Editing Indistinguishable particles (section)

=== Operator approach and parastatistics ===

The Hilbert space for <math>n</math> particles is given by the tensor product <math display="inline"> \bigotimes_n H </math>. The permutation group of <math> S_n </math> acts on this space by permuting the entries. By definition the expectation values for an observable <math>a</math> of <math>n</math> indistinguishable particles should be invariant under these permutation. This means that for all <math> \psi \in H </math> and <math> \sigma \in S_n </math>
: <math> (\sigma \Psi )^t a  (\sigma \Psi)   = \Psi^t a \Psi,</math>
or equivalently for each <math> \sigma \in S_n </math>
: <math> \sigma^t a \sigma = a </math>.
Two states are equivalent whenever their expectation values coincide for all observables. If we restrict to observables of <math>n </math> identical particles, and hence observables satisfying the equation above, we find that the following states (after normalization) are equivalent
: <math> \Psi \sim  \sum_{\sigma \in S_n} \lambda_{\sigma} \sigma \Psi</math>.
The equivalence classes are in [[bijective relation]] with irreducible subspaces of <math display="inline"> \bigotimes_n H </math> under <math> S_n </math>.

Two obvious irreducible subspaces are the one dimensional symmetric/bosonic subspace and anti-symmetric/fermionic subspace. There are however more types of irreducible subspaces. States associated with these other irreducible subspaces are called [[Parastatistics|parastatistic states]].<ref>{{Cite journal|last=Bach|first=Alexaner|date=1993|title=Classification of Indistinguishable Particles|journal=[[Europhysics Letters]]|volume=21|issue=5|pages=515–520|doi=10.1209/0295-5075/21/5/002|bibcode=1993EL.....21..515B|s2cid=250835341 }}</ref> [[Young tableau#Applications in representation theory|Young tableaux]] provide a way to classify all of these irreducible subspaces.