Klein transformation


In quantum field theory, the Klein transformation is a redefinition of the fields to amend the spin-statistics theorem.

Bose–Einstein

Suppose φ and χ are fields such that, if x and y are spacelike-separated points and i and j represent the spinor/tensor indices,
Also suppose χ is invariant under the Z2 parity mapping χ to −χ but leaving φ invariant. Obviously, free field theories always satisfy this property. Then, the Z2 parity of the number of χ particles is well defined and is conserved in time. Let's denote this parity by the operator Kχ which maps χ-even states to itself and χ-odd states into their negative. Then, Kχ is involutive, Hermitian and unitary.
Needless to say, the fields φ and χ above don't have the proper statistics relations for either a boson or a fermion. i.e. they are bosonic with respect to themselves but fermionic with respect to each other. But if you look at the statistical properties alone, we find it has exactly the same statistics as the Bose–Einstein statistics. Here's why:
Define two new fields φ' and χ' as follows:
and
This redefinition is invertible. Now, the spacelike commutation relations become

Fermi–Dirac

Now, let's work with the example where
.
Assume once again we have a Z2 conserved parity operator Kχ acting upon χ alone.
Let
and
Then

More than two fields

But what if we have more than two fields? In that case, we can keep on applying the Klein transformation to each pair of fields with the "wrong" commutation/anticommutation relations until we're done.
This explains the equivalence between parastatistics and the more familiar Bose–Einstein/Fermi–Dirac statistics.