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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Instantons in Minkowski spacetime? or only valid in Euclidean spacetime?

In the usual description of the instanton of nonabelian gauge theory in $D=4$ spacetime, we always (or just usually?) choose the $D=4$ Euclidean spacetime see for example https://en.wikipedia.org/wiki/...
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How does the electroweak interaction and QCD form $SU(2)$ and $SU(3)$?

I'm trying to get a foothold into quantum field theory from a mathematical background. I see the use of $SU(2)$ and $SU(3)$ in gauge theory and wonder the following questions to help me bring QFT ...
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Srednicki Chapter 83--Flavor symmetry change going from global $U$ symmetry for color group $SU(3)$ vs. $SO(3)$

I am writing about the first homework question in Srednicki Chapter 83 (83.1 part a). Please help me check my understanding. In other words, would my reasoning below be right? Forgive me too if ...
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How do we Wick rotate the Maxwell $U(1)$ gauge theory's field strength $F$?

How do we Wick rotate the Maxwell $U(1)$ gauge theory's field strength, say in 3 space and 1 time dimensions? Suppose we start with a Lorentz signature with coordinates $(x_0, x_1, x_2, x_3)$, then we ...
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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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Higher form symmetries and massive gauge fields

I have seen all kinds of questions and answers about how to identify a higher-form symmetries, but they all seem rather abstract. What I would like to do is investigate two very simple examples. ...
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Electric and magnetic charge quantization v.s. the radius of the $U(1)$ or $\mathbb{R}$ gauge parameter

We know (say from Griffiths E&M Problem 8.12) that the electric $q_e$ and magnetic charge $q_m$ (with a distance $\vec{z}$ apart) can store the angular momentum in the space: $$ \vec{L}=\int d^3 V ...
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Fock-Schwinger gauge in pure Yang-Mills theory and coordinate dependence of equations of motion

Let $(\Omega^\bullet (\mathbb{R}^n,\mathfrak{g}),d_A)$ be the Yang-Mills cochain complex on $\mathbb{R}^n$, where $d_A$ is the gauge covariant derivative, $d_A \circ d_A=0$. I was wondering: if we ...
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Can scale invariance symmetry in a conformal field theory (a theory with beta function=0) be localized?

Imagine one has a truly Scale-invariant theory (I mean not classical scale invariance but quantum mechanical, beta function vanishing one), can this symmetry become localized? If yes, what can be the ...
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Normalisation of QCD Lagrangian

In QCD, and more generally in representations of $\mathfrak{su}(N)$, there is a freedom to choose the normalisation of the generators, $$ \mathrm{Tr} \, \left[R(T^a) R(T^b)\right] = T_R \delta^{ab}.\...
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EOM, spin and mass of a particle described by a given Lagrangian

Consider the Lagrangian density $$L=\frac{1}{12}A^{\alpha \beta \gamma}A_{\alpha \beta \gamma}$$ and $B_{\alpha \beta}$, an antisymmetric two-indices, 4 dimensional, free field; moreover $A_{\alpha \...
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Wilson action equations of motion

Let $S_W$ be a Wilson action of $1\times 1$ plaquettes for a gauge group $G$: \begin{equation*} S_W = \beta a^4 \sum_P \left( 1-\frac{1}{N_G} \text{Re Tr}(U_P) \right), \end{equation*} where $\beta$ ...
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Trace properties of the gauge potential in non-Abelian gauge theory

I want to proof equation 69.18 in Srednicki's book "Quantum field theory", which reads: \begin{equation} A_\mu^a(x)=2\text{Tr}[A_\mu(x)T^a].\tag{69.18} \end{equation} $A_\mu(x)$ is the non-...
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Equation of motion of the curvature form $F$ in Yang-Mills theory

Following 4.2.1 in this document (Muharrem Küskü, The Free Maxwell Field in Curved Spacetime, 2001), I tried to adapt the method used (in particular equations 4.21 and 4.32) to Yang-Mills theory ...
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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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Why must the gauge potential create negative norm states?

This question concerns the quantisation of the EM gauge potential $A_\mu$. When the Gupta-Bleuler formalism is introduced, it is usually stated that the creation/annihilation operators satisfy $\...
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Mathematical Charges in Classical Physics, General Relativity and QFT

I have a very easy, and naive, question: given a field $\mathbf{A}$ on some vector space $V$, we can calculate how the flux or circulation of this field behaves. For example, we have Gauss's laws for ...
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Abelian theories with more than one charge

I have a question about the non-abelian character of QCD. In order to write a gauge-invariant Lagrangian, there must be a term with the strength tensor $X^{\mu\nu}_{a}X_{\mu\nu}^{a}$ where $$ X^a_{\mu\...
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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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Difference between field-antifield and light-cone quantisation

I have learnt field-antifield quantisation and know that it can be used for very general gauge theories - open and reducible. I have not got much into light-cone quantisation but I am unable to see ...
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Higher form symmetries and Yang Mills

I have been reading about higher-form symmetries, particularly how they are applied to non-abelian gauge theories. I have come across the claim that pure $SU(N)$ Yang Mills (i.e. with no quarks) ...
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Slavnov operator on the Lagrangian, does it always cancel? With example [duplicate]

I have a long Lagrangian when I apply the Slavnov operator all terms cancel except for the Gauge fixing term and the ghost term. I am using an unusual gauge fixing condition, $$F=(\partial_\mu + \frac{...
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Is this Lagrangian BRST symmetric?

I was doing a problem from a course online. And I calculated a Lagrangian from a non-linear gauge fixing condition, my answer agreed with the given answer online. But I think it might still be wrong. $...
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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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Doubt of gauge covariant derivatives: how can I derive it?

In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
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Gauge a global symmetry of a CFT

Suppose we have a CFT with a global symmetry, and let us denote the theory as $T$. Now we gauge the global symmetry, to obtain another theory $T'$. Does $T'$ also have to be a CFT? If not, what is the ...
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Could the Higgs field be the result of an effective field theory according to an underlying gauge field?

The quantum of the Higgs field, as far as I know, is considered as an elementary particle by the standard model. But it is peculiar in the sense that all the other fields of the standard model are ...
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Do Faddeev-Popov ghost contribute to vacuum polarisation?

I can imagine how one can draw a Feynman diagram for a boson self-energy with a ghost loop. My question is, shouldnt't the amplitude of that process be 0 as the ghosts are merely a mathematical tool?
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Why less gauge fixing conditions in Faddeev Popov method?

According to Dirac theory of constraints systems, to study the dynamics of gauge invariant observables, we can fix the gauge freedom by fixing it using gauge fixing conditions which are equal to ...
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Integration by parts of covariant derivatives in QED

I am reading Sidney Coleman's QFT ch. 27 (in particular Eq. (27.73)) where he said that we can use integration by parts to write the term in the action \begin{equation} \int d^4x (\mathcal{D}^{\mu} \...
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Assumed form for beta function

In a classical paper on Hierarchy of Interactions in Unified Gauge Theories, Georgi et al define the renormalization group equation $$ \mu \frac{\partial g(\mu)}{\partial \mu } =\beta(g(\mu)). $$ He ...
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Is unitarity equivalent to imposing Ward identity in $U(1)$ gauge theory?

I proved in a gauge theory lecture that unitarity violation implies ward identity violation in the simple $U(1)$ case. I was wondering if this statement can be reversed, i.e can we say that unitarity ...
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Mathematics for general gauge structure?

I am currently studying field-antifield (BV) quantization formulation from the review - arXiv:hep-th/9412228. This review gives a nice and quite involved treatment of general gauge structure and how ...
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Physical meaning of the photon propagator using gauge fixing term and path integral

I am considering the following generating functional in terms of functional integral for electromagnetic gauge fields \begin{equation} Z[J] = \int\mathcal{D}A_{\mu} \exp{\left(i\int d^4x \left(-\frac{...
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QFT/QM on loop space

I am reading "Gauging What's Real" by Richard Healey and the author argues for formulating electrodynamics/QED on loop space/the holonomy group, so that the real objects described by the ...
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Can the covariant derivative of General relativity be obtained from a $GL(4,\mathbb{R})$ transformation?

Is it possible to obtain general relativity as a gauge theory from the general linear group? The starting point is: $$ M'=GM $$ where $M',M,G$ are elements of $GL(4,\mathbb{R})$. I believe the second ...
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Gauge invariance of loop Diagrams

Say we have a gauge-fixed QED Lagrangian: $$\mathcal{L} = - \frac{1}{4}{F}_{\mu\nu}F^{\mu\nu}+ \frac{1}{2a}\left(\partial_\mu A^\mu\right)^2+\bar\psi_1(i\gamma^\mu D_\mu - m_1)\psi_1.$$ My question ...
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Does the structure constant of Yang-Mills field change sign under time reversal?

The time reversal of Abelian (electromagnetic) field strength is pretty straight forward. The electric field $F_{0i}$ is even under time reversion. The magnetic field $F_{ij}$ is odd under time ...
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Spontaneous Symmetry Breaking vs Addition of 'Mass terms' by hand [duplicate]

Let consider an $SU(2)$ doublet of bosons $\Phi =(\phi_A, \phi_B)$ which is described by the Lagrangian $$L = (\partial_{\mu}\Phi)^{\dagger}(\partial_{\mu}\Phi) +a^2 V(\Phi^{\dagger} \Phi) -\frac{\...
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How local gauge invariance explain charge conservation and electromagnetic force appearance?

Without electromagnetic coupling, the QM charged particle wave function is not invariant under a local gauge transformation — one with a phase that depends on space (space-time): \begin{equation} \psi ...
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When we use Lorenz gauge or Coulomb gauge, the result formula for electric $E$ and magnetic field $B$ is same or different?

Gauge condition can be chosen as you like or not? is the Lorenz gauge is the only one correct? If Coulomb gauge can obtained exactly same results as Lorenz gauge for the electromagnetic fields E and ...
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How to understand gauge fixing condition?

In Peskin & Schröder's book, they wrote that there is no propagator for an Abelian field due to gauge invariance of the action, and then proposed a gauge fixing condition: $$\partial_\mu A^\mu = \...
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What is the analogy between the gauge covariant derivative and the covariant derivative in General Relativity?

A particle in the Dirac field can be described with the following equation $$i\gamma^\mu\partial_\mu\psi-m\psi=0$$ This is if the particle is non-interacting. However, if we impose a local $U(1)$ ...
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Can a gauge transformation eliminate singularity of gauge potential?

Suppose I have a gauge potential $A$ which goes to infinity at some point $x_0$. Can I use a gauge transformation \begin{equation} A'=U^{-1}AU+U^{-1}dU,~~~U=\exp\{-i\alpha^a(x)T^a\} \end{equation} to ...
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Gauging R-Symmetry

I know that if one gauges the supersymmetry group, you get supergravity. You can then further gauge the R-symmetry and these are the so-called gauged supergravities. But I don't think I've seen anyone ...
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What books would you recommend to understand mathematically the Dirac matter

I am a math student who got interested in the topics above also i want to learn about the Dirac matter the spinors the Einsteins GR and the Yang-Mills Maxwell Anderson Higgs theories and models and ...
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Deriving a general gauge-invariant photon propagator

My understanding is that for a $U(1)$ gauge field $A_\mu$, the most general gauge-invariant kinetic term in the Lagrangian that can be written down which satisfies gauge invariance is something of the ...
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Positive Definiteness of Killing Form in Gauge Theory

This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (...
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109 views

Photon Path Integral and Lorenz Gauge

I am reading Srednicki's QFT book (http://web.physics.ucsb.edu/~mark/qft.html). In chapter 57, specifically in page 343, the book stated that there's a problem with the path integral because the ...

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