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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Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$

Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that $$D_\mu = \partial_\mu - igA_\mu, \tag1$$ you can built the kinetic term for the gauge potential $A_\...
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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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Notation and concepts of Yang Mills Theory

I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about ...
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What does $\mathcal{N}$ refer to in Gauge theories?

Context: I am a second-year (undergraduate) physics major applying for a summer research position. The investigator is working on Quiver Gauge Theories and in response to my inquiry email he told me ...
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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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What happens to the $U(1)$ factor in the $U(N)$ SYM gauge group of the AdS/CFT correspondence?

I'm learning about the AdS/CFT correspondence. I know that from the open string perspective, the dynamics on a stack of $N$ coincident $D3$-branes is given by a $\mathcal{N} = 4$ Super Yang Mills ...
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Higher form fermionic conserved currents

Higher form conserved currents have already been defined, such as those seen in Klebanov and Polyakov's work in 2002. There, the authors studied the $\text{AdS}_4$/CFT correspondence -- more ...
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What is the dimensionality of each part of a covariant derivative?

In the standard model, we have the following covariant derivative: $$D_\mu = \partial_\mu - ig_sG_\mu^a\lambda_a-igW_\mu^a\frac{\sigma^a}{2}-ig'B_\mu\frac{Y}{2}$$ If we let this work in on e.g. the ...
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Are Maxwell's equations “physical”?

The canonical Maxwell's equations are derivable from the Lagrangian $${\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ by solving the Euler-Lagrange equations. However: The Lagrangian above is ...
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Constrained Hamiltonian systems: spin 1/2 particle

I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action \begin{equation} S=-m\int ...
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Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-...
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$\theta$ Term In 2d for $U(1)$ and $SO(2)$

For the $U(1)$ gauge theory in 2d, there can be a theta term $$\frac{\theta}{2\pi}\int_{M} dA$$ where $A$ is the $U(1)$ gauge field and $\theta\sim \theta+2\pi$. However, it is known that $U(1)=SO(2)...
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Is it possible to construct a quiver diagram for electromagnetism?

I have been trying to learn about quiver diagrams and quiver gauge theory for a summer project. All of the lecture notes/papers on the topic give example diagrams that are mathematically simple but ...
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Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)

In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has: $$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
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Non-local field redefinition and effects on path-integral measure

Consider the partition function $$ Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
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If monopoles are excised points in a $U(1)$ bundle, how are they affected by other charges?

We currently understand electromagnetism as a U(1) gauge theory. If you take a point out the space manifold (base space) you can create magnetic monopoles with integral charges by making non-trivial ...
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A puzzle about Green's function and S-matrix

From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but ...
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Gauge fixing: Overcounting vs Inversion of Operator

In my studies (various books, and Lectures by Tobias Osborne) I've been told we gauge fix to stop the naive overcounting in the path integral. User @Marmot pointed out in a comment that if this was ...
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Why Coulomb gauge is a possible gauge choice?

In classical field theory we can get, that adding gradient of some scalar field to magnetic vector potential does not change the physics at all. So, we have such a symmetry: $\boldsymbol{A}\...
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Global $U(1)$ transformation properties of gauge fields

What are the Global gauge transformations of gauge bosons in Standard Model? To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as $$\...
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Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
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Is there a difference of sign conventions of Dirac Index between mathematics and physics?

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by $$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{...
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Derivatives in Poincare' gauge theory

I have been reading the lectures: http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf about Poincare' gauge theory. The Poincare' group is considered as semidirect ...
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Gauge fixing while preserving supersymmetry

In supersymmetric gauge theories, the vector potential is a part of a vector supermultiplet which is represented by a real superfield $V$. Expanded out in components, the Lagrangian for such a field ...
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Is any other gauge ever more useful than Lorenz gauge for practical calculations in classical EM?

Any problem in gauge theory can of course in principle be solved in any gauge (or, in the case of classical gauge theory, without using gauge potentials at all), but some gauges are much more useful ...
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Why demand local gauge invariance? [duplicate]

Ok, so I see how demanding local gauge invariance leads to gauge fields, and how quantizing those fields then leads to gauge bosons. But why do we take that step in the first place? What leads us to ...
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Do longitudinal and scalar have anything to do with Faddeev-Popov ghosts?

In his this book, Hatfield calls ghosts the negative states appearing in the covariant (Gupta-Bleuler) quantization prescription of the electromagnetic field (page 89). When discussing Yang-Mills ...
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Gauge transformation on vacuum

Suppose we have two electromagnetic vector potential operator that differ by a gauge transformation in free field theory. $A'_{\mu}= A_{\mu}-\partial_{\mu}\alpha$ is the state $A_{\mu}|0\rangle$ ...
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Gauge transformation on operators

Suppose we have two electromagnetic vector potential operator that differ by a gauge transformation. $A'_{\mu}= A_{\mu}-\partial_{\mu}\alpha$ Now suppose we have $\partial_{\mu}\alpha=F(x)\hat ...
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Vandermonde determinant factor in supersymmetric gauge localisation computation

I am trying to learn supersymmetric localisation computation in which the path integral of a supersymmetric gauge theory (placed on a sphere) can be localised exactly to a BPS configuration ...
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Quantizing first class constraints

Let $\gamma$ denote a first class constraint. Then if there exists a function on phase space $f(q,p)$ for which the Poisson bracket with the constraint does not vanish $\lbrace f, \gamma\rbrace \neq 0$...
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How does the photon arise from the $U(1)$ symmetry in QED?

I'm trying to study a bit of QED and I really don't get where the photon does arise. I know that we want our field theory to be gauge invariant under a $U(1)$ phase transformation, because that is ...
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On string-like excitations in (3+1)d discrete gauge theory

(3+1)d discrete $G$-gauge theory (untwisted Dijkgraaf-Witten theory) has both point-like and loop-like excitations; Point-like excitation is an electric charge labeled by an irreducible ...
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Energy of bound states in gauge theories

In gauge theories such as QED or QCD, are the rest energies, and therefore the invariant masses, of various bound states, such as those of positronium or charmonium, gauge-invariant observable ...
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Supersymmetry: spin of the superpartners

I'm currently working on my master thesis, and I need to know a bit of supersymmetry. I have been looking the theory and I have a basic knowledge about it. I have a problem with understanding the ...
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Is $F^{\mu\nu}F_{\mu\nu}$ equivalent to $A^{\mu}\nabla^{\alpha}\nabla_{\alpha}A_{\mu}$ for $U(1)$ gauge field lagrangian?

The two seem to yield the same equation of motion is why I asked. Where of course the standard notation for exterior forms applies $dA=F$. We all know how the field strength tensor plays into the ...
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Maxwell theory : How to justify the potential gauge fields if there are magnetic monopoles?

Without magnetic monopoles, the Maxwell equations are these (I'm dropping vector notations and all constants, for simplicity): \begin{align} \nabla \cdot E &= \rho_{elec}, \tag{1} \\[12pt] \nabla \...
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Why does gauge invariance HAVE to correspond to an observable?(Or is it the other way round)

Under the line integral of the geometrical Berry phase, a close-loop integral is gauge invariant as if we were to perform a gauge transformation of the initial state, with the end point of the path in ...
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Scattering matrix symmetries and standard model

I am not able to get around the following question (if it make sense): Suppose I can derive the scattering matrix S for any particle scattering process. Suppose that the standard model is actually ...
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Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
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Help with understanding the imposition of gauge conditions

Let $s$ be a positive integer and $h_{a_1\dots a_s}$ be a traceless and totally symmetric (real) field which is defined modulo gauge transformations of the form $$\delta_{\xi}h_{a_1\dots a_s}=\...
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What is the relation between the Gribov problem and color confinement?

I have heard that the Gribov problem is in some way related to color confinement (For instance: Gribov copies and confinement). Although I understand what both the Gribov and confinement problems are, ...
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Understanding Measure in Path integrals

I know this is still a current topic of study, so my question is less about mathematical underpinnings, and more about the transition from the "sum-over-all-possible-intermediate-points" measure $$\...
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Simplest example of a gauge theory?

I am not a physicist. I’ve read some informal stuff about gauge theory and gauge symmetries, however it’s pretty abstract to me. Can you give the simplest non-trivial example of a gauge theory? E.g. ...
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Why does Maxwell's equations $\partial_{\mu} F^{\mu \nu} = 0$ have 3 independent components (DOF) in $D = 4$?

And how can we generalize this to the statement that it has $D-1$ independent components in dimension $D$? I know that $F_{\mu \nu}$ has six independent components (because of antisymmetry), how do ...
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If $A^\mu$ is not determined uniquely by Maxwell's equations, what happens if we solve for it numerically?

Given a solution $A^{\mu}(x)$ to Maxwell's equations \begin{equation} \Box A^{\mu}(x)-\partial^{\mu}\partial_{\nu}A^{\nu}=0\tag{1} \end{equation} which also satisfies some specified initial conditions ...
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How do know that Goldstone Boson actually become the longitude degrees of freedom in W+,W- and Z boson?

In many Quantum Field Theory text books they says these about Spontaneous Breaking and Higgs mechanism like this In unitary gauge, the Goldstone Bosons are eaten by $W^\pm$ and Z and become their ...
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Notion of 'functional degrees of freedom' for the metric function in GR?

I have read through the numerous questions on 'degrees of freedom' in the metric tensor, and won't list them all here. However none of them address my question on 'functional' degrees of freedom in ...
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Quick question on choosing a gauge (E.g. Lorenz gauge)

I have been quite confused when I read about choosing a gauge. For example we have the gauge transformation $$ A_\mu\longrightarrow A_{\mu\prime}= A_\mu+\partial_\mu\alpha, $$ and we can choose any $...
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How does a quiver diagram represent the matter content in a guage theory?

So far I have understood that a quiver is a graph where every vertex is given a vector space and every edge represents a linear map between such vector spaces. I also think I understand what a ...