Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

All Questions

Tagged with
3
votes
1answer
82 views

What's the meaning of “inequivalent quantizations”?

The notion "inequivalent quantizations" is regularly used when topological terms are discussed. From what I've gathered so far, "inequivalent quantizations" means that there are different quantum ...
1
vote
0answers
27 views

What happens to large gauge transformations in gauges different from the temporal gauge?

There are already several questions regarding the meaning and definition of large gauge transformations. Discussions of large gauge transformations typically only happen in the context of the ...
6
votes
2answers
78 views

Particle on a circle with magnetic flux$.$

I am trying to understand the model studied in 1905.09315 §2, to wit, a $0+1$ dimensional theory with target space $\mathbb S^1$ with non-trivial magnetic flux: $$ \mathcal L=\frac12m\dot q^2-\frac{i}{...
1
vote
1answer
101 views

Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
4
votes
1answer
166 views

Do higher homotopy groups play any role in gauge theory?

As is more-or-less well-known, the magnetic monopoles of a gauge theory are classified by the first homotopy group of the gauge group, $\pi_1(G)$ (cf. Lubkin (1963)). The second homotopy group is ...
5
votes
1answer
95 views

What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
1
vote
0answers
100 views

Topological Charge in 2 dimensional U(1) Gauge Theory

The topological charge in 2 dimensions for $U(1)$ gauge fields is defined by $Q \propto \int d^2x ~\epsilon_{\mu\nu}F_{\mu\nu}$ with the field strength tensor $F_{\mu\nu}=\partial_\mu A_\nu - \...
1
vote
1answer
154 views

Why is the Standard Model gauge group a simple product?

The Standard Model gauge group is always given as $\text{SU}(3)\times\text{SU}(2)\times\text{U}(1)$ (possibly quotiented by some subgroup that acts trivially on all konwn fields). However, at the ...
5
votes
1answer
244 views

What's the difference between a gauge theory with group $G$ and one with its universal cover?

Consider a gauge theory with gauge group $G$, which is not simply connected. What is the difference between this theory, and one with gauge group $\tilde G$, the universal cover of $G$? Sharing the ...
7
votes
1answer
261 views

When is a Wess-Zumino term well-defined?

According to wikipedia, a Wess-Zumino term is well-defined when the Lie group (target space) $G$ is compact and simply connected, because that implies that $\pi_2(G)$ is trivial. But there are Lie ...
10
votes
1answer
366 views

How is the Chern-Simons action well-defined?

The Chern-Simons action $$ S = \int_M A \wedge \mathrm{d} A + \frac{2}{3}A \wedge A \wedge A $$ is not obviously gauge invariant. It is usually stated that under a gauge transformation, the action ...
4
votes
1answer
386 views

Axial anomaly in QCD VS axial anomaly in current algebra QCD

I would like to understand the distinction between an axial anomaly in QCD (Theta Vacuum: axion -> 2 gluons) and an axial anomaly in QCD of current (Chern–Simons term: pion->two photons, photon->three ...
1
vote
0answers
235 views

Trouble with Construction of Sphaleron S

Consider SU(2) YMH theory without fermions. Three-space is compactified by adding the sphere at infinity and configuration space is the space of all static finite-energy 3D gauge and Higgs fields in a ...
1
vote
0answers
67 views

Classical topological vacua

Suppose we have two vacuum potential $A_0$ and $A_1$ with winding number $n=0$ and $n=1$ respectively and related to each other by a large gauge transformation $U$ that is, $$A_1\mapsto U A_0 U^{-1} ...
1
vote
1answer
201 views

Aharonov-Bohm effect: a particle on a ring vs. a particle confined to the x-axis with periodic boundary conditions

Is there a fundamental difference between the Aharonov-Bohm effect in the two cases below? (a) A particle on a ring (polar coordinates) with a magnetic flux through the ring. (b) A particle confined ...
6
votes
1answer
168 views

Peculiarities of non-Abelian gauge groups: self-coupling and topology

There are two striking aspects of non-Abelian gauge groups (compared to their Abelian cousins): (1) The pure gauge parts of non-Abelian Lagrangians contain self-interaction terms that are trilinear ...
3
votes
5answers
1k views

Aharonov-Bohm effect and its topological connection

The topological explanation of Bohm-Aharonov effect assumes that the presence of a solenoid makes the configuration space non-simply connected. Now assume that the magnetic field inside the solenoid ...
1
vote
0answers
60 views

Why does Monopole in $D=3+1$ not drive a compact $U(1)$ gauge theory into a Confined Phase?

In compact $U(1)$ gauge theory, it has confined/deconfined phase. For $D=2+1$, monopole (2$\pi$-flux) event can always drive a system into a confined phase, as shown by duality mapping to a Sine-...
2
votes
1answer
73 views

Gauge potential over $\mathbb{S}^4$ vs. $\mathbb{R}^4$

NICHOLAS MANTON in "Topological Solitons" says "One may also regard the gauge potential as a connection on an $SU(2)$ bundle over $\mathbb{S}^4$, with field strength $F$. The fact that we can ...
0
votes
0answers
116 views

What is the topological explanation of the gauge covariant derivative?

Non-topologically speaking, a gauge covariant derivative is different to a regular derivative as it introduces a correction term that maintains invariance of the action of a Lagrangian under a gauge ...
8
votes
0answers
292 views

Is the QCD Lagrangian without a $\theta$-term invariant under large gauge transformations?

In his book "Quantum field theory", Kerson Huang states that we need to add the term $$\frac{i\theta}{32\pi^2}G_{\mu\nu}^a \tilde{G}_{\mu\nu}^a$$ to the Lagrangian, to make it invariant under large ...
12
votes
1answer
1k views

What defines a large gauge transformation, really?

Usually, one defines large gauge transformations as those elements of $SU(2)$ that can't be smoothly transformed to the identity transformation. The group $SU(2)$ is simply connected and thus I'm ...
2
votes
1answer
107 views

Connection between homotopic maps from $X\to Y$ and homotopic paths in $Y$ in the context of SU(2) Yang-Mills instantons

EDIT: I was reading little bit of homotopy theory in trying to understand the difference between homotopic maps from $X\to Y$ and homotopic paths in $Y$, and their significance in the context of SU(2) ...
5
votes
2answers
484 views

“Large” gauge transformation doesn't act as do-nothing transformation in QFT: looking for classical analog

The gauge symmetry in classical pure Yang-Mills theory with a gauge field $A_{\mu}$ requires an action $S$ to be invariant under continuous transformations $$ A_{\mu}(g) \to g(A_{\mu} + i\partial_{\mu}...
15
votes
1answer
689 views

Is there an argument for using the $\theta$-vacuum for a Yang-Mills theory that works regardless of the presence of fermions?

Consider a Yang-Mills theory, possibly including fermions. It has many possible vacua $\{|n\rangle\}$ labelled by integer winding number $n$, defined as the Maurer-Cartan topological invariant: for ...
0
votes
1answer
49 views

Rotating away a constant gauge field

In a few papers (see, for example, here, the bottom of the left column on the page 6, or here, the upper part of the page 5) I've met the strange calculations using the constant gauge field $$ A_{\mu}(...
0
votes
1answer
155 views

Vacuum Manifold of an $SU(2)$ Theory

I am reading Coleman's book "Aspects of Symmetry", specifically chapter 6 "Classical Lumps and their Quantum Descendants". He gives an Example 5 p. 209 for the topological solutions for an $SU(2)$ ...
8
votes
2answers
661 views

Yang-Mills potential and principal bundles

In section 2.7.2 of Bertlmann's "Anomalies in quantum field theory", it is stated that since a non-trivial principal bundle (based on a Lie group $G$) does not admit a global section, the Yang-Mills ...
2
votes
1answer
118 views

Left-right topology

Are there non-trivial topological solutions (in particular 't Hooft-Polyakov magnetic monopoles) associated with the (local) breaking \begin{equation} SU(2)_R \times SU(2)_L \times U(1)_{B-L} \to SU(...
1
vote
1answer
101 views

Some questions about gauge theory

Let's talk quantum mechanics. I have a charged particle moving on a sphere. It has a wave function $\psi$. At time $t=0$, there is no magnetic flux piercing the sphere. Instantaneously, I introduce a ...
2
votes
0answers
112 views

Why Dirac monopole is a topological defect in a $U(1)$ gauge theory? [duplicate]

How does $U(1)$ gauge group at long distances, give rise to magnetic monopoles? Also why is it said that Dirac monopole is a topological defect in a compact $U(1)$ gauge theory?
0
votes
1answer
217 views

Integrals of Chern class, $c_i$ in YM theories

I am a bit confused with the definition of the 1st (and 2nd by extension) Chern class in YM theories. I understand that in general $c_i \in H^{2i}(M,\mathbb{Z})$ where $M$ is a smooth manifold. Then, ...
11
votes
1answer
302 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, ...
23
votes
2answers
1k views

Does the existence of instantons imply non-trivial cohomology of spacetime?

Gauge theories are considered to live on $G$-principal bundles $P$ over the spacetime $\Sigma$. For convenience, the usual text often either compactify $\Sigma$ or assume it is already compact. An ...
5
votes
1answer
454 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? ...
20
votes
1answer
2k views

What is the conclusion from Aharonov-Bohm Effect?

What is the conclusion that we can draw from the Aharonov-Bohm effect? Does it simply suggest that the vector potential has measurable effects? Does it mean that it is a real observable in quantum ...
3
votes
1answer
357 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 ...
2
votes
0answers
195 views

Some questions on the Wilson loop in the projective construction?

Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ and $H'=\sum(\psi_i^\dagger\chi_{ij}'\psi_j+H.c.)$, we ...
8
votes
2answers
1k views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some $(k-1)$-...
5
votes
1answer
412 views

Aharonov-Bohm Effect in Torus

I had a very brief introduction to the Aharonov-Bohm effect in class. The lecturer introduced the notion that $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ gives identical energy spectrum and that the Hamiltonians ...
3
votes
2answers
403 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 $U(1)$ ...
14
votes
1answer
753 views

7 sphere, is there any physical interpretation of exotic spheres?

Basically an exotic sphere is topologically a sphere, but doesn't look like a one. Or more accurately: homeomorphic but not diffeomorphic to the standard Euclidean n-sphere The first exotic ...
6
votes
1answer
1k 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
507 views

What is the winding number of a magnetic monopole, and why is it conserved?

I had asked a similar question about a calculation involving the winding number here. But i haven't got a satisfactory response. So, I am rephrasing this question in a slightly different manner. What ...
9
votes
2answers
514 views

Chern-Simons Theory in 3D

For the CS theory on a three manifold $M$ with a gauge group $G$, it is said that the gauge field $A$ is a connection on the trivial bundle over $M$. Why the bundle should be trivial? I know that ...