Editing Klein transformation (section)

==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,

:<math>[\varphi_i(x),\varphi_j(y)]=[\chi_i(x),\chi_j(y)]=\{\varphi_i(x),\chi_j(y)\}=0.</math>

Also suppose χ is invariant under the '''Z'''<sub>2</sub> parity (nothing to do with spatial reflections!) mapping χ to &minus;χ but leaving φ invariant. Free field theories always satisfy this property. Then, the '''Z'''<sub>2</sub> parity of the number of χ particles is well defined and is conserved in time. Let's denote this parity by the operator K<sub>χ</sub> which maps χ-even states to itself and χ-odd states into their negative. Then, K<sub>χ</sub> is [[Involution (mathematics)|involutive]], [[Hermitian]] and [[unitary operator|unitary]].

The fields φ and χ 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 φ' and χ' as follows:

:<math>\varphi'=iK_{\chi}\varphi\,</math>

and

:<math>\chi'=K_{\chi}\chi.\,</math>

This redefinition is invertible (because K<sub>χ</sub> is). The spacelike commutation relations become

:<math>[\varphi'_i(x),\varphi'_j(y)]=[\chi'_i(x),\chi'_j(y)]=[\varphi'_i(x),\chi'_j(y)]=0.\,</math>