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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51 views

Nambu-Goto action and zweibein

Consider the Nambu-Goto action \begin{equation} S=\int d\sigma d\tau \sqrt{(\partial_\sigma X^\mu \partial_\tau X_\mu)^2-(\partial_\sigma X^\mu\partial_\sigma X_\mu)(\partial_\tau X^\mu\partial_\tau ...
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How do the symmetries of a system constrain gauge fields?

As a simple example, take a constant magnetic field $\vec{B} = (0,0,B)$. This is invariant under rotations about the $z$ axis. However, we can express $\vec{B}$ as the curl of a vector potential $\vec{...
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Calculating equation of motion in gauge theories: using ordinary derivatives or covariant derivatives?

For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the ...
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Why Hamiltonian of gravity is zero?

In paper Topological Gravity as the Early Phase of Our Universe there's statement: Hamiltonian of gravity would vanish by time reparameterization invariance. How to derive such result?
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Horizontal vector subsepace of electromagnetic connection in Minkowskian spacetime

For Minkowskian space-time $M$, the principal bundle for electromagnetism is $(M \times U(1), M, proj_{1}, U(1))$. I imagine there is a global gauge potential (since I can choose a global section, ...
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Relation between chemical potential and Gauge field [closed]

How the chemical potential is related to near boundary expansion of Gauge field?
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Deriving the Momentum Conjugate to $\mathbf{A}$ in Canonical Quantisation of Yang-Mills

In David Tong's lecture notes on Gauge Theory, in the section "Canonical Quantisation of Yang-Mills", the momentum conjugate to the field $\mathbf{A}$ is computed for the following ...
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What does projective space $\mathbf{S}^{2 N-1} / U(1) \cong \mathbf{C} \mathbf{P}^{N-1}$ mean?

In David Tong's lecture notes on Gauge Theory, in the section on 'Quantising the Colour Degree of Freedom', the following statement is confusing me, Since we already have the constraint (2.16), this ...
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“Axial” gauge in the $Z_2$ lattice gauge theory

I am reading the paper by Fradkin and Susskin on the lattice gauge theory (Order and disorder in gauge systems and magnets). In section III. C, where they were trying to introduce the duality ...
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Faddeev-Popov Gauge Fixing Procedure

I want to know that does $F^{a}[A_{\mu}] = 0$ condition used in Faddeev-Popov Quantization has unique solution $A_{\mu}$ or is it $F^{a}[A^{\theta}_{\mu}] = 0$ should have unique $\theta$ as solution ...
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Quantising Yang-Mills, analogy to Gauss' law

In David Tong's lecture notes on Gauge Theory, in the section on 'Quantising the Colour Degree of Freedom', the following action is discussed, $$ S_{w}=\int d \tau i w^{\dagger} \frac{d w}{d t}+\...
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Why are Lagrangians linear in $\dot{q}$ so ubiquitous? Gauge theory, Berry phase, Dirac Equation, and more

It seems to me that we encounter first-order equations of motion in some very special situations in physics. It is not clear to me what the connection is, and I am hoping to get some insight into what ...
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Yang-Mills massive ghosts

Is there any procedure to add a mass to Faddeev-Popov Lagrangian density of a pure Yang-Mills theory, other than just add it from nowhere?
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String coupled to Kalb-Ramond field under gauge transformation

I'm studying how a string coupled to a Kalb-Ramond 2-form $B_{\mu \nu}$ is affected by a gauge transformation of the K-R field, $\delta B_{\mu \nu} = \partial_{\mu} C_{\nu} - \partial_{\nu} C_{\mu}$ ...
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Reparametrization invariance of the particle in GR

In the article Antibracket, Antifields and Gauge-Theory Quantization, the relativistic particle in spacetime is studied. Its action is $$\int\text{d}\lambda\,\frac{1}{2}\left(\frac{1}{e}\dot{x}^2-m^2e\...
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Why does the Standard Model predict zero mass for all vector bosons?

This video from 37:33 argues that the Standard Model predicts zero mass for all vector bosons as follows: Gauge bosons must have gauge invariance. For a vector field $A$ define a transformation $\...
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Why should the electroweak Lagrangian have an $\rm SU(2)$ invariance?

The QED Lagrangian has a $\rm U(1)$ invariance so as to preserve electric charge, which has been empirically demonstrated to be conserved. The QCD Lagrangian has a $\rm SU(3)$ invariance so as to ...
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Phase term in Aharonov-Bohm effect

In $U(1)$ gauge, the transformation is given by $(c=1)$ $$e^{\frac{ie\int A_\mu dx_\mu}{\hbar}}$$ I know that this form comes from the phase picked up by electrons in Aharonov-Bohm effect. However, in ...
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Transverse traceless gauge in linearized GR

I'm reading about gravitational waves and I'm wondering how we know we can always go to the transverse and traceless gauge? Going to the de Donder gauge is fine, I can follow that, but then showing ...
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Quantization of Flux in Polyakov's 3D Compact QED

In his his book "Gauge Fields and Strings" Polyakov introduces the compact QED on a cubic lattice in 3D Euclidean space as: $$ S\left[ \left\{ A_{\mathbf{r},\mathbf{\alpha}}\right\} \right]=\...
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Deriving the non-abelian Aharonov-Bohm effect as a Berry phase

I am trying to derive the non-abelian Aharonov-Bohm effect by generalising Michael Berry's derivation to the case of non-abelian gauge field $A$. My derivation so far We require a degenerate ...
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Simplest potential to demonstrate Symmetry Breaking of $\rm SU(4)\times SU(2)_L\times SU(2)_R$ into $SU(3) \times U(1)_Q$?

In a 1974 Pati and Salam published the paper "Lepton Number as the Fourth Color", which suggested the gauge group $\rm SU(4)\times SU(2)_L\times SU(2)_R$ could be the fundamental symmetry ...
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Cauchy problem of classical Maxwell equations in Minkowski spacetime

I'm wondering a bit about the classical Maxwell equations in flat spacetime and their Cauchy problem. For the following, I suppose that the sources are already given and don't react to their own ...
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Massless spin 1 polarization

Reading Schwartz's book on Quantum Field Theory, one finds on page 120 that the forward polarization for a massless spin 1 field is $\epsilon_{f}\propto p_{\mu}$. This is equivalent to $A_{\mu}=\...
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Electromagnetic duality interacting with a complex scalar field

My question refers to example theory introduced in the book "Supergravity" from D.Z.Freedman & A. van Proeyen p.80. Its Lagrangian is given by $${\cal L}(Z,F) =-\frac{1}{4}(Im Z)F_{\mu\...
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Deriving the comparator to second order in Peskin and Schroeder

In section 15.1 of Peskin and Schroeder, expression (15.9) is given for the comparator $U(y,x)$ in an infinitesimal expansion to second order: $$U(x+\epsilon n, x)=\exp \left[-i e \epsilon n^{\mu} A_{\...
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Gauge cosmic strings and large gauge transformations

I've been going around in circles (hah) about how gauged cosmic strings work (I've been using Preskill's notes for the most part). The global string scenario makes sense to me, since different points ...
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Haag's comment on the relation between fields and particles

I am very confused by the statement made in Haag's, Local Quantum Physics: Fields, Particles, Algebras (page 46): ... the idea that to each particle there is a corresponding field and to each field a ...
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Can Maxwell's equations be generalized to all fields?

For having studied both classical and quantum optics, I regard Maxwell's equations as the grand "cheat sheet" from which (almost) all optical/photonic phenomena can be derived. Yet, I also ...
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What is the connection between vertex/spin models and gauge theory?

In the usual formulation of lattice gauge theories, one considers gauge variables on the links of a lattice (often hypercubic) taking value in some representation of a gauge group, $U_{ij} \in G$. The ...
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Horizontal Gauge Symmetry?

Some physics literature says Horizontal Symmetry of gauge theory, such as this paper, available also at arXiv. What does this Horizontal Symmetry of gauge theory or Horizontal gauge group mean? Does ...
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Anticommutator of gauge covariant derivatives

I must convert some dimension-6 operators I've obtained to the SILH base (ref: this, "Review of the SILH basis", CERN presentation by R. Contino). In this conversion I've got operators such ...
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How do generators of the Lie algebra correspond to gauge fields?

I’m tackling physics recreationally from a pure math perspective. Right now I’m looking at just the outline of gauge theory. The Wikipedia article explains that gauge fields correspond to generators ...
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Topologically massive $p$-form gauge fields in arbitrary dimensions

$\newcommand{\d}{\mathrm{d}}$In $d=2p+1$ dimensions one can have topologically massive $p$-form abelian gauge fields $A\in\Omega^p(X_{2p+1})$ by considering a Maxwell–Chern–Simons action: $$S[A] = \...
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Batalin-Vilkovisky (BV) form of the Chern-Simons Action

As seen in Section 4 of Chapter 5 of Costello, K. "Renormalization and Effective Field Theory", or in section 5.2 $L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky ...
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Local gauge transformation in Fock space of charged particles

I'm currently fiddling around with gauge-phase-transformations in Fock space. Especially, I'm trying to write a local gauge-phase-transformation as an operator in a basis-independent way. Here is what ...
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Gauge fixing and instanton calculation

I am reading Cheng&Li's book "Gauge theory of elementary particle physics". In section 16.2, I am confused by some assumptions. Suppose we have a $SU(2)$ gauge theory in $\mathbb{R}^4$ $$...
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Choice of representation $R$ in the $\rm SU(2)$ Yang-Mills action $\frac{1}{g^2} \mathrm{Tr}_{R} (F\wedge \star F)$

Usually we write the Yang-Mills theory with gauge group $G$ as $$\frac{1}{g^2} \mathrm{Tr}_{R} (F\wedge \star F)$$ But here we need to choose what $R$ is. There are several cases one may expect: $R$ ...
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Boundary condition of gauge field for finite Euclidean action

I am reading the book "Gauge theory of elementary particle physics" by Cheng & Li chapter 16 and I am confused by some statements. In Euclidean 4D spacetime we have a $SU(2)$ gauge teory ...
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Gauge transformation vs field excitation

I think I'm fundamentally misunderstanding something. Say I have a gauged Lagrangian for a complex scalar field $\phi$ with no SSB: $$\begin{equation} \mathcal{L} = (D_{\mu}\phi)(D^{\mu}\phi)^{\dagger}...
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Light composite fermion as a bound state formed by $SU(4)$ gauge force attractions

In this paper https://inspirehep.net/literature/152400, in eq.(3.4), it claims that the MAC (most attractive channel) in $SU(4)$ gauge theory will attract fermions in $[1]_4$ and fermions in $[3]_4$ ...
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Breaking/Tumbling gauge theory and composite fermions

There is some statement learnt from this paper Tumbling gauge theories by [Raby, Dimopoulos, Susskind (1979)]: Given 4d SU(5) gauge theory with fermions in the representation $$\bar 5\oplus 10$$ Add a ...
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Why Einstein action is not Yang-Mills action for gauge theory of Poincaré algebra?

It is well known, how to construct Einstein gravity as gauge theory of Poincare algebra. See for example General relativity as a gauge theory of the Poincaré algebra. There are Construction of ...
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Transformation law of a matrix of scalars

If I have a theory with a $2 \times 2$ matrix $\Phi$ of scalars that transforms under a gauge group $SU(2)_L \times SU(2)_R$ as $\Phi \rightarrow U_L \Phi U_R^{\dagger}$, how does $\Phi$ transform ...
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Do all continuous gauge theories use Lie groups?

The article about gauge theory on Wikipedia contains the sentence "Lie group". How can we prove that the gauge transformations that are given in an article form a Lie group? I give you an ...
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How can we know a gauge theory is not anomalous?

Say we have a putative 4d gauge theory coupled to fermions of various representations. In order for this theory to be consistent, we need to check that no there are both no triangle anomalies and no ...
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96 views

What was the motivation for thinking the weak interaction could be described by a Yang-Mills theory?

In some sense, describing the strong force using an $SU(3)$ Yang-Mills theory makes perfect sense: Yang-Mills theories describe massless bosons, of which the gluon is clearly a member, while the two ...
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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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$1+1D$ $U(1)$ 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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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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