Bosonization

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In theoretical physics, one often studies two-dimensional conformal field theory. It has many very special properties. One of them is the equivalence of fermionic elementary fields and bosonic elementary fields. Bosonization also appears in 2D theories which aren't conformal field theories.

The process of replacing fermions by bosons is called bosonization. Two complex fermions \psi,\bar\psi are written as functions of a boson φ

\psi = :\exp(i\phi):,\qquad \bar\psi = :\exp(-i\phi):

while the inverse map is given by

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

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

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