will have integer eigenvalues for $Y=diag(1/N,1/N,\...
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Magnetic Dirac operator rotationally invariant?
Let $\omega=e^{2\pi i/3}$ be the third root of unity.
We consider the two-dimensional Dirac operator with homogeneous magnetic field
$$ H = \begin{pmatrix} 0 & 2 D_{\bar z} -Bx_2 \\ 2 D_{z} -B_{...
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1answer
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Gauge fixing in the classical $U(1)$ gauge theory
My question concerns the gauge fixing in classical v.s. quantum $U(1)$ gauge theory. I will ask about the gauging fixing in quantum $U(1)$ gauge theory in a separated Phys-SE post.
For the classical $...
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1answer
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Chern-Simons term for a non-abelian gauge multiplet
In equation (20.9) of Freedmann and Van Proeyen's Supergravity, it is stated that for the following Chern-Simons term:
$$S_{\mathrm{CS}} = C_{IJK}\int A^I\wedge F^J \wedge F^K$$
to be invariant under ...
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3answers
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What does it mean to be “gauging” a symmetry?
I read this and other similar questions, but they all address the problem of gauging a global symmetry (implying that one could also gauge a local one).
This confused me a lot: in my mind gauge and ...
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Why and where Poincaré gauge theory fails?
I would like to post this question because I have seen no one post it in this explicit way.
From what I have seen, Poincaré gauge theory uses connections (gauge fields) like the spin connection and ...
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Why is the group of gauge transformations on the frame bundle isomorphic to $\text{Diff}(M)$?
Consider the frame bundle $LM \to M$ for given Lorentzian manifold $M$. The group $\mathcal{G}$ of gauge transformations of the second kind are automorphisms $\phi:LM \to LM$ covering the identity $\...
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What is wrong with my counting of electromagnetic field degrees of freedom?
When we go from the field variables $({\vec E},\vec{B})$ to the potentials $(\phi,{\vec A})$, the number of degrees of freedom describing any electromagnetic field is reduced from $6$ ($3$ components ...
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Can a QFT be anomaly-free on spacetimes that are boundaries but still have an anomaly on other spacetimes?
If $D$ is the Dirac operator for some dynamic spinor fields in background gauge and gravitational fields, then the partition function is supposed to be $\mathrm{det}(D)$. For this to make sense, we ...
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Anomalies in QFT: why do we require smooth dependence on the background fields?
If $D$ is the Dirac operator for some dynamic spinor fields in a background gauge field $A$, then the partition function is supposed to be $\mathrm{det}(D)$. But if the coupling to the gauge field is ...
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Large gauge transformations and global symmetries
In gauge theories, states in Hilbert space that are related by local gauge transformations are identified as the same physical state. This is necessary because otherwise the Hilbert space contains ...
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$\mathbb{Z}_2$ gauge theory and disorder
I am confused about basics of $\mathbb{Z}_2$ (and likely other) gauge theories and plain disorder. Let
$$H=H_F + h\,H_{EM}$$
$$H_F = -t\sum_l (c^\dagger_l \sigma^z_{l,l+1} c_{l+1} + h.c.)$$
be (the '...
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$\mathcal{N}=1$ SUSY gauge theories in 2D?
The following introduction explains what I mean by $\mathcal{N}=1$ (often also called (1,1)) supersymmetry in 2D.
We can define a superfield
$$\Phi(x,\theta)=\phi(x)+\bar{\theta}\psi +\frac{1}{2}\bar{\...
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Is this the most general form for a gauge transformation?
From my understanding, a gauge transformation in QFT is a local transformation in the fields under which the action is invariant. I usually write it, for a theory with scalars, fermions, and vector ...
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Check gauge invariance of non-Abelian theory
I want to check that
$$\mathcal L=\mathcal L_{free}+ \mathcal L_{int}=\bar{\psi}_i(i\gamma^\mu\partial_\mu-m)\delta_{ij}\psi_j+g\bar{\psi}_i\gamma^\mu A_\mu^at_{ij}^a\psi_j$$
is invariant under the ...
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2answers
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Initial condition of the wave function while solving the Schrodinger equation
To calculate the evolution of a wave function under Schrodinger equation, we need to set an initial condition. In this context, can we associate a unique wave function to a given physical (observable) ...
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How to write the gauge-invariant anomalous Nambu Green's function for 2D square lattice with uniform $\pi$ flux?
For the free fermion system in two-dimensional square lattice, we add the $\pi$ flux in each plateau:
$$H=-t \sum_{\langle i, j\rangle} e^{i A_{i, j}} c_{i}^{\dagger} c_{j}+h . c .$$
where
$$\sum_{\...
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1answer
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Gauge covariant derivative, - how do I get the field?
Suppose I wish to create a gauge covariant derivative from
$$
\psi(x)\to e^{ia(x)}\psi(x)
$$
I first note that the usual derivate is not covariant:
$$
\partial_x(e^{ia(x)}\psi(x))=i(\partial _xa(x))e^{...
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1answer
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Gauge fixing and transformation
given the gauge choice that div A = some value/function.
i am completely fine that in the context of electromagnetism that by setting the divergence of this to be anything it has no effect on the curl ...
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Is the Bohm-Aharonov effect a proof of real $U(1)$ local gauge transformations “taking place” (instead of being just a math procedure)?
Under a local gauge (phase?) transformation of the field operator for electrically charged fields, $\psi \rightarrow e^{\mathrm{i}\phi(x_{\mu})}\psi$, where $e^{\mathrm{i}\phi(x_{\mu})} \in U(1)$, the ...
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1answer
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Why the $R_{\xi}$-contribution to the Lagrangian disappears when computing physical observables?
In QED for example, you add the term $$\mathcal{L}_{GF}=-\frac{1}{2\xi}(A_{\mu}A^{\mu})^{2}$$ so you can compute the photon propagator. The question is basically, why you can compute physical ...
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Are there any non-gauge invariant physical theories that describe anything from the “real world”?
I have learnt QFT and the Standard Model and we always had gauge invariance there (in my lectures at least...) I wonder if there are any theories which are not gauge invariant that describe a bit of &...
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Standard Model: Problem with Masses of Elementary Particles?
In his book "Modern Particle Physics", Mark Thomson explains two problems with masses of elementary particles in the SM:
(i) If we take the QED Lagrangian $\mathcal L = \bar{\psi}\left( i\...
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1answer
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Proving that the kinetic energy of the gluon fields $G_{\mu}^{a}$ is $SU(3)$-invariant?
In QCD, we introduce the gluon-fields $G_{\mu}^{a}$ when defining the covariant derivative $D_{\mu} \equiv \partial_{\mu} + ig_{S}G_{\mu}^{a}T^{a}$ to make the free-particle Lagrangian $
\mathcal L_{\...
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2answers
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Isn't Stress-Energy Tensor of Maxwell field in presence of charges gauge variant?
Versions of this question have been asked on this site before but have not directly addressed by concern. In the $(+---)$ convention the EM lagrangian in the presence of charge sources is
$$
\mathcal{...
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1answer
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Rotational invariance of the the elastic deformation in spherical symmetry
I write the elastic deformation $E$ for an incompressible material in spherical symmetry (Have a look at here Eq.(5.32) ). Since we are incompressible, $\mathbf{det} \,E=1$, so in spherical ...
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Proving that free-particle Lagrangian is not invariant under $SU(3)$ local gauge invariance?
I would like to show that the free-particle Lagrangian $\mathcal L = \bar{\psi}\left( i\gamma^{\mu}\partial_{\mu} - m\right)\psi$ is not invariant under the $SU(3)$ local gauge invariance ...
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$U(1)$ Local Gauge Invariance: What do $q$ and $\alpha(x)$ mean?
When deriving the existence of the photon, we start with the free Lagrangian $\mathcal L_{\text{free}} = \bar{\psi}\left( i\gamma^{\mu}\partial_{\mu}-m\right)\psi$ and require $U(1)$ local gauge ...
