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Questions tagged [gauge-theory]

A gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of *continuous symmetries* of these internal degrees of freedom, *i.e.* the predicted physics under any transformation in this group on the degrees of freedom. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.

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Commutation relation in quantized electromagnetic field theory

I have a question regarding a proposed problem (Problem 4.8) in Rodney Loudon's book "The Quantum Theory of Light". Let $U(t)$ be an operator defined by $$ U(t)=\exp\left\lbrace\frac{i}{\hbar}\int\...
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What is the topological data for $(\mathbb Z_n)_p$ theories?

Consider the 3d TQFT described by the Lagrangian (Dijkgraaf-Witten with gauge group $\mathbb Z_n$ at level $p$): $$ \mathcal L=\frac{n}{2\pi} B\wedge\mathrm dA+\frac{p}{4\pi}A\wedge\mathrm dA $$ with $...
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Is it enough to assume $F_{\mu\nu}\to 0$ at infinity but not $A_\mu$ to derive the equation of motion?

Suppose the the Lagrangian $\mathscr{L}$ of the free electromagnetic field is augmented with the term $$F_{\mu\nu}\tilde{F}^{\mu\nu}=\partial_{\mu}(\epsilon^{\nu\nu\lambda\rho}A_\nu F_{\lambda\rho}).$$...
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Isn't there a unique vacuum of the Yang-Mills quantum theory?

The theta vacua$^1$ of a Yang-Mills quantum theory are given by $$|\theta\rangle=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}|n\rangle.$$ In Srednicki's Quantum Field Theory, he claims that the ...
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Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
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Gravity from algebra

Can someone provide me a reference that describes the gauging of the Poincare algebra to obtain Einstein's relativity? "It is well known that Einstein’s formulation of gravity can be obtained by ...
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GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?

Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version: $$ D = d+A, $$ where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection ...
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Commutators in Gupta-Bleuler formalism for quantization of the electromagnetic field

In the Gupta-Bleuler formalism we have for the canonical momenta $$\pi_\mu=F_{\mu0}-g_{\mu0}\partial_\alpha A^\alpha. $$ Every resource I find online say that the equal time canonical commutation ...
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Commutator of Gauge Transformations for Yang-Mills Theory

Following the conventions of "Quantum Field Theory and the Standard Model" by Schwartz, we have that for Yang-Mills Theory, an infinitesimal gauge transformation acts like $$\delta_{\alpha} A = d\...
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Conserved currents in Yang-Mills theory: gluon current vs. quark current

In Yang-Mills theory there are two currents we can construct. There is the well-known quark current related to the global $SU(3)_C$ symmetry, $$j{}^{\mu\,A}_\text{quark} = \overline{\psi}{}^i \gamma{}^...
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Counting degrees of freedom in the Higgs mechanism for different gauges

I am wondering how to count the degrees of freedom (dof) for a massive gauge field in different gauges. I've been reading some other answers, but haven't found a solution yet. I am looking at the ...
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Infinitely many conserved currents in any QFT?

So I have the following curiosity: Consider for example, in QED, the quantity $$ j^\mu\equiv\partial_\nu (\lambda(x) F^{\mu \nu}) $$ where $\lambda(x)$ is an arbitrary scalar function of spacetime, ...
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Is this theory equivalent to QED?

I've found the following Lagrangian $$\mathcal{L}=i\bar{\psi}\gamma^\mu\left(\partial_\mu-ieA_\mu -ieA'_\mu\right)\psi-m\bar\psi\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}F'_{\mu\nu}F'^{\mu\nu}.$...
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How does the Lorenz gauge eliminate the scalar component of the vector field?

Wikipedia states that by using the Lorenz gauge, $\partial_\mu A^\mu=0$, we eliminate the scalar part (spin-0) of the vector potential that previously had spin-1 and spin-0 components${}^1$. However,...
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Can we gauge away curvature in supergravity?

In locally supersymmetric theories we can make a supertransformation which rotates one field into another. With enough supersymmetries could one not make a supertransformation at each point which made ...
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Why do we require that functions which parametrize gauge transformation are smooth?

A local $U(1)$ transformation is given by \begin{equation} f(x) = e^{i\epsilon(x)} \qquad \text{with} \qquad \epsilon(x) \in C^\infty \, . \end{equation} Why do we require that the functions in ...
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The meaning of gauge-fixing in covariant quantization of the electromagnetic field

I am having trouble wrapping my head around the idea behind the covariant quantization for the electromagnetic field that is usually done in textbooks (I'm currently following Mandl & Shaw and ...
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The location of an object is gauge dependent. Therefore, it's not measurable?

The location of an object $x$ depends on how we choose our coordinate system. If we move the zero point, $x$ also changes. However, since we have translational invariance, we can always do such shifts ...
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Hamiltonian Structure of Chern Simons Electrodynamics

I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ...
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Gauge Invariance in Electrodynamics

I am studying Electrodynamics and I have been introduced to the concept of Gauge Invariance. This was introduced by noting that $E$ and $B$ amount to 6 six degrees of freedom and the Maxwell ...
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Does a source term for electric charge necessarily break global $U(1)$ symmetry?

The conservation of electric charge in, e.g., quantum electrodynamics $$\mathcal{L} = -\frac{1}{4}F^2 - A \cdot J + \mathcal{L}_\mathrm{matter}(J)$$ can be derived using the invariance under global $U(...
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Are there more general gauge transformations than simple phase shifts?

Usually, in the context of a $U(1)$ gauge theory, we only consider gauge transformations of the form \begin{equation} \Psi(x) \to \mathrm{e}^{i\epsilon(x)} \Psi(x) \, . \end{equation} Are there ...
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Why are transformations that only change something within a finite region redundancies?

I'm trying to build some intuition for a very particular definition of the notions global and local gauge symmetries. The definition goes as follows and appears, for example, in "Quantum Field Theory -...
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What's the physical meaning of the gauge invariant quantity $\partial_\mu \varphi(x) - A_\mu(x)$?

A famous locally gauge invariant quantity is $$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , $$ which is interpreted as the measurable electric and magnetic field strengths. Now, ...
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Is it possible to derive the correct QED Lagrangian without demanding local gauge invariance?

Usually, the correct interaction term $A_\mu \bar{\Psi} \gamma_\mu \Psi$ in the Lagrangian is derived by demanding local gauge invariance. Is there any other argument that fixes the form of the ...
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What is the relationship between velocity-dependent potentials and non-Abelian gauge fields?

My (limited) understanding of non-Abelian gauge fields is that they arise from the construction of a theory using a non-Abelian Lie group (as a generalization of the Abelian group underlying E&M) ...
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Can we visualize the standard model fermions as a 5-dimensional matrix with only the first 3 dimensions gauged?

Standard model fermions are usually represented by columns. However the column can carry different connotation depending on the matrix operator acting on it. For example, the Dirac spinor 'column' and ...
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156 views

What is the Noether charge associated with the the color $SU(3)$ symmetry of QCD?

A version of the Noether's theorem applies to local gauge symmetries. What is the Noether's charge associated with a non-abelian gauge symmetry such as the color $SU(3)$ and how is that derived? I ...
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201 views

Can Yang-Mills field strength be defined as covariant derivative squared?

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as $$ \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\...
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Local charge current for gauge field and conservation of charge

Motivation: It is a well known fact that the gravitational field (in General Relativity and direct generalizations of it) has no local energy-momentum density. Usually there are two reasons stated, ...
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Are the nonphysical degrees of freedom in Yang-Mills theory analogous to the worldsheet metric in the Polyakov formalism?

The Polyakov string action on a flat background (in the Euclidean signature) $$S_{P}[X,\gamma]\propto\int_{\Sigma}\mathrm{d}^2\sigma\,\sqrt{\text{det}\gamma}\,\gamma^{ab}\delta_{\mu\nu}\partial_{a}X^{...
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Is fixing the gauge the same thing as performing a Lorentz transformation?

Let's say I have a moving charged particle, with constant velocity. Its electric field is given by (generally): $$ \mathbf{E} = -\nabla\phi - \frac{\partial \mathbf{A}}{\partial t}. $$ If I perform ...
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What is the physical meaning of Wilson loops?

I'm a mathematician trying to get some very basic physical intuition on gauge theories, so I apologize if what follows is really naive. My first super elementary question is: Am I right to think ...
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World-volume (higher) gauge theory on D-branes

It is well known that if one constructs ordinary WZW type sigma model for string action, it is possible whenever they find a cocycle in appropriate Chevalley-Eilenberg algebra. If I understand it ...
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Number of degrees of freedom of a photon

I am learning Quantum Field Theory. There they have shown that, for the four vector $A$, even though it has 4 components, it only has 2 degrees of freedom because the other 2 corresponds to gauge ...
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Is it possible to directly derive the $K$ matrix for a topological order described by a gauge-theory Hamiltonian?

To be concrete, Let us consider a $Z_2$ gauge theory in the deconfined phase coupled to matter field, \begin{align} S_{Z_2}=\beta\sum_{\vec r\mu}\phi(\vec r)U_\mu(\vec r)\phi(\vec r+\vec e_{\mu}) + K \...
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What is gradient-flow modified operators in thermal gauge theory?

This is going to be a very "soft-question". I do not fully understand German, so I am not sure if "Korrelationsfunktionen mit Gradientfluss modifizierten Operatoren bei endlicher Temperatur in ...
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How to identify higher-form symmetries?

A $q$-form symmetry is a symmetry that naturally acts on objects whose support is a $q$-dimensional surface (ref.1). For example, what we usually call a "regular" symmetry, is actually a $0$-form ...
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Gauge transformations and Covariant derivatives commute

I would like to understand the statement "Gauge transformations and Covariant derivatives commute on fields on which the algebra is closed off-shell" which was taken from section 11.2.1 (page 223)...
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Confusion Between Associated and Principle-G-Bundles

I realize there have been similar questions on stack before, but none of them have answered what I'm after. -My question is really whether I can import wholesale everything from the principle bundle ...
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Is it possible to couple an odd number of Dirac fermions, at finite density, to a massless gauge field in 2+1d?

In a beautiful paper by A. N. Redlich (PRL $\bf{52}$, 18 (1984)) on the parity anomaly, the author indicates that an odd number of Dirac fermions can never be coupled to a massless gauge field in 2+1d ...
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Factorization of exponential of broken generators in parametrization of scalar multiplet in non-abelian SSB

Describing abelian symmetry breaking in his book on gauge theories, after favouring a vacuum (whose expectation value is $v$) from the symmetric continuum, Quigg parametrize the complex scalar as $$ ...
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Are internal string states analogues of internal particle states?

It is sometimes said that in string theory elementary particles correspond to modes of vibration of a string. Likewise in QFT, several particle fields take their values in a higher dimensional space,...
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$USp$ global symmetry in $d=3$, ${\cal N}=4$ supersymmetric QFT

Define a 3-dimensional QFT with $N=4$ supersymmetry (4 supercharges), and the field content is $g$ $N=4$ hyper-multiplets (that are in a representation $R$ of some group $G$). Each hyper-multiplet is ...
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Showing that $U(1)_R$ charge is non-anomalous in SUSY QCD when $r=\frac{F-N}{F}$

I'm trying to show that the value of the R-charge $r$ for which the R-symmetry is non-anomalous is given by $r=\frac{F-N}{F}$. To do this we must calculate the triangle diagrams for the quarks $\...
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What is the effect of including additional representations in the action of a lattice gauge theory?

I'm reading Introduction to Quantum Fields on a Lattice by Jan Smit. When introducing the lattice gauge-field action as a sum over plaquettes, Smit says that in general the action should include a sum ...
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331 views

Vector Potential of a rotating Spherical Shell

I have done all the calculation in finding the vector potential leading up to the equation 5.68. But then Prof. Griffiths goes back to the original figure and mentions that the coordinate of the point ...
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Which of the Wightman axioms are not incorporated by four dimensional quantum Yang-Mills?

I am trying to understand the quantum Yang-Mills existence problem but the best I have seen so far is the statement that there is no known interacting relativist field theory in four dimensions which ...
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Small confusion about the Aharonov-Bohm effect

I am mostly aware of the Aharonov-Bohm effect's (AB effect) physical interpretation, as well as the corresponding mathematical/differential geometric interpretation. What does confuse me slightly ...
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How to understand the non-abelian DBI action?

I'm reading chapter 7 in "String Theory and M-theory, A Modern Introduction" by Becker, Becker and Schwarz. It says that the understanding of the square root of the determinant is to compute the ...