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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39 views

Why do not we consider the topological term in Abelian gauge theory?

The second Chern form $\epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}$ is topological in 4-dimensional spacetime. However, we usually only consider this term in non-Abelian gauge theory, but not ...
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Normalization of the Chern-Simons action in the Dijkgraaf-Witten paper

I am trying to understand the seminal paper "Topological gauge theories and group cohomology" by Dijkgraaf and Witten. They consider an oriented three-manifold $M$, compact Lie group $G$ and a $G$-...
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25 views

Induced “ungauged” Chern-Simons terms from a massive Dirac fermion

It is a well-known fact that a massive Dirac fermion minimally coupled to a gauge field $A_\mu$ induces a Chern-Simons term when integrating out the fermion: \begin{align} i\bar{\psi}\gamma^\mu(\...
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Factorizing four generators into two commutators

Let $t_a$ be generators of any given Lie algebra such that $[t_a, t_b]=iC^{c}_{ab}t_c$. Let $A_{a\mu}$ be gauge bosons associated with this Lie algebra. Here $\mu$ is the spacetime index and $a$ is ...
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Are there fermionic bound states that are gauge neutral to all gauge forces?

Puzzle: In our standard model particle physics, are there fermionic bound states that are gauge neutral to all gauge forces? Here we concern strong SU(3) color, electromagnetic U(1) EM, hyper U(1), ...
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What causes here an apparent violation of Elitzur's theorem?

Elitzur's theorem [Ref. Andreas Wipf, Statistical approach to quantum field theory] states that A local gauge symmetry cannot break spontaneously. The expectation value of any gauge non-invariant ...
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Are neutrons gauge neutral to all gauge forces?

Are neutrons gauge neutral to all gauge interactions? Neutron has mass, so it does couple to gravity. However, if we focus on the strong, electromagnetic EM, and weak forces, are there gauge ...
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Can we dispense with the Manifold in General Relativity?

I am studying Quantum Gravity by Rovelli. In chapter 2, the author describes the path that Einstein followed to arrive to General Relativity (GR). At the end of the discussion of the hole argument, ...
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59 views

Why does this amplitude not vanish by the Ward identity?

Consider the process $e^-\rightarrow e^-\gamma$ depicted in the following Feynman diagram. The spin-averaged amplitude with linearly polarised photons is $$\overline{|M|^2}=8\pi\alpha\left(-g^{\mu\nu}...
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Regarding instantons in 2D Abelian gauge theory

I am trying to understand the analogues of instantons in a $U(1)$ gauge theory in 2D Euclidean spacetime. If we follow the same arguments as the 4D case and say that the the gauge field must tend to ...
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Why does gauge invariance have physical consequences?

My understanding is that gauge invariance occurs when the description of a physical field as a mathematical field (i.e., function whose domain is space-time) contains a redundancy: there are ...
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What does it mean to say we have a QFT?

When can we say that the particular action/ Lagrangian we write is an QFT? Does it have to be perturbatively renormalisable? Not allow for negative norm states? The gauge theories we write in 4D, ...
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Definition of gauge freedom in electromagnetism and general relativity

The freedom we have in choosing the vector potential $\vec{A}$ in E&M is referred to as the gauge freedom, whereas in general relativity (GR), we refer to the freedom to choose any coordinate ...
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Consistency condition for Yang-Mills on a Torus

So I was recently studying 't Hooft's paper on self-dual solutions to Yang-Mills on $\mathbb{T}^4$. So the basic idea is that you consider a box with periodic boundary conditions and then you impose ...
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The Higgs mechanism in a quiver gauge model

I've been reading chapter 20 in Peskin and Schroeder about the Glashow-Weinberg-Salam Theory for a $U(1)$ gauge symmetry with gauge coupling strengths $g$, and a complex scalar $\phi$, specifically ...
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1answer
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Why are magnetic monopoles hard to find (if exist)?

I understand the Yang-Mill perspective of $U(1)$-gauge theory. In that, you can easily write down the field of a Dirac magnetic monopole. What interests me is the fact that it's so hard to find (if ...
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Finding the Feynman rules and invariant amplitude of a non-abelian gauge theory [closed]

I am having trouble with finding the invariant amplitude from a given Lagrangian. I am given a Lagrangian of a non-abilian gauge field and a complex scalar field. It is given by, $$ \mathscr{L}=\...
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2answers
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Gauge symmetries not from promotion of global symmetries

The most intuitive example of a gauge symmetry is such where you take a theory that has some global symmetry, and ask what needs to be done for this symmetry to be local. This procedure involves the ...
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Is a pseudo-Goldstone boson always a pseudoscalar particle?

There are several examples of pseudo-Goldstone bosons which are CP-odd particles, such as the pion, as well as many axion-inspired models. If we invert the logic, Are all pseudo-Goldstone boson of ...
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Fermion operators: why $\rm SU(2)$ symmetry and not $\rm U(2)$ symmetry?

Let us consider operators $c_{\uparrow}$ and $c_{\downarrow}$ which destroy a fermion with spin up and a fermion with spin down, respectively. These operators can be found, for example, in the Hubbard ...
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't Hooft Anomaly Equivalent Definitions

I've seen a 't Hooft anomaly defined in two ways. Roughly, a theory has a 't Hooft anomaly when Once the theory is coupled to a background gauge field $A$ (so study eg the partition function $Z[A]$), ...
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Is there a name like “color” for points on the associated bundle of the electroweak $SU (2)$-bundle?

It is said that quarks have "color", where a color corresponds to a point in the fiber $C_x$ of the associated bundle bundle $C \to \mathbb {R}^{3+1}$ of the strong $SU (3)$ principal bundle with the ...
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Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented. Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
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Degrees of freedom of a photon/electromagnetic field, meaning of degrees of freedom [duplicate]

as I understand it a photon field $A^\mu$ has two physical degrees of freedom corresponding to the two polarisations. I was wondering why, for example, energy isn't considered a degree of freedom. I ...
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1answer
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Gauge fixing, Lorentz invariance and positive definite metric of Hilbert space

Updated 0n ${\bf 02.04.2020}$ $\large{\bf Context}$ In the first $3$ minutes of this video lecture (based on the presentation here) on the subject matter of Goldstone theorem without Lorentz ...
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Field Strength and Source terms

This question is related to my recent unanswered question, but it was too complicated so please let me make this new question at first. First, I consider a field strength which is expressed as \...
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Question about gauge symmetry confronted in Schwartz‘s book

This picture is from Schwartz book on QFT on page 131. I cannot understand that: What does the orange underlined sentences mean? How is Equation 8.108 derived? Could anyone kindly make some further ...
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“Gauge Freedom” in GR

When we derive the equations for propagating waves in GR, we have to make a gauge choice to get something unique. I understand that in electromagnetism, the gauge is not in general something ...
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How do the massive gauge bosons eat up the would-be Goldstone bosons?

In the literature one often reads that the would-be Goldstone bosons of the spontaneously broken $SU(2)\times U(1)$ symmetry are eaten up by the longitudinal polarisation vectors of the now massive ...
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General relativity as a gauge theory of the Poincaré algebra

Let the Poincaré algebra be given without any factors of i as $[P_\mu,P_\nu]=0$, $[M_{\rho \sigma},P_\mu]=\eta_{\sigma\mu}P_\rho-\eta_{\rho\mu}P_\sigma$, $[M_{\mu\nu},M_{\rho\sigma}]=\eta_{\nu\rho}...
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Is it possible to derive a non-mass gap from the Yang-Mills action? [duplicate]

The action: $$J=∫Tr(F∧⋆F).$$ Is it possible to find the non-mass gap property from the Yang mills action? If so, how?
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Field strength invariance on curved manifold

Promoting a global gauge symmetry $\Psi\rightarrow\Psi^\prime=U\Psi$ to a local symmetry requires the definition of a gauge covariant derivative as ${D_\mu}^\prime\Psi^\prime=UD_\mu\Psi\Rightarrow{D_\...
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How do we guess that photons have zero mass from the quantization of the EM field?

I found that to guess that photons have zero mass from the quantization of the electromagnetic field I have to take into count the gauge invaiance of the field. Is it right or do I just have to ...
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How can QED have degenerate vacua without superselection?

This question is based on Andrew Strominger's lecture on the IR structure of field theories, in particular Section 2.11 The usual story with spontaneous symmetry breaking is a follows. You have ...
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50 views

Why does Yang Mills Theory only work for massless particles? [duplicate]

Why does the non-abelian element of the Yang Mills theory (SU(3), SU(2), etc.), inherently imply a 'non mass gap' and long range forces? Please explain as simple as possible, I haven't seen a clear ...
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Physical content of $[D_\mu, D_\nu]=ieQF_{\mu\nu}$ and $[D_\mu, D_\nu]=igT^a G^a_{\mu\nu}$

For the abelian QED theory, $$[D_\mu, D_\nu]=ieQF_{\mu\nu}$$ where $D_\mu=\partial_\mu+ieQA_\mu$ is the gauge covariant derivative in QED and $F_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu$ is ...
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1answer
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Physical meaning of minimal coupling assumption

In SM, both for the gauge field $A_{\mu}$ associated with photon and for the gauge fields $W^{j}_{\mu}$, to restore the invariance of the lagrangian after the changing of the global transformation to ...
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1answer
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Must my gauge transformation be defined on all space?

Consider the set-up of the Aharonov-Bohm effect with a solenoid of radius $a$ aligned along the $z$-direction and uniform magnetic field $B_0$ within. One possible vector potential, in cylindrical ...
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1answer
66 views

Theta-terms in 3+1D QCD and 1+1D QED / $CP^1$ models

It is well known that topological $\theta$-terms in gauge theories are total derivatives and vanish after integration over the Lagrangian (or Hamiltonian) density, unless there are nontrivial boundary ...
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Why do we use affine groups in gauge theory? What is the purpose?

When we study General Relativity in the frame of gauge theory, what's the importance of affine group?
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Gauge ghosts & unphysical states in gauge theory

I have a general question about a statement from Wikipedia about ghost states as occuring in gauge theory: "In the terminology of quantum field theory, a ghost, ghost field, or gauge ghost is an ...
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What does adding a gauge fixing term $-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$ really mean?

Given any electric and magnetic field (or $F_{\mu\nu}$), there is always some freedom in defining what $A_\mu(x)$ should be. In fact, there are infinite choices for $A_\mu(x)$. This is because for an ...
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1answer
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Why does symmetry spontaneously break? [closed]

Why does symmetry spontaneously break? For example,in the case of the mexican hat potential, as far as I can tell, a particle that is at the unstable equilibrium ought to be able to continue staying ...
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1answer
90 views

Gauge bosons of an Abelian gauge field massless

In https://en.wikipedia.org/wiki/Photon#The_photon_as_a_gauge_boson is stated "...The quanta of an Abelian gauge field must be massless, uncharged bosons, as long as the symmetry is not broken; hence,...
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Lagrangian density of a free electromagnetic field [duplicate]

How do you derive the result for the lagrangian density of a free electromagnetic field
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Local Symmetry or gauge transformation of second kind in QED

While ensuring the gauge invariance of the lagrangian $$\mathcal{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}+\bar{\psi}(i \not \partial-m) \psi-e \bar{\psi} \gamma^{\mu} A_{\mu} \psi$$ we consider the ...
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If gravity is a gauge theory, what is the Lie group? [duplicate]

Here I asked a question. In one curious comment, I see a statement that gravity is a gauge theory. However, my definition (based on what I read till date) of a gauge theory is a field theory which is ...
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Missing factor in Dimensionally reduced Yang Mills Ghost Field

I'm trying to calculate the ghost field in the background field gauge for the dimensionally reduced Yang Mills action in this paper. I am using the expression from Srednicki's book, chapter 78. The ...
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1answer
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Vacuum expectation value (VEV) of a Gauge theory - Spontaneous Symmetry Breaking (SSB) - Higgs Mechanism

I am dealing with a sort of scalar QED with a term of SSB \begin{equation} \mathcal{L}=\left|D_{\mu} \phi\right|^{2}-\frac{1}{4}\left(F_{\mu \nu}\right)^{2}-V\left(\phi^{*} \phi\right) \end{equation} ...
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1answer
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Parity Invariance Complex Scalar Field Lagrangian

I am trying to prove the parity invariance of some terms in a complex scalar field Lagrangian, for example $m^2 \; \phi^* \phi$ or $\partial_{\mu} \phi \;\partial^{\mu} \phi^*$. So what I want to ...

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