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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 $V=V_1\otimes V_2 \otimes ... 4 For SU(N), the adjoint representation can be obtained from a tensor product of the fundamental representation with its dual and projecting out the scalar. Thus we can replace the adjoint representation indices$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$$is non-abelian. Here T_a are the generators of the ... 2 Possible answer to your first bullet: Stokes theorem. If G=U(1), then F=dA (not dA+[A,A]) and so \int_{\partial D}A=\int_{D}dA=\int_D F. Further, if memory serves, U(1) is the only Abelian Lie algebra (thinking of the Cartan classification) so G is Abelian means G=U(1) anyway. EDIT: Coordinate expression for the Holonomy. Hol(A,C) is defined ... 2 Perhaps the resources you have seen are confusing "force" with "energy", which is a common misunderstanding that frequently leads to mistakes in terminology. Energy must be applied to overcome the color force and increase the distance between quarks. The formation of a new quark/anti-quark pair when the gluon field becomes too stretched may provide an ... 1 I just found that the solution of SU(2) matrices is really simple. When there is no hopping term, the projected spin state of the above d+id mean-field Hamiltonian indeed has TR symmetry. Because there exist global SU(2) matrices G_i which implement the TR transformation, say G_i=i\tau ^x. 1 If the comparator was not unitary, you could not expand it in terms of Hermitean generators of SU(2), which is required in order to construct the non-Abelian covariant derivative, as it is done in Peskin and Schroeder. 1 Alright, let's go on a thrilling tour through the theory of representations. Notation in the following is the same as in the OP, except that we call the trivial representation \boldsymbol{1}, as is canon. We start from my expression in the question$$ I := \int \alpha(g)^i_{i'}\beta(g)^j_{j'}\gamma(g)^k_{k'} = \sum_\rho \sum_{\mu = ...