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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8answers
362 views

What is the physical interpretation of inducing a local symmetry?

In QFT we upgrade global symmetries to local symmetries and in order to keep the Lagrangian invariant we must add another gauge field. This produces the forces in the standard model. I understand the ...
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2answers
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$1+1D$ $U(1)$ gauge theory is a quantum mechanical system

In article Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory there is statement (page 13): Ordinary $1 + 1$-dimensional $U(1)$ gauge theory is effectively a quantum ...
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1answer
61 views

How to derive the hidden symmetry behind linearized gravitation equations?

I am trying to derive the "gauge-like" symmetry of linearized gravitation equation, after deriving the latter heuristically from Newton's universal of gravitation. I am roughly following ...
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Mass terms for scalar lagrangians?

First off, a pre-question: if I got this wrong, then probably the whole reasoning is wrong as well. Studying the lagrangian for a two-particle scalar field with a quartic interaction in the context of ...
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0answers
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Lectures by Susskind versus Zee on the ${\rm SU}(5)$ grand unified theory notations

I was comparing two lectures about ${\rm SU}(5)$ grand unified theory. a lecture of Susskind He showed how to write $$ (5 \times 5)_{asym}=\bar{10} $$ as a lecture of Zee showed how to write $$ (\...
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Singularities in gauge-fixing conditions and topological defects

I am studying the 't Hooft's paper "Topology-of-the-gauge-condition-and-new-confinement" https://doi.org/10.1016/0550-3213(81)90442-9 and there are several points which I would like to ...
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1answer
92 views

How does the gauge-fixing term break gauge invariance?

Given the gauge-fixing term $\mathcal{L}_\mathrm{gf}=-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$ and the gauge transformation $A_\mu\mapsto A_\mu+\partial_\mu\alpha$, how does the term break gauge ...
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1answer
62 views

What's the difference and relations between $SU(N)$ Schwinger boson and $CP(N\!-\!1)$ non-linear sigma model?

There are two ways when dealing with spin system(Heisenberg model): non-linear $\sigma$ model and Schwinger boson. Non-linear $\sigma$ model When taking large $S$ limit, the quantum fluctuation of ...
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0answers
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Why are gauge transformations required to be *smooth*?

Question Suppose you perform a gauge transformation $f(x)$ that is only $n$ times differentiable, for any $n$. Can the discontinuity in the $(n+1)^{th}$ derivative change any observable? Clarification ...
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1answer
91 views

$SU(2)$ doublets from the transformation law of a matrix of scalar fields

If we have a $2 \times 2$ $SU(2)_L$ and $SU(2)_R$ matrix $\Phi=\begin{bmatrix} a & c \\ b & d \end{bmatrix}$, where a, b, c and d are four complex Klein-Gordon fields, that under a gauge ...
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1answer
136 views

When does a VEV induce another VEV?

If we have a model with more than one Higgs-doublet, when do the VEV of a scalar field of one of those doublets must induce a nonzero VEV on a scalar field of another of those doublets?
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Physical meaning of mass renormalization

In the case of charge renormalization, one can present a neat and nice physical idea that brings a physical ground to it called "Vacuum Polarization". Which can be even extended to non-...
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2answers
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2 coinciding D-branes leads to a $U(2)$ gauge theory

I'm having trouble understanding how two D-branes leads to a $U(2)$ gauge theory from David Tong's notes, chapter 7, pages 191-192. I am learning group theory and I understand that a 'charge' is a ...
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1answer
30 views

Open gauge algebras apart from supergravity theories

Does anyone know of a gauge system that is not a model of (super-)gravity where the gauge algebra fails to close off-shell?
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To which symmetry group is the original symmetry group of a theory broken when we add to the theory a scalar field with a VEV?

If one has a gauge theory with a specific symmetry group and we add to it a scalar field with a non-zero VEV, how do we know in general to which symmetry group will the original symmetry be ...
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114 views

Why would one want to study the geometry of Lie groups?

Lie groups are commonly used in theoretical physics and mathematical physics. They are useful tools to study simple systems such as the harmonic oscillator. They are also crucial in representation ...
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1answer
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What is the gauge boson mass term for a theory with a matrix of scalars?

If we have a theory with gauge group $SU(3) \times SU(2)$ with a set of six complex scalar fields grouped in $\Phi=\begin{pmatrix} a & d\\ b & e\\ c & f \end{pmatrix}$, where, for instance,...
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Spin orbit coupling and effective gauge field

In the context of solid state system, the spin-orbit (SO) coupling from low-energy expansion of Dirac equation is $$H_{SO} = \frac{1}{2 m^2 c^2} (\vec{s}\cdot (\nabla V \times \vec{p}))$$ My question: ...
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2answers
140 views

Is electric field operator in Schrödinger picture time-dependent?

We know that in the Schrödinger picture, operators are time-independent if they do not have explicit time-dependence. So do electric field and vector potential field operators have time dependence in ...
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Bundle to be considered in the Dirac monopole

When dealing with the Dirac magnetic monopole from the point of view of the gauge theory, most authors consider a principal bundle over $S^2$ under the pretext of being homotopic to $\mathbb{R}^3 \...
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1answer
118 views

Symmetry group describing the electroweak interaction

I have a question about the spontaneous symmetry breaking (SSB) and its effect on the group symmetries of the Standard Model. If I understand correctly, before SSB (at high temperatures/energies) the ...
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49 views

Global and gauge charges

I have seen several questions regarding the difference between global and gauge charges, but I don't really get the physical implications. The sQED lagrangian is: $\mathcal{L}=-\frac{1}{4}F_{\mu \nu}^...
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1answer
71 views

Is there a physical mechanism explaining the link between Gauge-Bosons and local Gauge-Invariance?

Imposing local gauge-invariance naturally couples e.g. a charged fermion-field to the electromagnetic field. To my understanding local gauge-invariance is imposed because a gauge in one system should ...
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196 views

What is Noether's Second Theorem?

I have been unable to find a short statement of Noether's second theorem. It would be helpful to have the following: A short mathematical statement of the theorem. Does it imply a conservation law ...
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1answer
117 views

Is the Covariant Derivative a phenomenological attempt?

I am trying to self study QFT and i am very confused about the covariant derivative. When we require our theory to be invariant under local gauge transformations we kind of "guess" that we ...
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Is it possible to have a compact abelian $U(1)$ lattice gauge theory on a non-compact manifold?

We have a compact lattice gauge theory if we let $A_{i}(n)\in[-\pi,\pi]$, and if we identify $A_{i}(n)\sim A_{i}+2\pi$. A simple lattice gauge theory in 2+1D then has an action $$S=\sum_{x}1-\cos(F_{\...
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60 views

Resources on BRST and BV quantisation for local quantum field theories

This is a reference request, to ideally a textbook, monograph, set of lecture notes or lecture videos, on the topics of BRST quantisation and the Lagrangian BV formalism. My constraints are as follows:...
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1answer
155 views

Winding number is an integer

In computing the variation of the action in Chern-Simons, and in other contexts, we get the following expression that is named the winding number, where $U$ comes from a gauge transformation: $$ W[U] =...
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Winding number of center symmetries

Suppose we have a center symmetry transformation that acts on the connection as: $$A \mapsto \ ^gA = \Omega^{-1} (A+d)\Omega $$ and satisfies $$ \Omega(t+\beta,x) = h\Omega(t,x)$$ Suppose $W[U]$ ...
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Chern-Simons Path integral restricting to small gauge transformations

How does one compute the Chern-Simons path integral in 2+1 dimensions considering only small gauge transformations? Is this even a well-defined theory?
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Chern-Simons theory with only small gauge transformations

Usually when one derives the variation of the Chern-Simons action in 2+1 dimensions, one has a term that is proportional to the winding number. Then one argues that the coupling constant must be an ...
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0answers
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Expectation value of Polyakov loop

Assuming a pure Yang-Mills theory, how exactly does one get that, for appropriate $\beta$ for confinement, the expectation value of the Polyakov loop $<\Phi>$ equals zero? I do not seem to get ...
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1answer
183 views

Why is this Chern-Simons theory gauge invariant?

I am reading the lecture notes in https://arxiv.org/abs/hep-th/9902115 and in it, it says that the Lagrangian $$\mathcal{L}_{\mathrm{CS}}=\frac{\kappa}{2} \epsilon^{\mu \nu \rho} A_{\mu} \partial_{\nu}...
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3answers
140 views

The form of gauge invariance

When imposing local gauge invariance for a simple lagrangian describing a free Dirac fermion field: $$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi$$ Is there a particular ...
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1answer
75 views

Batalin-Vilkovisky quantization

Batalin-Vilkovisky (BV) quantization is way of quantizing a theory, which is apparently more powerful than BRST quantization. It has been used, for example, for string field theory, in the closed ...
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1answer
89 views

Two-field Symmetry Breaking unitary gauge

Let's consider the following theory: $$L= -\frac{1}{4}F_{\mu \nu}F^{\mu\nu} +{1\over 2} |D_\mu \Phi|^2 +{1\over 2}|D_\mu \chi|^2 + \lambda_1\bigl(|\Phi|^2-\frac{v_1^2}{2}\bigr) +\lambda_2\bigl(|\chi|^...
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1answer
170 views

What is the reason for turning global symmetries into local symmetries?

For example a simple complex scalar field theory has a global $ U(1) $ symmetry where the field $ \psi $ can be replaced by $ e^{ i \alpha } \psi $, where $ \alpha $ is just some real constant, ...
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2answers
158 views

Fermion current and gauge interactions in Lagrangian density

I believe the answer to this is quite simple, perhaps so simple that I cannot find it in any book. Usually, if there is a gauge symmetry in the theory, we add an interaction term in the covariant ...
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1answer
64 views

Better understanding of Gauge-Invariance

This is a long question and the main points are emphasised in bold. Consider a non-Abelian SU(N) gauge theory. $t_a $ is an Hermitian generators of SU(N) so that $$U = e^{i\alpha^a(x)t^a} \tag{1}$$ is ...
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1answer
95 views

Question about Faddeev-Popov gauge-fixing in Schwartz textbook

I am trying to understand equation (25.91) from Schwartz's Quantum Field Theory textbook. The goal is to gauge-fix the path integral for Quantum chromodynamics using the Faddeev-Popov trick. Briefly, ...
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1answer
272 views

How do we know from representation theory that a massless spin-1 particle has only two polarizations?

In chapter 8.2.3 of Schwartz' textbook "Quantum Field Theory and the Standard Model", the author states the following, Finally, we expect from representation theory that there should only be two ...
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55 views

Regarding Gauge fixing in General relativity

I have been reading about gauge fixing in general relativity but I have failed to understand what does choosing a particular gauge really mean? Is it just a choice of a coordinate system where a ...
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0answers
43 views

Do fermions have groups like how gauge bosons have gauge groups?

As I understand it, the Lie groups $U(1)$, $SU(2)$, and $SU(3)$ correspond to the electromagnetic, weak, and strong forces respectively (ignoring electroweak mixing) and the generators of their ...
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1answer
114 views

Gauge fixing conditions in general relativity

Is there a limit to gauge fixing conditions we can impose in gravity ? I have seen two gauge fixing conditions. The DeDonder gauge $\partial_\mu g^{\mu\nu}$ and then in 3+1 formalism the gauge fixing ...
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1answer
106 views

Gauge invariance of Yang-Mills lagrangian

I am trying to show gauge invarince of the Yang-Mills lagrangian $$\mathcal{L}= -\frac{1}{4}F_{\mu \nu }^{a}F^{\mu \nu ,a}+\sum_{i,j}^{N}\overline{\psi}_{i} (\delta _{ij}i\partial_{\alpha}\gamma^{\...
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2answers
43 views

Understanding the Leibniz rule and dynamical variables

On the following question Derivation of Maxwell's equations from field tensor lagrangian We try to calculate the equation of motion of $(1)$ where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu ...
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1answer
141 views

Kaluza-Klein metric and Ricci scalar?

The metric is \begin{equation} ds^2 = G^D_{MN}dx^M dx^N = G_{\mu\nu}dx^\mu dx^\nu + G_{dd}(dx^d + A_\mu dx^\mu)^2. \end{equation} Then \begin{equation} G^D = \begin{bmatrix} G_{\mu\nu} + G_{dd}A_\mu ...
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3answers
142 views

Are all vector-bosons gauge-bosons?

All QFTs that I come across have vector fields appearing as gauge-bosons. Is there any problem with vector fields that are not gauge-bosons? I am not so concerned about the theory producing results ...
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1answer
79 views

Eliminating residual gauge in BRST quantization of Yang-Mills theory

I would like to know if there is a procedure to completely fix a gauge, which I believe we must do in order to make sense of the path integral? In chapter 74 Sredniki introduces the Lagrangian $$ \...
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2answers
130 views

$U(1)$ gauge invariance

I am looking at some exercises in an online course in QFT and there is a question about the $U(1)$ gauge invariance of this operator: $$i\bar{\psi}\sigma^{\mu\nu}\gamma_5(\partial_{\mu}A_{\nu})\psi$$ ...

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