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7
votes
1answer
244 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 ...
4
votes
1answer
89 views

Expressing an adjoint representation Wilson line in terms of the fundamental representation

I'm working out some calculations with Wilson lines, defined as path-ordered exponential integrals of a gauge field: $$U = \mathcal{P}\exp\biggl(ig\int_{-\infty}^{\infty}\mathrm{d}x^\mu T^c ...
6
votes
1answer
117 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 ...
7
votes
1answer
215 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 ...
5
votes
1answer
120 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) ...
5
votes
1answer
113 views

Question about equation 2.27 from Pachos's Introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
3
votes
0answers
142 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 & ...
3
votes
0answers
79 views

Homeomorphism between the space of all Ashtekar connections and spacetime?

Excerpt from an essay of mine: Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, where $\mathcal{M}$ is spacetime. Then, let ...
3
votes
1answer
54 views

an Abelian complex statistical phase from exchanging non-Abelian anyons?

We have some discussions in Phys.SE. about the braiding statistics of anyons from a Non-Abelian Chern-Simon theory, or non-Abelian anyons in general. May I ask: under what (physical or mathematical) ...
1
vote
1answer
42 views

High-energy scattering matrix a become wilson line

I would like to ask why the scattering matrix $S^{ab}(x)=\left\langle gluon\: b|gluon\: a\right\rangle$ can be approximated for fast particle interacting with a traget field $A^{-}$ by wilson line ...
1
vote
0answers
44 views

Some question on the definition of flux in the projective construction?

Here I have some confusing points about the definition of flux in the projective construction. For example, consider the same mean-field Hamiltonian in my previous question, and assume the $2\times 2$ ...
2
votes
0answers
82 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 ...
2
votes
0answers
115 views

Is the $SU(2)$ flux defined in the context of Projective Symmetry Group(PSG) an observable quantity?

The $SU(2)$ flux defined in the context of PSG is as follows: Consider the mean-field Hamiltonian $H_{MF}=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ description of a 2D lattice spin-model, the ...
7
votes
2answers
218 views

Physical consequences of non-abelian non-trivial holonomy

The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Significance) can be well described and explained in terms of holonomy of the $U(1)$ connection of the ...
6
votes
2answers
601 views

Intuition for gauge parallel transport (Wilson loops)

I'm looking for a geometrical interpretation of the statement that "Wilson loop is a gauge parallel transport". I have seen QFT notes describe U(x,y) as "transporting the gauge transformation", and ...
6
votes
1answer
192 views

Functional Derivation of Holonomy

I would like to know how to take the functional derivative of the holonomy, or Wilson line. I have tried it and I will show what I have done below, but before I wanted to say that I also have seen ...
7
votes
2answers
838 views

Some questions about Wilson loops

Let $G$ be the gauge group whose Yang-Mill's theory one is looking at and $A$ be its connection and $C$ be a loop in the space-time and $R$ be a finite-dimensional representation of the gauge group ...
7
votes
2answers
941 views

Wilson/Polyakov loops in Weinberg's QFT books

I wanted to know if the discussion on Wilson loops and Polyakov loops (and their relationship to confinement and asymptotic freedom) is present in the three volumes of Weinberg's QFT books but in some ...