I have two questions about the $Symmetrization \ Postulate$:

In a system with $N$ identical particles, physical states aren't arbitrary states in $V^{\otimes n}$. Rather, they're totally symmetric (belong to $Sym^NV$), in which case the particles are said to be bosons, or they're totally anti-symmetric (belong to $Anti^NV$), in which case they're said to be fermions. (From MIT OpenCourseWare.)

First, it is said that a postulate is something technically can't be derived. In this case, is it because this postulation gives a statistical definition of bosons and fermions, which is something that more advanced theories based on so that it cannot be proved? Second, I've seen a Venn diagram showing the relations between $Sym^NV$, $Anti^NV$, and the total space $V^{\otimes N}$. When $N = 2$, the union of those two subspaces is just the total space. However, when $N$ is greater or equal to 3, there's something else in $V^{\otimes N}$ which lies in neither $Sym^NV$ nor $Anti^NV$, those states are called partial-symmetry or partial-antisymmetry states, which could be found through Young tableaux. Thus I'm wondering since those partial (or can I replace 'partial' by 'mixed'?) (anti-)symmetry states are not physically realizable, why are we interested in figuring them out, and what's their significance?



The symmetry of Bose wavefunctions or antisymmetry of Fermion wavefunctions under interchange of the particles is something deduced from experiment. It is an observed property of the real world and not something that can be "proved" by any mathmatical argument.

The mixed symmetry cases are also interesting in the real world because they are relevant when we exhange write $\psi= \psi_{\rm spatial}\otimes \psi_{\rm spin}$ and interchange only the labels on the spatial part. This resulting mathematics has much use in atomic spectroscopy.


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