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. ...

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When considering local phase transformations are we forced to use covariant derivatives?

When considering local phase transformations $e^{i\theta(x)}$ of the fields $\phi$ and $\phi^*$ corresponding to \begin{equation} \mathcal{L}=\partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi \end{...
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99 views

Modified gauge fixing condition in Faddeev-Popov approach

Which gauge fixing conditions are allowed to choose for this approach? For example (following the steps of Peskin in 9.4) I can choose a "modified" lorenz gauge ( for a abelian theory): $$ (\...
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28 views

Construction of $\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$ [closed]

First, Consider a $\phi$ as a coordinates on a copy of $Z= C^N$ Then, I know \begin{align} |\phi_1|^2 + |\phi_2|^2 + \cdots |\phi_N|^2 = r \end{align} which describe $S^{2N-1}$. Implementing $U(1)$ ...
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341 views

Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
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119 views

Gauge group topology

The fundamental difference between spinors and tensors is that spinors are sensitive to the homotopy classes of paths through the rotation group $SO(3)$: \begin{equation} \pi_1(SO(3)) = \mathbb{Z}_2, ...
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Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
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Finding the gauge transformtation of a Lagrangian [duplicate]

I am asked to find the gauge symmetry of the following Lagrangian: $L = -\frac{1}{4} F^2_{\mu \nu} + (\partial_{\mu} \phi_1 - m_1 A_{\mu})^2 + (\partial_{\mu} \phi_2 - m_2 A_{\mu})^2$ Then I have to ...
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225 views

How can I prove that the axial gauge is a valid Gauge fixing condition?

I am studying classical electrodynamics and I have been introduced to the concept of gauge transformations and gauge fixing conditions. Right know I am trying to prove that some conditions are valid ...
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61 views

Equal-time commutation relations, Feynman propagator for gauge parameter $\lambda = 1$, physical meaning

Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F_{\mu\nu}F^{\mu\nu}\right),\text{ where }F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$The ...
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145 views

Intuition behind Nekreasov's instanton partition function. What do the partitions represent exactly?

I am struggling to understand many things behind Nekrasov's solution. Firstly I want to understand the following In this theory, $a$ represents VEVs the Higgs scalar. So, is the gauge field of the ...
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93 views

Feynman propagator for arbitrary values of the gauge parameter $\zeta$

For the choice $\zeta = 1$ the Lagrangian can be brought into a particularly simple form upon integration by parts in the action integral. Equation$$\mathcal{L}' = -{1\over4}F_{\mu\nu}F^{\mu\nu} - {1\...
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73 views

Three gauge bosons vertex

I was told that two $Z$ bosons could not decay to one (virtual) $Z$ boson at any loop level. Is it true? if so, why? Does it also hold for photons? Could we generalise the statement to "There cannot ...
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1answer
95 views

What is a dual field?

Can you give me an intuitive, physical understanding of a "dual field"? For example, the Hodge dual of the gluon field strength matrix $F$ is $\tilde{F}_{\mu \nu}=\epsilon_{\mu \nu \alpha \beta} F^{\...
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120 views

Gauge covariant derivative of a creation operator

Suppose we define the (gauge) covariant derivative or as $$\tilde{\nabla}=\nabla+ie\textbf{A},$$ where the vector potential $\textbf{A}$ has a matrix structure where only the diagonal has nonzero ...
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1answer
100 views

Yang and Mills' (and others') justification for local gauge invariance

In most physics textbooks, local gauge invariance is simply postulated---you start with a global symmetry, e.g. the global phase, then allow it to depend on the spacetime point, make the necessary ...
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174 views

BRST quantization and norm

States with definite ghost number have zero norm (since ghost number is anti-hermitian and has real eigenvalues). E.G. when quantizing relativistic point particle, physical spectrum turns out to ...
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1answer
68 views

Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
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178 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \...
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1answer
58 views

Phase diagram of gauge + matter theories

I am looking for some notes/reviews on confinement and Higgs phases suitable for Fermionic/Bosonic matter coupled to Abelian ($Z_2$ or $U(1)$ etc) gauge fields. The purpose is to understand issues ...
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1answer
120 views

Charge not conserved in scalar QED? [duplicate]

Since conservation of charge seems to be a well known concept, I am hoping that I am missing something and that the conclusion is incorrect. However, I have been unable to disprove this. Let me ...
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319 views

Polarization Sums in QCD for the calculation of parton model splitting functions

Before i state the actual problem, here's a premise. In the case of a Spin 1 massive particle it's possible to demonstrate that $$\sum_{\lambda=0,\pm1}\epsilon_{\lambda}^{* \ \mu}\epsilon_{\lambda}^{\...
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84 views

Do lattice gauge theories with discrete gauge groups have sensible continuum limits?

In lattice gauge theories the only gauge invariant observables are constructed from Wilson loops and local field strength observables are reconstructed as zero size limits of Wilson loops. Furthermore ...
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160 views

Is there any $SU(\infty)$ gauge theory in quantum field theory?

The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there ...
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88 views

A question to gauge fixing in nonabelian gauge theories

In quantum gauge theories it is usual to fix the gauge with the equation $\partial^\mu A_\mu = 0$ where $A_\mu$ is the gauge connection. From this gauge fixing condition the remaining gauge degree of ...
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69 views

About equivalence of two ways of “derivation” of Standard model

Two ways of SM derivation I know two methods of SM lagrangian "derivation". The first one, which I will call as Weinberg way, is based on approaches of SM as theory with spontaneusly broken $SU(2)_{...
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36 views

Properties of vector potential

Given the definition of vector potential as, $ A_{\mu}= \sum_{k} \partial_{\mu}\theta F^{k}$ , where F are the generators and $\theta$ are the parameters of the symmetry group. I have two questions ...
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Gauge transformation and Special relativity

While explaining gauge theories, a book makes a comment that the U(1) transformation definition, $ U= e^{i q \lambda(x)}$ is analogous to a special relativity transformation in freely falling elevator....
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1answer
29 views

Vector potential in gauge transformation

While applying Gauge transformation, $\psi\prime = U \psi$ , where $ U= e^{i q \lambda(x)}$ , transformation law for "Vector Potential" comes out to be : $$ A_{\mu}\prime= UA_{\mu}U^{-1}-\dfrac{i}{q}(\...
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1answer
133 views

Dyon condensation and generalized Meissner effect

In section 2.B of Metlitski and Vishwanath's paper: "Generally condensation of a dyon with charges $(q,m)$ gives rise to an analogue of a Meissner effect for the gauge field combination $q\vec{b}-2\...
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157 views

When do gauge theories have protected gapless excitations?

Goldstone's theorem states that a system in which a continuous symmetry is spontaneously broken necessarily has gapless excitations. (A hand-waving "proof" of Goldstone's theorem can be given by ...
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92 views

What S means in S-duality?

As I know, there are many dualities related to S-duality. For example, Montonen-Olive duality, Seiberg duality. and so on. so, I wonder that what "S" means in the term "S-duality". If this is a stupid ...
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25 views

Expansion of non-abelian heat kernel operator $Q$

For abelain $U(1)$ gauge theory ($F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$) we can expand heat kernel operator $Q$ as \begin{align} Q &= -(\partial - iA)^2 + m^2 \\ & = - \...
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Are correlators constructed out of Wilson loops singular in pure Yang-Mills?

If I have some gauge invariant function of two Wilson loops (such as $\left<\text{Tr}W_1 \text{Tr}W_2\right>$) does the expectation value diverge when the loops coincide the same way $\left<\...
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64 views

$U(1)$ gauge symmetry in superfluid

The conventional superfluid phase in a Bose-Hubbard ground state has $U(1)$ symmetry. In the presence of spin-orbit coupling (SOC), the superfluid ground state has non-uniform phases. Why do people in ...
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1answer
72 views

Is local chiral symmetry qualitatively the same as gauge symmetries?

I am confused by the role that local chiral symmetry plays in chiral perturbation theory. For the case of chiral QCD with three quark flavors, the Lagrangian is invariant under global $SU(3)_L\times{}...
4
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1answer
309 views

Einstein-Yang-Mills Connections

I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations. Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = \omega(r)\...
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287 views

How to include Berry connection in Hamiltonian?

When we calculate Berry connection, $A(R)=i<\psi(x,y)|\frac{d}{dR}|\psi(x,y)>\hat{R}$ corresponding to the Berry phase of any system, the gauge potential is related to the $R$ of the parameter ...
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490 views

Non-inertial frames in Lagrangian mechanics?

Building on this Phys.SE post I am interested in how non-inertial frames can be considered in Lagrangian mechanics. My understanding is that changing the reference frame causes a transformation of the ...
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323 views

Is there some no-go theorem for $D=9$ Kaluza Klein QCD+EM?

While QCD is a typical product of AdS/CFT and some other research trends in extra dimensions, I have never found in the literature an example producing the non-chiral part of the standard model, ...
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1answer
179 views

Georgi-Glashow model and the VEV of the scalar field

Consider the Georgi-Glashow model, an $SU(2)$ gauge theory with a real scalar in the adjoint (thus a 3-vector in the colour space) $\phi$. The Lagrangian is $$ L = -\frac{1}{4g^2} F_{\mu \nu}^{\, a} ...
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61 views

Coupling an electric charge to a gauge field. How is it done in this setup?

In page 9 of Tachikawa's N=2 susy dynamics for pedestrians it says that an electric particle with charge $n$ in the first quantised setup (in what sense first quantised?), Wick rotated to Euclidean ...
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205 views

Charged CFT observables and AdS/CFT

I have a simple question regarding the holographic dictionary when mapping operators on the CFT side to those in AdS. One piece of the dictionary is that a global symmetry maps onto a gauge symmetry ...
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1answer
110 views

Is EM interpreted in a principal or vector bundle?

I've read in a few places that EM is a $U(1)$-principal bundle; but is this correct? Isn't it rather an associated vector bundle using the adjoint representation of $U(1)$?
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Massive Gauge Bosons without Higgs Effect

In a possible theory like our Standard model but without a Higgs i.e.: $$ \mathcal{L}=i\bar{\Psi}_f\gamma_\mu D^\mu\Psi_f-\text{Tr}[G^b_{\mu\nu}G^{b\,\mu\nu}] $$ where $b,f$ run over the typical ...
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181 views

Consequences of local and global anomaly

Are the physical consequences of anomalies associated with a local symmetry is different from that of a global symmetry? If yes, why? We have global anomaly in the standard model but not local anomaly?...
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1answer
97 views

Dimensional reduction of SUSY theories

I know that if one reduces 10 dimensional $\mathcal{N}=1$ SYM theory to 4 dimensions one gets $\mathcal{N}=4$ SYM. There are other examples also. I have two related questions regarding this fact. ...
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1answer
40 views

Gluon have colour-anticolour; what about weak bosons?

Gluons can be red-antiblue, or green-antired, etc. What about weak interaction bosons? (Say before symmetry breaking, to make matters simpler.) Is there a similar "weak charge" structure of charge-...
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75 views

How are quadruple gluon vertices related to $SU(2)$ and $SU(3)$?

I once read that the non-commutativity of the Lie Groups $SU(2)$ and $SU(3)$ is the reason that the weak and strong interactions have Feynman diagrams with quadruple vertices, where four gauge bosons ...
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460 views

How do gauge boson interact with elementary particles?

We know that gauge bosons are the force carriers of fundamental interactions, but how do the gauge bosons themselves interact with particles?
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89 views

Why must superpartners have the same gauge quantum numbers?

The title leaves it quite clear, why must superpartners have the same gauge quantum numbers?