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 $\phi$

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

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.

See also

• Luttinger Liquid