Pregeometry (model theory)

2014-01-23T16:31:56Z

138.51.209.11: /* Examples */

{{Unreferenced|date=December 2009}}
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,

:<math>[\phi^i(x),\phi^j(y)]=[\chi^i(x),\chi^j(y)]=\{\phi^i(x),\chi^j(y)\}=0.</math>

Also suppose χ is invariant under the '''Z'''2 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 '''Z'''2 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 [[Involution (mathematics)|involutive]], [[Hermitian]] and [[unitary operator|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:

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

and

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

This redefinition is invertible (because Kχ is). Now, the spacelike commutation relations become

:<math>[\phi'^i(x),\phi'^j(y)]=[\chi'^i(x),\chi'^j(y)]=[\phi'^i(x),\chi'^j(y)]=0.\,</math>

==Fermi–Dirac==
Now, let's work with the example where

:<math>\{\phi^i(x),\phi^j(y)\}=\{\chi^i(x),\chi^j(y)\}=[\phi^i(x),\chi^j(y)]=0</math>

(spacelike-separated as usual).

Assume once again we have a '''Z'''2 conserved parity operator Kχ acting upon χ alone.

Let

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

and

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

Then

:<math>\{\phi'^i(x),\phi'^j(y)\}=\{\chi'^i(x),\chi'^j(y)\}=\{\phi'^i(x),\chi'^j(y)\}=0.</math>

==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 statistics|Bose–Einstein]]/[[Fermi–Dirac statistics]].

{{DEFAULTSORT:Klein Transformation}}
[[Category:Quantum field theory]]

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Pregeometry (model theory)