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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76 views

Getting Slavnov-Taylor identity

Let's have generating functional in path integral form for gauge $SU(n)$ theory with interaction: $$ \tag 1 Z[J] = \int DB D\bar{\Psi}D\Psi D\bar{c}Dc e^{iS}. $$ Here $$ S = S_{YM}(B, \partial B) + ...
2
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0answers
35 views

Gauge formalism in rigid body mechanics

When doing calculations in rigid body mechanics, it is necessary to choose an origin to calculate torques and angular momenta. However, the underlying dynamics does not depend upon the choice of that ...
4
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1answer
109 views

Is gauge connection unique?

In QFT, given a gauge group and matter field, is the form of the gauge field unique? In other words, given a principal G-bundle and its associated vector bundle, is the construction of the principle ...
4
votes
1answer
109 views

Is there a method which quantizes non-abelian gauge theories without path integrals formalism?

In the most QFT books there is a method of quantization of non-abelian theories through path integral methods. But I want to learn also the other methods without using of this formalism. Does anyone ...
7
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1answer
237 views

Gauge Field Tensor from Wilson Loop

It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The ...
5
votes
2answers
158 views

Questions about the degree of freedom in General Relatity

I'm confused about the number of degrees of freedom in General Relatity. There are two ways to count it. However, they are contradictory. For simplicity, we consider vacuum solution. First, ...
3
votes
2answers
99 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 ...
7
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2answers
402 views

Faddeev-Popov Ghosts

When quantizing Yang-Mills theory, we introduce the ghosts as a way to gauge-fix the path integral and make sure that we "count" only one contribution from each gauge-orbit of the gauge field ...
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0answers
44 views

U(1) local gauge transformation for Dirac spinor field

How can we define U(1) local gauge transformation for Dirac spinor field?, like scalar fields?
3
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2answers
104 views

Showing that Coulomb and Lorenz Gauges are indeed valid Gauge Transformations?

I'm working my way through Griffith's Introduction to Electrodynamics. In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric ...
3
votes
1answer
47 views

How do we know what type of gauge field to add to a theory?

I've been watching Leonard Susskind's particle physics lectures and in one lecture, he discusses a very simple gauge theory. We have a complex scalar field $\phi(x)$ with Lagrangian $$\mathscr{L} = ...
5
votes
1answer
92 views

Why do we need to prove the gauge invariance of QED (or all of the gauge theories) on the Feynman diagrams language?

Let's have the QED lagrangian. It has explicit gauge invariance, so, by the naive thinking, all of the EM processes must satisfy the property of gauge invariance. So why do we need to recheck of gauge ...
4
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2answers
114 views

Why are the “coupling constants” constant?

The coupling constants (in the gauge theory) fix an inner product on the lie algebra of the gauge group and we use it to define strength of the fields. we are using ad-invariant inner products which ...
1
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0answers
42 views

compact and non-compact gauge theory [closed]

From the answer to this question, I see that the gauge theory has to be compact if the charges need to be quantized. I am not sure in what sense these are necessary and sufficient. So here is a ...
2
votes
1answer
128 views

What exactly is a gauge anomaly?

In lots of papers I read about gauge anomalies. For example, avoiding gauge anamolies in the MSSM is the reason for introducing an extra Higgs doublet. Gauge anamolies in the Standard Model are ...
4
votes
3answers
258 views

What is the basis of gauge theory?

I’m learning about gauge concepts. I’ve always had the idea that by looking at a phenomenon from different viewpoints, that symmetries could be derived – in fact, that was what an equal sign ...
1
vote
1answer
82 views

Has a metric formulation of electromagnetism ever been attempted? [duplicate]

I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such ...
2
votes
3answers
104 views

About constraints of the first class and electrodynamics

Let's have some theory in hamilton formalism and let's assume that it has the constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx ...
3
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83 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 ...
5
votes
3answers
201 views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the ...
3
votes
1answer
86 views

Yang-Mills Lagrangian invariant under BRST

In equation 16.47 in Peskin & Schroeder, it is claimed that $$ -\frac{1}{2}g^2f^{abc}f^{cde}\left(A_{\mu}\,^{b}c^{d}c^{e}+A_{\mu}\,^{d}c^{e}c^{b}+A_{\mu}\,^{e}c^{b}c^{d}\right) ~=~ 0 \tag{16.47}$$ ...
6
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1answer
264 views

History of the names “Feynman-gauge” & “Landau-gauge”. How arised & how settled?

Warning: Students, stay away from antiquities. The aim to learn is to survive. Hi. Today the nomenclatures Feynman gauge and Landau gauge seem established, but could you explain the history? It's ...
6
votes
1answer
113 views

The 6-j symbol and intersecting Wilson loops, redux

This is a quite specific question continuing the problems I have with computing the expectation value of intersecting Wilson loops I laid out here. Using the tools from the answer there, I quite ...
6
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0answers
90 views

Faddeev Popov Gauge Fixing in Electromagnetism

Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. ...
8
votes
2answers
159 views

Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics

How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics. I need an clear argument that involves the ...
7
votes
1answer
199 views

Intersecting Wilson loops in 2D Yang-Mills

I am currently trying to understand 2D Yang-Mills theory, and I cannot seem to find an explanation for calculation of the expectation value of intersecting Wilson loops. In his On Quantum Gauge ...
8
votes
2answers
159 views

Why gauge $SU(N)$ and not $SO(N)$?

When building models people typically gauge $SU(N)$ but rarely try to gauge $SO(N)$ (the only example I know about is $SO(10)$, but even that isn't quite $SO(10)$ but actually its double cover). At ...
6
votes
1answer
57 views

How many coupling constants if my gauge group has many factors?

I am reading a review article where $U(1)\times{}SU(2)\times{}SU(3)$ gauge transformations are considered. It says that when such a gauge transformation is done the gauge fields $A^{\alpha}_{\mu}$ ...
4
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0answers
67 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)? ...
1
vote
1answer
111 views

A formula in Sung-Sik Lee's paper

I want to ask if anyone has gone through the derivation of the second equality in the following formula which comes from http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.165102.
2
votes
2answers
75 views

Gauge symmetry for p-forms

It is well known that the Lorentz invariance of the S-matrix implies Gauge redundancy for 1-forms,'photons'. Does this argument go through to p-forms? That is does lorentz invariance of s-matrix of ...
5
votes
3answers
334 views

Is there any relationship between gauge field and spin connection?

For a spinor on curved spacetime, $D_\mu$ is the covariant derivative for fermionic fields is $$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$$ where $\omega_\mu^{ab}$ are the spin ...
2
votes
1answer
188 views

Equations of motion for the Yang-Mills $SU(2)$ theory

I have an exercise for Yang-Mills theory. I can't find answer anywhere. Derive equations of motion for the Yang-Mills theory with the gauge group $SU(2)$ interacting with $SU(2)$ doublet of scalar ...
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0answers
17 views

Lattice Gauge and Spin Network

I see the similarity between the Lattice Gauge and Spin Network . (For example , the both theories depict the node part as quantum (the latter is explained as spin)) Are there any other ...
5
votes
1answer
127 views

Equivalency of Gauge Conditions

How is the Lorenz gauge condition $\partial_\mu \overline{h}^{\mu \nu}=0$ equivalent to the harmonic gauge condition $\Box x^\mu=0 $?
2
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50 views

Transformation Law for Covariant Derivative in $SU(2)$ Yang-Mills

In page 488 of Peskin and Schroeder, it is stated (emphasis mine): It is not difficult to check using (15.27) and (15.21) that, even for finite transformations, the covariant derivative has the ...
5
votes
1answer
117 views

Why is the Yang-Mills Comparator unitary?

In chapter 15.2 of Peskin, the comparator is defined, as some object $U\left(y,\,x\right)$ which transforms as: $$ U\left(y,\,x\right) \mapsto V\left(y\right) U\left(y,\,x\right) ...
1
vote
0answers
57 views

path integral quantization of EM field derived from canonical quantization?

In Peskin's QFT book page 294, he formally addressed the quantization of EM field, $$propagotor_{EM}=\frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$ Now that we have the functional integral quantization ...
0
votes
0answers
24 views

Gauge Higgs Unification [duplicate]

http://arxiv.org/abs/1003.6023 Gauge Higgs unification . Does gauge higgs unification has been proved ? If so , what is it exactly meaning ?
5
votes
1answer
93 views

Does the LSZ reduction method prove gauge-independence in massless gauge theories?

I've been working my way through L. Baulieu's excellent paper [Perturbative gauge theories, Physics Reports, Volume 129, Issue 1, December 1985, Pages 1-74]. Towards the end, he goes on to prove that ...
2
votes
1answer
100 views

What is the Physical Significance of Tr(A) w.r.t. Matrix Representations in Group Theory

I've seen the post on mathoverflow.SE asking almost the same question, and I have indeed flipped through said answers, but most are in a more general context ie quantum mechanics and do not provide a ...
0
votes
0answers
105 views

Loop Quantum Gravity and Gauge theory

Are there any connection between Loop Quantum Gravity and Gauge theory? If so, how does the gauge theory is described? When exactly does the spin network of the foam is created in the era around the ...
2
votes
1answer
52 views

Qustion about the appearance of $\Delta_{FP}[A_{\mu}]$ in the path integral of gauge field

Why is the Faddeev-Popov quantization of a $U(1)$ gauge field not the naive solution $$\int {\cal D}A \, \, \delta\left[F(A_\mu) \right]\exp \left\{ -\frac{i}{4}\int \mathrm{d}^4 x \, ...
5
votes
1answer
158 views

Will all physical quantities unchanged by this transformation?

I am reading an article about Bloch-Floquet state. My questions is in Part II.B and Appendix A of this paper, I will describe them below. The original Schordinger equation we consider is: ...
9
votes
1answer
184 views

A graphical proof that the $SU(2)/\mathbb{Z}_2$ vortex is non-orientable

The text, see [1], compares the vortex solutions of a spontaneously broken symmetry $U(1) \rightarrow 1$ and $SU(2)\rightarrow U(1) \rightarrow \mathbb{Z}_2$. The vortices can be classified by ...
2
votes
1answer
119 views

$SU(2)$ gauge symmetry

Take the Lagrangian with one fermion: $$ \mathcal{L} = -\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi$$ where the gauge covariant derivative $D_\mu = ...
4
votes
1answer
93 views

$U(1){\times}U(1)$ local gauge invariance derivative

In QED and the basic Higgs mechanism, there is a local gauge transformation where a scalar field $\phi$ is transformed as: $e^{i\theta\eta(x)} \phi$ The partial derivative of this however makes the ...
0
votes
1answer
83 views

Electric Magnetic potential and Lorentz transform [closed]

I have heard that the scalar potential and the magnetic vector potential in the electromagnetic four potential become the four vector by the Lorentz transform. Thereafter, the Lorentz transform leads ...
2
votes
1answer
44 views

A question about propagator of Maxwell field in different gauge

The propagator of Maxwell theory is different, depending on the gauge fixing procedure used. Then why will the S-matrix elements be the same for the same process in different gauges?
0
votes
0answers
29 views

dual variables for lattice fermions

I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I ...