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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3answers
123 views

Spontaneous symmetry breaking to subspace not giving massless bosons

I'm currently trying to understand spontaneously broken in general and have stumbled upon a weird result which doesn't seem to correspond to my knowledge about broken gauge symmetries. Suppose we ...
4
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1answer
274 views

The BRST construction for YM with or without auxiliary field

I'm learning BRST symmetry for Yang-Mills theory and I see that there are two ways of writing BRST differential. In some books (for example Ryder's and Ramond's textbooks) BRST differential acts as ...
4
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1answer
176 views

What is the constraint on the Gauge Potential in the Covariant Gauges?

One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term \begin{equation} -\frac{(\partial_\mu A^{\mu})^2}{2\xi} \end{equation} to the Lagrangian. ...
4
votes
1answer
211 views

About the gauge formalism in statistical quantum field theory

I would like to understand a bit more the aspects of the gauge theory in statistical field theory. In particular, I would like to understand how the replacement $\tau \rightarrow it/\hbar$ is ...
4
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1answer
148 views

To what extent correlation functions determines the theory (and lagranian)

In other words, does a finite set correlation functions sufficient to determine a theory? Is there a chance correlation functions are more fundamental then the lagrangian?
4
votes
1answer
190 views

Conserved topological charge for d=3 Yang-Mills. G=U(2)

Consider a pure Yang-Mills lagrangian density $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu}$$ with gauge group $U(2)$. Take the generators for $U(2)$ to be $t_0$, $t_i \ i=1,...,3$ with ...
4
votes
1answer
238 views

Local guage symmety implies causality

I read in a QFT book that local gauge symmetry implies causality. Could someone please explain that statement and why it's true? Thank you.
4
votes
1answer
425 views

Yang-Mills instanton

How can instanton solution to Yang-Mills theory with gauge group $SU(3)$ or $SU(N)$ be obtained? For $SU(2)$ it is explained in textbooks but what about more general color gauge groups? EDIT: How ...
4
votes
1answer
753 views

Noether current for the Yang-mills-higgs lagrangian

I am trying to calculate the Noether's current, more specifically, the energy density of the Yang-mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, ...
4
votes
1answer
240 views

Gauge symmetry description for $\phi^4$?

That is a follow-up to this question: Gauge symmetry is not a symmetry? Ok, gauge symmetry is not a symmetry, but ... ... a redundancy in our description, by introducing fake degrees of freedom ...
4
votes
1answer
471 views

Gauge invariance and the form of the Rarita-Schwinger action

in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field ...
4
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3answers
181 views

About constraints of the first class and electrodynamics

Consider a theory in the Hamiltonian formalism and assume that it has constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx 0$ of ...
4
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1answer
203 views

What are type system examples of local gauge transformation- and field strength-like objects?

This is essentially a follow up motivated by this answer to my question about the gauge transformation interpretation of identity types. A field $$\psi:\mathcal M\to\mathbb C^n$$ is a section of the ...
4
votes
1answer
189 views

Does the projected spin state of the $d+id$ mean-field Hamiltonian on a triangular lattice has time-reversal(TR) symmetry?

Consider the following $d+id$ mean-field Hamiltonian for a spin-1/2 model on a triangular lattice $$H=\sum_{<ij>}(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$$, with $\chi_{ij}=\begin{pmatrix} 0 & ...
4
votes
1answer
350 views

What determines the spin of fields in gauge field theories?

I understand that gauge bosons transform as the adjoint of their respective symmetry groups, but what determines the spin of the field? Can you have some gauge group where the adjoint is spin zero?
4
votes
1answer
450 views

About the gauge invariance of Chern-Simons' theory (in local coordinates)

I am aware of the differential form language proof of the fact that for arbitrary gauge transformations the Chern-Simons' term shifts by a WZW term (on the boundary). But I am getting confused if ...
4
votes
1answer
648 views

Wilson loops and gauge invariant operators (Part 2)

These questions are sort of a continuation of this previous question. I would like to know of the proof/reference to the fact that in a pure gauge theory Wilson loops are all the possible gauge ...
4
votes
2answers
448 views

Diff(M) and requirements on GR observables

This question is kind of inspired in this one: Diff(M) as a gauge group and local observables in theories with gravity The conundrum i'm trying to understand is how is derived the (quite) ...
4
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0answers
105 views

Why the field strength tensor for locomotion at low Reynolds number may be written like that?

I've been studying locomotion at low Reynolds number for some time now and it has been a quite tough problem. I've already asked two questions about the problem here, and now there is this question ...
4
votes
0answers
56 views

Reality of the action in QFT

Following Ramond, 1.5 Field Theory, it is mentioned that the classical Lagrangian density in (workable for HEP) QFT theories has to be Real, otherwise total probability is not conserved. Can someone ...
4
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0answers
49 views

What is the status of gauged gravity [duplicate]

The Standard Model of elementary particles is a gauge theory with gauge group $SU(3)\times SU(2)\times U(1)$, which is really a successful theory. We might be able to quantize gravity similarly. ...
4
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0answers
98 views

Can you gauge a $U(1)_L$ symmetry?

I recently calculating the one loop correction for the propagator of a gauge boson, $\hspace{5cm}$ I assumed arbitrary left and right couplings, $ g _L $ and $ g _R $. I found that the one loop ...
4
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0answers
166 views

Chern-Simons on a lattice and the framing anomaly

Can someone make or refer me to the argument for why $U(1)$ Chern-Simons theory in three dimensions cannot be defined by a lattice action? (Unlike Dijkgraaf-Witten theories, which are defined on the ...
4
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0answers
138 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
4
votes
0answers
75 views

Why does strong interaction increase with distance?

I read numerous times that strong interaction increases with distance. But how can one actually derive the force-distance relation from the lagrangian (quark field + gluon field + gauge coupling)? ...
4
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0answers
102 views

Unitary gauge for non-abelian case

I'm reading Chapter 19 of Mandle and Shaw's Quantum field theory. In the first section it is explained that one can go with a $SU(2)$ followed by a $U(1)$ transformation from ...
4
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0answers
51 views

Quiver and Gauge theory

i want to know how to construct a quiver of a Gauge theory specified by groupe g with rank=r ?
4
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0answers
130 views

Noether current for the Lagrangian without Lorentz invariance

I am reading an article by Watanabe & Murayama. It gives a proof on the counting of Nambu–Goldstone bosons without Lorentz invariance. I am trying to derive all the equations to get a better ...
4
votes
0answers
100 views

Is there a critical order of the Abelian gauge theory in (2+1)D

In (2+1)D spacetime, it is known that the $U(1)$ gauge theory is always confined (according to Polyakov), while the $\mathbb{Z}_2$ gauge theory can support a deconfined phase. Now consider a generic ...
3
votes
2answers
295 views

Under which representation of U(1) transform electron and photon gauge field?

I know that under $SU(2) \times SU(2)$, the left-handed electron transforms under $ ( \frac{1}{2},0 ) $ representation and the vector gauge field $A_\mu$ under $ ( \frac{1}{2},\frac{1}{2}) $. Since ...
3
votes
1answer
301 views

Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed

I am preparing myself for an exam and I got stuck with the following problem. If I wanted to calculate the vector potential $A$ on a sphere (not off or in), where some points are removed, how would I ...
3
votes
4answers
983 views

First class and second class constraints

Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second ...
3
votes
1answer
603 views

Large gauge transformations

I would like to understand what is the importance of large gauge transformations. I read that these gauge transformation cannot be deformed to the identity, but why should we care about that?
3
votes
2answers
350 views

What is conformal gauge?

I often see in physics articles on gravity such notion as conformal gauge and Weyl transformation. They use Conformal gauge to change coordinates to transform metrics from arbitrary $$ds^2=g_{\mu ...
3
votes
3answers
1k views

Counting degrees of freedom of gauge bosons

Gauge bosons are represented by $A_{\mu}$, where $\mu = 0,1,2,3$. So in general there are 4 degrees of freedom. But in reality, a photon (gauge boson) has two degrees of freedom (two polarization ...
3
votes
1answer
160 views

What is the four-dimensional representation of the $SU(2)$ generators?

Recently, I have been learning about non-Abelian gauge field theory by myself. Thanks @ACuriousMind very much, as with his help, I have made some progress. I am trying to extend the Dirac field ...
3
votes
1answer
329 views

What is a covariant derivative in gauge theory?

I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
3
votes
1answer
538 views

Noether First and Second Theorem

I have this question related to the the Noether's Theorems. I want to know a rigorous enough enunciation of this theorem, the context is Classical Field Theory without fancy geometrical structures ...
3
votes
2answers
413 views

primary constraints for constrained Hamiltonian systems

I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian ...
3
votes
1answer
65 views

What is the phase of a gauge coupling?

We typically take gauge couplings to be real and positive. Why do we impose these two conditions? I assume this is a requirement because gauge theories without positive couplings are unphysical or is ...
3
votes
2answers
328 views

Why does the $\pi$-flux state have time-reversal symmetry?

It's known that the $\pi$-flux state of the antiferromagnetic Heisenberg model on the square lattice is an important concept. The $\pi$-flux state is described by the (simplified) mean-field ...
3
votes
3answers
183 views

A question about an identity in deriving Born-Infeld action

I have a question in David Tong's Example Sheet 4 Problem 5b, how to verify the last equation (*) on p.2? (There is a solution for example sheet 3, but seems to be no solution for example sheet 4.) ...
3
votes
1answer
194 views

Four-gauge-boson vertex in non-Abelian gauge theories

In Peskin & Schroeder's book page 524, the following diagram is calculated for the gauge boson self-energy in order $g^2$: In dimensional regularization, its contribution is given by ...
3
votes
2answers
217 views

Proof of quantization of magnetic charge of monopoles using homotopy groups

Suppose we place a monopole at the origin $\{{\bf 0}\}$, and the gauge field is well-definded in region $\mathbb R^3-\{0\}$ which is homomorphic to a sphere $S^2$. Then the total manifold is $U(1)$ ...
3
votes
1answer
650 views

Lattice QCD and string theory

I've heard the claim that some aspects of string theory are used to improve Monte-Carlo simulations of lattice QCD, for example by people working at the LHC. I know a bit about Monte-Carlo methods in ...
3
votes
1answer
89 views

What are the definition and examples of topological excitation?

I read topological excitation in wiki, while it's too brief. What is the precise definition of topological excitation? And can give me some examples and explain why they are topological excitation? ...
3
votes
2answers
153 views

Covariant derivative applied to a vector vs. applied to a matrix?

I know there are (say) two different definitions/representations of the covariant derivative: one is the covariant derivative applied to a vector $F$, which reads as $$DF=\partial F+iAF$$ ...
3
votes
1answer
104 views

How to see the ground state degeneracy (GSD) from a $BF$ theory in $2+1$ $d$?

I have seen many times the $BF$ theory has non-trivial ground state degeneracy (typically on torus), but I can not see how the conclusion come out. Recently I found a paper by Hansson, Oganesyan and ...
3
votes
2answers
127 views

Utility of gauge four-potential $A_{\mu}$ (as opposed to electric and magnetic fields ${\bf E}$ and ${\bf B}$) in E&M?

The action for an electromagnetic field with source charges is given by $$S= \int \left\{ \frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu} - J^\mu A_\mu \right\}dx$$ By setting $dS=0$ and taking the Lorenz ...
3
votes
2answers
1k views

Polarization vectors of massive and massless particles

I read from Mandl & Shaw that when quantizing massless vector particles such as photons in Lorentz gauge, there are 4 linearly independent polarization vectors (2 of them being able to "gauged ...