bosonisaatio

Bosonisation is a technique in theoretical physics, particularly in two-dimensional quantum field theories, that relates fermionic fields to bosonic fields. It is a powerful tool that simplifies the analysis of certain quantum systems by allowing a change of variables. In essence, it states that in one spatial dimension (plus one time dimension, making it a 2D spacetime), a theory of interacting fermions can be equivalent to a theory of free or interacting bosons, and vice-versa.

The core idea is that fermionic operators, which obey anti-commutation relations, can be expressed in terms of bosonic fields, which obey commutation relations. This transformation is not a simple one-to-one mapping but involves more complex relationships, often involving exponential functions of the bosonic fields. The equivalence arises because the defining properties of quantum fields in 1+1 dimensions lead to a degeneracy of the vacuum state that can be naturally described by bosonic excitations.

One of the most well-known examples is the equivalence between the Thirring model and the sine-Gordon model. The Thirring model describes interacting fermions, while the sine-Gordon model describes a scalar bosonic field with a cosine potential. Bosonisation provides a way to map the solution of one model to the other. This technique has been instrumental in understanding phenomena such as the fractional quantum Hall effect, the Luttinger liquid model of interacting electrons in one dimension, and certain aspects of string theory. The mathematical underpinnings of bosonisation involve concepts from conformal field theory and operator algebra.