Klein transformation
In quantum [field theory], the Klein transformation is a redefinition of the fields to amend the spin-statistics theorem.
Bose–Einstein
Suppose and are fields such that, if x and y are spacelike-separated points and i and j represent the spinor/tensor indices,Also suppose is invariant under the Z2 parity mapping to but leaving invariant. Free field theories always satisfy this property. Then, the Z2 parity of the number of particles is well defined and is conserved in time. Let's denote this parity by the operator Kχ which maps -even states to itself and -odd states into their negative. Then, Kχ is involutive, Hermitian and unitary.
The fields and above don't have the proper statistics relations for either a boson or a fermion. This means that they are bosonic with respect to themselves but fermionic with respect to each other. Their statistical properties, when viewed on their own, have exactly the same statistics as the Bose–Einstein statistics because:
Define two new fields and as follows:
and
This redefinition is invertible. The spacelike commutation relations become
Fermi–Dirac
Consider the example where.
Assume you have a Z2 conserved parity operator Kχ acting upon χ alone.
Let
and
Then