In theoretical condensed matter physics and quantum field theory, 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 C. 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 section may be confusing or unclear to readers. (August 2020)

A pair of chiral fermions ${\displaystyle \psi _{+},{\bar {\psi }}_{+}}$, one being the conjugate variable of the other, can be described in terms of a chiral boson ${\displaystyle \phi }$

$${\displaystyle \psi _{+}=:\exp \left(+i\int _{\infty }^{z}\partial _{++}\phi \right):,\qquad {\bar {\psi }}_{+}=:\exp \left(-i\int _{\infty }^{z}\partial _{++}\phi \right):}$$

where the currents of these two models are related by

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

where composite operators must be defined by a regularization and a subsequent renormalization.

• Holstein–Primakoff transformation