##### Wiktionary

**bosonization**

n. The act, process or result of bosonize.

##### Wikipedia

**Bosonization**

In theoretical condensed matter physics, **Bosonization** is a mathematical procedure by which a system of interacting fermions in (1+1) dimensions can be transformed to a system of massless, non-interacting bosons. The method of bosonization was conceived independently by particle physicists Sidney Coleman and Stanley Mandelstam; and condensed matter physicists Daniel Mattis and Alan Luther in 1975.

The basic physical idea behind bosonization is that particle-hole excitations are bosonic in character. However, it was shown by Tomonaga in 1950 that this principle is only valid in one-dimensional systems. Bosonization is an effective field theory that focuses on low-energy excitations. This is done for Luttinger liquid theory.

Two complex fermions $\psi,\bar\psi$ are written as functions of a boson *ϕ*

$$\bar\psi_-\psi_+ = :\exp(i\phi):,\qquad \bar\psi_-\psi_+ = :\exp(-i\phi):$$

while the inverse map is given by

$$\partial\phi=:\bar\psi\psi:$$

All equations are normal-ordered. The changed statistics arises from anomalous dimensions of the fields.