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9

The original papers by Gerard 't Hooft himself are quite readable. On the Phase Transition Towards Permanent Quark Confinement A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories Whenever I open these papers, I'm always awestruck.


6

Consider the finite dimensional unitary representations $\alpha,\beta,\gamma$ of the given compact group $G$ on corresponding vector spaces $V_1,V_2,V_3$. Let $|i\rangle_j,i=1,\dots,n_j$ be an orthnormal basis of $V_j$ where $dim V_j=n_j$. Then $\{|i\rangle_1\otimes|j\rangle_2\otimes|k\rangle_3\}$ forms an orthonormal basis of $V=V_1\otimes V_2 \otimes ...


5

This is the definition of the gauge field. Suppose you have an SU(2) symmetry, for definiteness, consider isospin. So the notion of "proton" and "neutron" define two axes in isospin space, and you might want to say that it is arbitrary which two linear combinations of proton and neutron are the right basis vectors. So that someone defines one basis of ...


4

For SU(N), the adjoint representation can be obtained from a tensor product of the fundamental representation with its dual and projecting out the scalar. Thus we can replace the adjoint representation indices $a, ...$ by double indices $i \bar{j}$ and write this relation as: $$W^{i \bar{j}}_{k \bar{l}} = U^i_k U^{\dagger j}_l - \frac{1}{N} \delta^i_k ...


3

Non trivial holonomies have been proposed for quantum computation, see this article http://arxiv.org/pdf/quant-ph/0007110v2.pdf The basic idea is this: Suppose you have a sistem prepared in the ground state of an Hamiltonian $H(\lambda)$, where $\lambda$ is a set of parameters. If you slowly change this parameters the state evolves remaining in the ground ...


3

There exists an extensive literature for discretization of the abelian and the non-abelian gauge theories, known as lattice QED and lattice QCD, respectively. Here we will only sketch the main idea. Let us for simplicity use Euclidean signature $(+,+,+,+)$. A small Wilson-loop $$\tag{1} W~=~{\rm Tr}{\cal P}e^{ig\int_{\gamma}A}$$ lies approximately in a ...


3

It is simple to describe mathematically. First I will recall what the equation for the Aharonov-Bohm phase means, and then I will describe (without proof) its relationship to parallel transport for $G$-bundles, which I define. The gauge potential $A$ is a connection on some principal $G$-bundle, where $G$ is the gauge group. Principal $G$-bundles over a ...


3

If you put a non-Abelian anyon and its anti particle on a sphere, then moving the non-Abelian anyon around its anti particle only induces an Abelian phase. Also, twisting a non-Abelian anyon by 360$^\circ$ only induces an Abelian phase as well, which define the (fractional) spin of the non-Abelian anyon.


2

The Aharonov-Bohm effect for nonAbelian gauge fields is subtle, even the definition of flux and charge is more complicated than Abelian cases. Both charge and flux can be nonAbelian. A flux is defined as a conjugation class of the gauge group G, and a charge is a (irreducible) representation of (subgroup of) the gauge group. However, (in 2D) a general ...


2

I'm going to wave my hands wildly here, but I think the point is that you are treating the particle semi-classically, and the amplitude for going from point a to point b is equal to the 1-particle Feynman path integral, which is dominated by the classical action. The kinetic part contributes a simple phase, while the minimal coupling with the field $A$ ...


2

1) Let us write the Wilson-line of a simple open curve $\gamma: [s_i,s_f]\to \mathbb{R}^4$ as $$\tag{1} U(s_f,s_i) ~=~ \mathcal{P}\exp \left[ i\int_{\gamma} A_{\mu}~ dx^{\mu} \right]. $$ 2) The path-ordering $\mathcal{P}$ becomes important if the gauge potential $$\tag{2}A_{\mu}~=~A^a_{\mu} T_a$$ is non-abelian. Here $T_a$ are the generators of the ...


2

Possible answer to your first bullet: Stokes theorem. If $G=U(1)$, then $F=dA$ (not $dA+[A,A]$) and so $\int_{\partial D}A=\int_{D}dA=\int_D F.$ Further, if memory serves, $U(1)$ is the only Abelian Lie algebra (thinking of the Cartan classification) so $G$ is Abelian means $G=U(1)$ anyway. EDIT: Coordinate expression for the Holonomy. $Hol(A,C)$ is defined ...


1

If the comparator was not unitary, you could not expand it in terms of Hermitean generators of $SU(2)$, which is required in order to construct the non-Abelian covariant derivative, as it is done in Peskin and Schroeder.


1

Alright, let's go on a thrilling tour through the theory of representations. Notation in the following is the same as in the OP, except that we call the trivial representation $\boldsymbol{1}$, as is canon. We start from my expression in the question $$ I := \int \alpha(g)^i_{i'}\beta(g)^j_{j'}\gamma(g)^k_{k'} = \sum_\rho \sum_{\mu = ...



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