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In quantum field theory, the Klein transformation is a redefinition of the fields to patch up 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 (nothing to do with spatial reflections!) 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 (even though the number of χ particles itself depends on the choice of which splitting into a free Hamiltonian and an interacting Hamiltonian we make in the interaction picture, which doesn't even exist for interacting theories (the number is typically infinite)). 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 (because Kχ is). Now, the spacelike commutation relations become : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Klein transformation」の詳細全文を読む スポンサード リンク
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