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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De Rham current associated with knot in abelian CS theory on a generic manifold
I'm studying TQFT and I'm stucked on this part of the paper of my teacher:
My teacher didn't explain a lot about it and I've never followed an advanced course on differential geometry or algebraic ...
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Constraints Generating Gauge Transformations and BRST
Given a gauge-invariant point particle action with first class primary constraints $\phi_a$ of the form
$$S = \int d \tau[p_I \dot{q}^I - u^a \phi_a]$$
we know immediately, since first class 'primary' ...
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Gauge invariance of simplified weak interaction
I am having difficulties with a homework set. We are given the following lagrangian for a simplified weak interaction between an electron $\psi$, neutrino $\chi$, and a massive (complex) vector-boson $...
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Action for scalar field with externally specified EM fields and gauge
Consider the following Lagrangian for a complex scalar field $\Psi$ along with an electromagnetic environment,
$$\mathcal{L} = ([\partial_\mu - i e A_\mu] \Psi)^* [\partial^\mu - i e A^\mu]\Psi - V(\...
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Anderson Higgs mechanism and superconducting phase fluctuations
In the context of superconductivity, in the Anderson Higgs mechanism (see, for instance, https://doi.org/10.1119/1.5093291 (PDF), or https://doi.org/10.21468/SciPostPhysLectNotes.11 (PDF), the gauge-...
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Problem evaluating an anticommutator in supersymmetric quantum mechanical gauge theory
I am trying to reproduce the results of a certain paper here. In particular, I'm trying to verify their eqn 5.31.
The setup is N = 4 gauge quantum mechanics, obtained by the dimensional reduction of N ...
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Correlation function of three gauge fields with a derivative
I am trying to calculate the following correlation function
\begin{equation}
gt^at^bf^{bed}\langle \partial_\mu A^a_\nu(x)A^e_\alpha(y) A^d_\beta(y) \rangle
\end{equation}
where $A_\mu=t^aA^a_\mu$ are ...
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Understanding a supersymmetric quantum mechanical gauge theory model
I'm studying this paper on supersymmetric ground state wavefunctions. In section 5 "quantum mechanical gauge theories", it says:
"We begin with the ${\cal N} = 2$ gauge theory which ...
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What is an Intuitive example of a Gauge Symmetry?
Can anyone give an intuitive example of what a gauge symmetry is? I am new to this concept and would like to understand it better!
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Why can Principal $G$ Bundles be Trivialized when $G = SU(N)$?
Reading about TQFT one usually comes about the fact that over 3-manifolds, Simply Connect Lie Group-bundles can be trivialized, yet it is a bit hard to find a clear answer online. Why is that the case?...
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Basic Question on Differential Forms (Chern-Simons Level Quantization)
I came across the following post regarding the boundary term in Chern-Simons theory (specifically the level quantization of the theory). I am new to differential forms so the following questions may ...
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Gauge Boson Self-Interactions with covariant derivative
Self-Interactions of the unphysical gauge bosons $W_1, W_2, W_3$ are written within the gauge term
$L_\mathrm{Gauge}=-\frac{1}{4} W_{\mu \nu} W^{\mu \nu}$
with $W_{\mu \nu}= \partial_\mu W_\nu - \...
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Why are physical states not eigenstates of BRST charge?
In many texts in quantum field theory or string theory, it is stated that the BRST charge $Q$ must annihilate physical states because the states are required to be BRST invariant. Since $Q$ generates ...
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What sort of QED-like theories can have non-quantized charge?
It is often said that the existence of a single monopole would force electric charge to be quantized, due to Dirac's argument. However, one can write down theories like QED that, independently of the ...
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Fermionic and bosonic degrees of freedom of a vector superfield
I am currently studying supersymmetry with the SUSY primer of Stephen P. Martin (https://arxiv.org/abs/hep-ph/9709356) and there seem to be not equally many bosonic and fermionic degrees of freedom (...
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Holonomy expansion for path deformation
A path deformation by $\epsilon^{\mu}(s)$ induces a variation of the connection $A'(s)=A(s)+\Delta A(s)$. I'm trying to obtain the first-order expansion of the holonomy $H_{\gamma}(A)=Pe^{i\int_{\...
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How to calculate the component term in BFSS matrix model?
I'm reading articles about BFSS, and confused by the calculation.
The Hamiltonian is
$$
H=\frac{g^2}{2}TrP_{I}^{2}-\frac{1}{4g^2}Tr[X_{I},X_{J}]^2
-\frac{1}{2}Tr\psi_{\alpha}\gamma_{\alpha \beta}^{I}[...
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How does spin $j$ matter contribute to the running of the gauge coupling?
The one-loop beta function $\beta(g)$ for the gauge coupling $g$ with gauge group $G=SU(N_c)$, in a theory with $n_f$ spin-1/2 fermions in a representation $R_f$ of $G$, and $n_s$ complex scalars in a ...
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Can a symmetry of a Lagrangian be of order higher than the order of the Lagrangian itself?
In book "Quantization of gauge systems" by Henneaux and Teitelboim, one of the exercises (see picture) was to show that any symmetry of Lagrangian $L$, if $L$ depends only on the generalized ...
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Calculating gauge propagator in minimally coupled, non-relativistic fermion system
For context, I am trying to derive Eq. 4.1 of $T_c$ superconductors">this paper. Consider the action
$$S[\psi^\dagger, \psi, a] = -\int d\tau \int d^2r \sum_\sigma \psi^\dagger (D_0-\mu_F-\frac{1}{...
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Understanding notation in $-\overline{U}_LmU_R+\text{h.c.} =-\overline{U}mU$
Suppose $U$ is a field with left handed and right handed parts $U_L,U_R $, respectively.
When discussing Lagrangians, I keep finding the following simplification step
$$-\overline{U}_LmU_R+\text{h.c.} ...
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Performing integration by parts on a relativistic Lagrangian [duplicate]
I have a modified Lagrangian for an electromagnetic field:
$$L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - \frac{\xi}{2} (\partial_\mu A^\mu)^2.$$
All the symbols and variables have their usual meanings. I ...
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Representation of nonabelian Wilson line in terms of fermionic fields
Context:
The coupling action of a particle of charge $q$ to a $U(1)$ gauge field is given by
\begin{equation}
S = q \int d \tau A_\mu \left( X \right) \frac{dX^\mu(\tau)}{d \tau} = -i \ln W_q,
\tag{...
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Concluding that local $SU(2)$ symmetry implies that we have charged and uncharged fields
In Samoil Bilenky's (2nd. ed) "Introduction to the Physics of Massive and Mixed Neutrinos," he constructs the $SU(2)$ Yang-Mills model.
He starts by introducing the doublet (section 3.2, ...
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QCD in Coulomb Gauge Kernel Expansion
I was re-reading the Hamiltonian QCD in Coulomb gauge section in Particle Physics and Introduction to Field Theory by T.D. Lee and I was trying to understand better the form of the Coulomb kernel that ...
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Broken symmetry and three-photon vertex
I know that loop-level three-photon vertex in QED is zero since the contribution from fermion and antifermion cancel each other.
Also, from what I know this has something to do with gauge invariance ...
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Why is it justified to focus on gauge transformations constant at spatial infinity in QCD instantons?
In the context of Yang-Mills theories and QCD instantons, much of the literature and conventional treatment hinges on the consideration of gauge transformations that remain constant at spatial ...
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Instanton solution in a gauge theory with higher derivatives
If one has the following higher derivative gauge field lagrangian
$\mathcal L \propto F_{\mu\nu}^a \:\Delta^n \:F^{a\mu\nu} $
where $\Delta$ is the covariant laplacian operator acting on the 2-form $F$...
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What is the gauge covariant derivative?
What is the gauge covariant derivative in layman's? Does it describe the kinetic energy of the gauge fields?
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What is the reverse operation of gauging a global symmetry?
As far as I understand, gauging a global symmetry means taking a model with a global symmetry and transforming it into a model such that the previous symmetry group is now the gauge symmetry of your ...
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Dressing an operator by Wilson line in Quantum Electrodynamic
I am reading a paper arXiv:1507.07921 which introduce gravitational dressing. The paper compare it to dressing in QED.
Consider the scalar QED lagrangian
$$\mathcal{L}=-\frac{1}{4}(F^{\mu\nu})^2-|D_\...
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Physical states in Gupta-Bleuler quantization
I'm reading Timo Weigand notes for Gupta-Bleuler quantization of free EM field.
On page 109, Author has made the following statements.
The Gupta-Bleuler condition for physical state is
$$|\vec{p},\...
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Resolution of Singularities in Seiberg-Witten Moduli Space
Consider the low-energy effective action for $\mathcal{N} = 2$ super Yang-Mills with gauge group $SU(2)$. This was famously studied by Seiberg and Witten in their pioneering paper. They found that ...
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In the equation $B = ∇ \times A$. When $B =0$ it is said $A$ is not necessarily equal to zero. My question is how can $A$ exist without $B$?
How can one understand intuitively that without magnetic field, the potential can still exist?
Also $\nabla^2 \Phi=−\rho/\epsilon_0$. If charge density is $0$,$\Phi$ is non zero. How can potential ...
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Why did Polyakov choose the particular expression for "distance" between metrics in the space of parametrized curves? (Eq. 9.23)
In Chapter 9 of Polyakov's book Gauge Fields and Strings, 1987 , he studies measures in the space of metrics and diffeomorphisms. My question is - how does one come up with Eq. 9.23?
For some context: ...
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The BRST variation of the gauge fixing condition
Following Polchinski volume I, p 126 onwards, The BRST variation of fields $\phi^{i}$ is given by
$$\delta_{B} \phi_{i} = - i \epsilon c^{\alpha} {\delta}_{\alpha}\phi_{i} \; .\tag{4.2.6a}$$
My ...
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Trouble reconciling these two views on gauge theory
Very generally speaking, I view gauge theory as asking what local symmetries leave our theory invariant and then seeing the consequences. Thus, taking a look at the Lagrangian for electromagnetism, we ...
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Is there a way to visualise / understand intuitively the curvature in the $U(1)$ circle bundle responsible for the electromagnetic force?
In general relativity we have embedding diagrams of different slices of spacetimes. These can be quite helpful to understand the geometry of a given pseudo-Riemannian manifold (especially when the ...
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Gauge invariance in QED with just fermion transformations
I've got myself confused about a basic question. If we have a gauge-invariant operator $\mathcal{O}$ whose expectation value is
\begin{equation}
\left\langle\mathcal{O}\left(x_1, \ldots, x_n\right)\...
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When we solve the Maxwell equations for $(\phi,{\bf A})$ in a gauge, will the solution $(\phi,{\bf A})$ automatically obey the gauge condition?
As the title of the question suggest; how you could determine if a gauge fixing is a condition or a requirement. Let me explain.
Imagine you are working with Maxwell's Equations. By the definition of ...
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Expression of Bianchi identity in associated bundle
Theorem 5.14.2 of Mathematical Gauge Theory by Hamilton states that the curvature form $F^A_M\in\Omega^2(M,Ad(P))$ satisfies the third form of the Bianchi identity $d_AF^A_M=0$ where $F^A_M$ is the ...
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Secondary constraint imposed gauge fields
I am asking about aspects of quantizing electromagnetic field followed in Weinberg's Quantum theory of fields Volume I, in section 8.2 . I am able to understand the primary constraint that arises from ...
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Characteristic classes and index theorems for physicists
Since characteristic classes and index theorems are occasionally used in the QFT (for example, when discussing instantons or quantum anomalies), I want to learn more about them. Is there any good ...
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Stueckelberg mechanism for interacting QFTs
The Stueckelberg mechanism or "trick" (see e.g. Section 4 of https://arxiv.org/abs/1105.3735) is basically a method to take the massless limit of massive gauge theories in a smooth way, ...
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Why is the canonical momentum operator for a charged particle not observable in an E&M background?
From the Wikipedia article on the momentum operator (https://en.wikipedia.org/wiki/Momentum_operator#Canonical_commutation_relation) the following operator (in position representation)
$$\hat{\bf{P}} ...
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Why can't we gauge the Lorentz group? (Or can we?)
One of the (many different, somewhat independent) routes to gauge theory is to start from a global symmetry of some kind and "gauge" it, which involves promoting it to a local symmetry and ...
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Partial gauge fixing a point to infinity in conformal field theory [duplicate]
While deriving the structure of a 4 pt. function in CFT, we write the conformal block with respect to the cross ratios
$$ u = \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \; v = \frac{x_{14}^2x_{23}^2}{...
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Faddeev-Popov Method for Gauge Fixing in CFT (Light-ray Operators)
I was attempting to go through the paper by Petr Kravchuk and David Simmons-Duffin: https://arxiv.org/abs/1805.00098 where I encountered the following
Just below E.4, it is mentioned that for the ...
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Gauge transformation with complex parameter in quantum mechanics with minimal substitution
This follows from the question Can an Electromagnetic Gauge Transformation be Imaginary?
It's about the Hamiltonian $$H=\frac{(p-A)^2}{2m}$$
in units where $c=e=\hbar=1$. The question regarded a gauge ...
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Quantum Mechanics with non-Abelian gauge fields
In "conventional" quantum mechanics, in the presence of electromagnetic fields, we use minimal substitution in the Hamiltonian in the simplest case. I.e.
$$H=\frac{\vec{p}^2}{2m}\rightarrow \...
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