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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.


5

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 $|i>_j,i=1,\dots,n_j$ be an orthnormal basis of $V_j$ where $dim V_j=n_j$. Then $\{|i>_1\otimes|j>_2\otimes|k>_3\}$ forms an orthonormal basis of $V=V_1\otimes V_2 \otimes V_3$. An element ...


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

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

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 ...


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

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

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.



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