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The QM texts I have gloss over this.

A typical approach is to define the symmetric and anti-symmetric kets that serve as a basis for the eigenspace containing ##|ω_1,ω_2⟩## and ##|ω_2, ω_1⟩## where ##ω_1## and ##ω_2## are eigenvalues of an operator Ω corresponding to a measurement that can be made on either of the identical particles, and then say that the system state after a measurement of Ω for both particles must be either one of the symmetric or antisymmetric kets, according to whether the particles are bosons or fermions.

But they don’t say anything about how, if at all, the state space that is subject to this adjustment differs from the raw direct product space.

My guess is that, if there are two identical particles then the state space ##\mathscr{V}## is a quotient space of the unrestricted state space ##\mathscr{V}_0## that is obtained as the direct product of the state spaces for the individual particles, but quotient over what subspace?

Again I have a guess. That is that we take the quotient of ##\mathscr{V}_0## over the subspace ##\mathscr{S}## of ##\mathscr{V}_0## generated by ##\cup_{\omega_1,\omega_2\in Eig(\Omega)} \mathscr{G}_{\omega_1 +\omega_2}## where ##\mathscr{G}_{\omega_1 +\omega_2}## is the two-dimensional eigenspace of the ##\mathscr{V}_0##-operator ##\Omega^{(2)}## that measures ##\Omega## for both particles, corresponding to eigenvalue ##(\omega_1 +\omega_2)## of ##\Omega^{(2)}##, and ##Eig(\Omega)## is the set of eigenvalues of ##\Omega##.

However I have a feeling that this quotient may remove too much from the state space, not leaving enough distinguishable states.

I would like to work out what, if anything is the correct quotient space, and then (1) work out how the quotient changes if the indistinguishable particles change between being all bosons and being all fermions and (2) extend it to the case of more than two indistinguishable particles.

Does anybody have any suggestions? Am I completely on the wrong track? I've googled things like 'indistinguishable particles quotient space' with no luck. Is the space still a Hilbert Space once it contains indistinguishable particles?

Thanks for any suggestions.