(vii) Boundaries and impurities

One may also add boundaries or defects to the models so far described in infinite space and exploit the integrability (factorizability) in a similar fashion as explained in the previous sections. How to treat boundaries in statistical mechanics, that is how to extend the Yang-Baxter equations, is known since about twenty years [36 ]. Ten years ago we undertook the first attempt to extend these ideas to QFT [66 ] by formulating the bootstrap principle for systems with reflecting boundaries. We solved the proposed equations for a particular class of ATFTs [ 67 ] and also allowed the boundary to have degrees of freedom [ 68 ]. Since then an entire industry has developed working according to the principle: to each bulk theory one can add a boundary and solve it. Of course it is interesting to have more and more solutions and essentially everything said in (vi) carries over to this situation. Concerning the general direction of this activities, there are motivations from string theory (an open string has naturally a boundary), but I am not aware of concrete investigations and results in this direction. With regard to applications in condensed matter physics the situation is different, boundaries are interesting (quantum wires are finite see (viii)), but possible more relevant are defects and impurities. One can then treat impurities according to the same factorization principles of IQFT and introduce besides a reflection also a transmission amplitude. It is clear that the most interesting situation will be one in which both types of amplitudes are simultaneously non-vanishing. We showed, however, [33 ] that even in the most general set up with degrees of freedom in the boundaries this is only possible for theories whose bulk theory is free, possibly obeying some exotic statistics. In [ 25 ,26 ] we constructed various solutions for simultaneously non-vanishing reflection and transmission amplitudes for a variety of single multiple defect systems. In particular the solutions of double defects were very interesting as they allow for resonances which resemble the behaviour of unstable particles. Concerning boundaries one should start organizing the existing solutions and seek for more generic (Lie algebraic) structures analogue to what has been said in (vi).