Bosonisation

Bosonisation is a theoretical technique in quantum field theory that relates fermionic systems to bosonic systems. It is particularly powerful in one spatial dimension, where a system of interacting fermions can be exactly mapped to a system of free or weakly interacting bosons. This transformation allows for the study of strongly interacting fermionic systems, which are often intractable by other methods, by employing the simpler and more well-understood bosonic framework.

The core idea of bosonisation is that in one dimension, the collective excitations of a fermionic system can be described as waves of density. These density waves behave like bosons. The mathematical machinery of bosonisation involves defining bosonic fields that represent these density fluctuations and then showing that the fermionic operators can be expressed in terms of these bosonic fields. This results in a dramatic simplification of the Hamiltonian describing the system.

Bosonisation has found wide applications in various fields of physics. It is crucial for understanding phenomena in condensed matter physics, such as the Luttinger liquid model, which describes one-dimensional metals. It is also employed in quantum chromodynamics and string theory. The technique provides a unique perspective on the behaviour of quantum systems and offers elegant solutions to complex problems.