# Bosonization

In theoretical condensed matter physics and particle 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. [1] 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.[1] In particle physics, however, the boson is interacting, cf, the Sine-Gordon model, and notably through topological interactions,[2] cf. Wess–Zumino–Witten model.

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.[3] Bosonization is an effective field theory that focuses on low-energy excitations.[4] This is done for Luttinger liquid theory.

Two complex fermions ${\displaystyle \psi ,{\bar {\psi }}}$ are written as functions of a boson ${\displaystyle \phi }$

${\displaystyle {\bar {\psi }}_{-}\psi _{+}=:\exp(i\phi ):,\qquad {\bar {\psi }}_{-}\psi _{+}=:\exp(-i\phi ):}$[5]

while the inverse map is given by

${\displaystyle \partial \phi =:{\bar {\psi }}\psi :}$

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