# 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.

## References

1. ^ a b Gogolin, Alexander O. (2004). Bosonization and Strongly Correlated Systems. Cambridge University Press. ISBN 0-521-61719-7.
2. ^ Coleman, S. (1975). "Quantum sine-Gordon equation as the massive Thirring model" Physical Review D11 2088; Witten, E. (1984). "Non-abelian bosonization in two dimensions", Communications in Mathematical Physics 92 455-472. online
3. ^ Sénéchal, David (1999). "An Introduction to Bosonization". Theoretical Methods for Strongly Correlated Electrons. CRM Series in Mathematical Physics. doi:10.1007/0-387-21717-7_4.
4. ^ Sohn, Lydia (ed.) (1997). Mesoscopic electron transport. Springer. arXiv:. ISBN 0-7923-4737-4.
5. ^ In actuality, there is a cocycle prefactor to give correct (anti-)commutation relations with other fields under consideration.