# 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 electromagnetic field.

What happens physically is that after parallel transport along a loop, the wave function acquires a phase difference, physically invisible, but that can lead to observable interference.

Now, a "phase" in a $U(1)$ gauge theory can generalize in two different ways if we go to non-abelian gauge theories:

1) It could remain a phase, leading to no physical difference (except interference);

2) It could become a more general $G$ transformation, where $G$ is the gauge group (like $SU(2)$ or $SU(3)$), and for example change the color of a quark after a loop.

What does quantum field theory predict? What would happen if we set a strong (or weak) equivalent of the Aharonov-Bohm effect (despite the obvious experimental difficulties)?

If 1) is true: wouldn't it mean that the meaningful (curved) part of the holonomy is in fact Abelian? If 2) is true: wouldn't it violate (for quarks) conservation of color charge?

Any reference would be good, as long as it gets to the point.

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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 particle may carry both flux and charge. For given gauge group, there are fixed number of elementary particles, they are classified by so called supersecltion sectors.

Suppose a charged particle C carries a nontrivial flux which is non commuting with another flux F, when C moves around F, both C and F will be transformed by a group action. For this reason, there will be no AB effect (or no interference) in this case since we can tell which way the particle C comes to the screen by measuring its flux (different ways result in different flux). Non trivial AB effect can be oberved only when the flux carried by C commutes with F.

For reference, see Preskill's lecture notes about topological quantum computation, it is friendly for beginers (like me):

http://www.theory.caltech.edu/~preskill/ph219/topological.pdf

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 state of the changing Hamiltonian (Adiabatic theorem). By performing a loop on the space of parameters the sistem returns in its original state but for a phase given by a dinamic contribution and by a geometrical one (the holonomy). Therefore you can control the sistem by controlling some classical parameters (magnetic fields and so on). The non triviality of the holonomy then allows to implement non trivial operations.

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/quant-ph/0007110 – Qmechanic Mar 28 '14 at 11:31
• @user3376924 Wow, this is very, very interesting! Thank you! – geodude Mar 28 '14 at 11:50