Quantum Graphs : Physics of Quantum Singularity (5-2)
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Duality and Aholonomy 2
There is an occurence of this type of duality in high energy physics, in which fermionic and bosonic field theory are proven to be mathematically identical when coupling strengths are inverted. This could be both useful in technical sense and also in fundamental sense. Strong coupling system is hard to solve in general, and existence of a mapping to a weak coupling theory could considerably fascilitate the calculation. Theory with fermion-boson duality is in high demand, since it gives a hope of understanding one of the fundamental symmetry of the universe. In fact, the memory of the sensation caused by the renormalizable Seiberg-Witten model with fermion-boson duality should be still fresh.
For us, however, the most interesting aspect of our example is the fact that this formerly exotic fermion-boson duality in it perfect form is observed in a naive looking low energy quantum mechanics in the simplest of settings.
The first well-known example of the fermion-boson duality is the celebrated Coleman's proof of equivalence between bosonic \phi^4 field theory and fermionic massive Schwinger filed thery. The model discussed here may be regarded as its reduced version in the quantum mechanical setting.
In low dimensional manybody electron theory, there has been a well known concept of Tomonaga-Luttinger liquid in which strong coupling limit fermionic system is known to display all the characteristics of free bosonic system. Our model with point interaction could be viewed as a limit model in which exact bosonization is achieved for all coupling strength.
The duality we have discussed is a profound but a rather arcane exotica whose immediate impact is not easy to determin. We shall now look intoanother type of quantum phenomenon found in our model whose exotic feature is visually apparent.
It is the phenomenon called "aholonomy". It referres to the non-reccurence of original state after the cyclic variation of system parameter. Since the system parameter determines the state almost uniquely in quantum mechanics, it necesarily has to come about in a very subtle fashion.
The first discovered among this type of phenomena is of course the Berry phase, which we have already covered. There, the wave function comes back to its original shape with an added phase. A "non-abelian" extension to phenomena has been found later for the systems with degenerate eigenststes, for which cyclic motion of parameters bring the original states back as a set, as it should, but one state can turn into the other (or some mixture of two) within the degeneracy.
But thinking again, we notice that at each fixed value of the parameters, we have a series of quantum eigenstates at various energy, and there is no a priori reason that one particular state cannot be turned into another state after the cyclic variation of the parameter, given that the entire series of states are obtained back as a set after such variation.
This is what we shall see soon.
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