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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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Comparator operator in QFT

In Peskin and Shroeder, for a local $U(1)$ transformation, the comparator operator is expanded as: \begin{equation} U(x+\epsilon n, x) = 1 -ie\epsilon n^{\mu}A_{\mu} + \mathcal{O}(\epsilon^2) \tag{15....
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What happens to the symmetry after gauge fixing?

Given a theory with gauge symmetry. After gauge fixing where does the symmetry go? Does the gauge symmetry turn into a global symmetry? For example there is a way to quantize fields theory with BRST ...
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Spinning Particles in Background Gauge Fields

A simple model for a spinning particle is $$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$ with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\...
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Gauge transformation of wave function of a system of stationary charges

Let's say we have a system of $n$ stationary charges interacting via Coulomb potential. Let's ignore possible external electromagnetic fields. Moreover the system is quantum, and its wave function is $...
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Calculating $S$-matrix in string theory

To calculate string $S$-matrix, we mainly use Faddeev-Popov gauge fixing method, as in chapter 6 of Polchinsky's book 《string theory》. But in section 6.2, 'tree level amplitude', I didn't find ...
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Is this a gauge symmetry?

Imagine a hypothetical action: $$S=\int \left(\frac{\partial}{\partial t}\phi(x,t)\right)^2 d^3x dt$$ Now we have a symmetry of the action: $$\phi(x,t)\rightarrow \phi(x,t)+\chi(x).$$ At time $t$, $\...
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Global anomaly for discrete groups

We know that: a global anomaly is a type of anomaly: in this particular case, it is a quantum effect that invalidates a large gauge transformations that would otherwise be preserved in the ...
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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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Gauge invariance or global invariance, which one makes theory renormalizable?

We know that gauge theory is renormalizable, due to the Ward-Takahashi identity (for non-Abelian theory, it is Slavnov-Taylor identity), which reflects the conserved current of gauge symmetry. But ...
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Diffeomorphic but physically inequivalent spacetimes

In the last few years there has been a considerable endeavor in understanding the asymptotic symmetries of quantum gravity on Minkowski Spacetime. This has been tied to a study of the BMS group that ...
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Potential functions for separation and isochronic gauges

Most potentials in physics are expressed as a radius or another geometric norm/gauge. I am looking to understand the significance of the choice of potential functions for force/pressure separation in ...
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Anyon condensation – what's the precise definition?

Say I have an anyonic system modelled as a Chern-Simons system with group $G$. If the centre of $G$ is non-trivial, one may also study the system described by $G/\Gamma$, where $\Gamma$ is a discrete ...
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Yang-Mills Feynman rules

Good morning/evening. In Peskin & Schroeder chapter 16 on gauge invariance, the gauge boson self interaction vertex rules are given. For three gauge bosons, the relevant interaction term in the ...
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Do the equations of motion simply tell us which degrees of freedom are superfluous?

A massless spin $1$ particle in 4D has 2 degrees of freedom. However, we usually describe it using four-vectors, which have four components. Hence, somehow we must get rid of the superfluous degrees ...
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Is general relativity resulted from diffeomorphism invariance?

Any action expressed as the integral of a 4-form in 4-dimensional spacetime is diffeomorphism invariant. For example the following 4-form topological (Pontryagin) action $$ S = \int F\wedge F $$ is ...
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Gauge fixing with vector potential: Coulomb gauge

There is something I would like to clarify with gauge fixing. In E.M, we can introduce the potential vector. As $div(\vec{B})=0$ we know that we can write $\vec{B}=\vec{curl}(\vec{A})$. But as $\...
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Coulomb gauge fixing and “normalizability”

The Setup Let Greek indices be summed over $0,1,\dots, d$ and Latin indices over $1,2,\dots, d$. Consider a vector potential $A_\mu$ on $\mathbb R^{d,1}$ defined to gauge transform as $$ A_\mu\to ...
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How does gauge-fixing really work?

Leaving technical issues like Gribov copies and residual gauge freedom aside, how do gauge fixing conditions like the Coulomb condition \begin{equation} \partial_i A_i =0 \end{equation} or the axial ...
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Why does Lorenz gauge condition $\partial_\mu A^\mu =0$ pick exactly one configuration from each gauge equivalence class?

For a vector field $A_\mu$, there are infinitely many configurations that describe the same physical situation. This is a result of our gauge freedom $$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + ...
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Question about the Lie group $SU(3) \times SU(2) \times U(1)$ and the concept of manifold

I don't know if this question is a duplicate, so I'll delete if is. Well, I'm in the very beginning of the study of contemporary topics such as gauge theories, I would say that I'm in a "science ...
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Delta function conversion into gauge-fixing Lagrangian in the path integral

So, I am at the moment working on gauge-fixing a path integral. The procedure involves adding a delta function $\delta g$ to the path integral (together with the faddeev-popov determinant, but that is ...
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Why does Coulomb gauge condition $\partial_i A_i =0$ pick exactly one configuration from each gauge equivalence class?

There are infinitely many configurations of a vector field $A_\mu$ that describe the same physical situation. This is a result of our gauge freedom $$ A_\mu (x_\mu) \to A'_\mu \equiv A_\mu (x_\mu) + \...
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Showing that Coulomb and Lorenz Gauges are indeed valid Gauge Transformations?

I'm working my way through Griffith's Introduction to Electrodynamics. In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric ...
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Dualities involving Supersymmetric QED in $3+1$d

Most of the supersymmetric dualities in $3+1$d involve only non-Abelian gauge theories, like $SU(N)$ $\mathcal{N}=1$ SQCD, etc. Are there examples of dualities which involve supersymmetric QED (i.e. ...
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Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
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Connection between gauge invariance and Lorentz invariance

This question is presented in the context of Weinberg's QFT book treatment, in particular considering the electromagnetism chapter. It begins in chapter 5 where Weinberg argues that in order to have ...
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Mass dimension and Abelian super-gauge transformation

A vector superfield is defined by postulating an invariance under a linear transformation in the space of vector superfields: $V \longrightarrow V + i\Lambda - i\Lambda^{\dagger}$ where $i\Lambda - ...
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Does gauge invariance of scalars/fermions in the adjoint representation induce the existence of Wilson loop and (then) covariant derivative?

First, for the known case of $U(N)$ gauge invariance we have scalars (it works for fermions too) transforming as (fundamental representation) $$ \phi(x)\to V(x)\phi(x), \ \ V(x)\in U(N) $$ So then we ...
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Angular momentum in the Maxwell field theory/Chern-Simons theory?

I'm trying to calculate the angular momentum in the chern simons theory. But equivalently, I was trying a calculation of angular momentum in the Maxwell field theory, which will hopefully be ...
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Count degrees of freedom of gauge tensors

For degrees of freedom (dof) it is said that spin-1 massless boson like photon has 2 dof in 4d, like U(1) gauge theory. it is said that spin-2 massless boson like photon has 2 dof in 4d, like ...
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Does a gauge group $G$ determine the Principal $G$-bundle?

I'm trying to understand the mathematical underpinnings of gauge theories in the language of principal $G$-bundles and associated vector bundles. Not long ago, I had assumed that the physical choice ...
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Hodge dual and QED

I was studying the paper Topological massive gauge theories in three dimensions by Deser, Jackiw and Templeton. In the paper, they use Hodge dual for some reason which I don't understand at all. So I ...
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Chiral anomaly: gauge covariance and regularization

I am looking at the treatment of the chiral anomaly in Fujikawa and Suzuki's "Path Integrals and Quantum Anomalies." To illustrate the quantum breaking of chiral symmetry (section 4.3), they start ...
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Ward identity prohibits mass of photon

On wikipedia one can read the following statement: The photon and gluon do not get a mass through renormalization because gauge symmetry protects them from getting a mass. This is a consequence of ...
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To which type of particles gauginos are supposed to couple?

I wonder about to which type of particles gauginos couple in general. I admit my knowledge in supersymmetry is very limited. Let's take an example: The photino. If it behaved similar to the photon, it ...
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Is a translation a gauge symmetry?

A gauge symmetry is a symmetry (often local) which doesn't affect the physical quantities or the Lagrangian. A translation of the whole Universe 3 meters to the left won't affect anything physically. ...
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Transformation of Yang-Mills fields

I am trying to recover the transformation properties of a Yang-Mills field and I'm not sure if I am wrong or if I am misunderstanding what is meant by a Yang-Mills field. Suppose I had a principle $G$...
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Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
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Gauge fixing of Polyakov Action

In the Gauge fixing of Polyakov action we do general coordinate transformation where we take the transformation stated below $$h_{\alpha\beta} = e^{\phi(\sigma)}\eta_{\alpha\beta}.$$ But here the ...
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Does the temporal gauge condition uniquely determine a gauge in case of non-Abelian gauge theory?

For a $U(1)$-gauge theory, we can fix $A_0 = 0$ by choosing a temporal gauge. Can we do the same for all of the gauge components of the $SU(2)$ gauge field, i.e., $W^a_0 = 0$ for $a \in \{1,2,3\}$? ...
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Gauge anomaly in Polyakov string and Faddeev-Popov method

I am currently trying to gain a better understanding of the gauge fixing procedure used in chapter 5 of David Tong's notes. Since the central charge of the Polyakov action for, say, the bosonic ...
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Field redefinition changing energy of one particle states

Consider Free scalar field theory $$ S[\psi^*,\psi] = -\int dx^4 (\partial_\mu \psi^* \partial^\mu \psi + m^2 \psi^* \psi) $$ Upon usual quantisation $$ \hat{\psi}(x) = \int \frac{d^3p}{(2\pi)^3 2 E_p}...
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What are emergent gauge fields in condensed matter physics?

My background: I have a very little knowledge about topological insulators. Medium level knowledge of Quantum mechanics and linear algebra. Almost no knowledge about Field Theories. I have studied ...
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$1/4$ coefficient in QED Lagrangian [duplicate]

What is the reason 1/4 coefficient in the tensor multiplication of the electromagnetic field strength? $$\mathscr{L} = -\, \frac{1}{4} \, F_{\mu \nu} \, F^{\mu \nu}. \tag{1}$$
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Do Lie groups in bundle automorphisms have corresponding principal bundles?

Given some theory described by fields from a vector bundle $E$, with section $\phi \in \Gamma(E)$, we get that transformations on the fields are part of the automorphisms on that bundle, ie, $\Phi \in ...
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Is there a specific structure to the automorphism set of a field theory with respect to internal v. spacetime symmetries?

I'm trying to work out what it means exactly for a field to be transformed, without referring to gauges for now. As far as I can tell, from a rigorous perspective, a transformation of the field is an ...
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What classifies gaugings?

Gauging a global symmetry $G$ introduces several free parameters to the theory. For example, In $d=4$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, ...
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Why define $D_\mu = \partial_\mu -ieA_\mu$ with the electric charge $e$?

If $D_\mu = \partial_\mu - ieA_\mu$ then the QED Lagrangian is invariant under $$A_\mu \to A_\mu + \frac{1}{e}\partial_\mu\alpha(x)$$ $$\psi \to e^{i\alpha(x)}\psi$$ However if $D_\mu = \partial_\mu -...
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Self-interaction of gauge bosons in electroweak theory

As one learns in QFT, in Yang-Mills theories non-Abelian gauge transformations give rise to self-interactions of the gauge fields in the quadratic field strength term. In QCD this produces the 3- and ...
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Should the energy-momentum tensor be invariant under gauge transformations?

For example, consider the electromagnetic theory given by \begin{align} I=-\frac{1}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu}, \end{align} where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The action ...