Spatial Wavefunction Symmetries and Identical Particles

Question

I was reading this and it mentions in the 3-electron section, that for a spacial wave function to be symmetric under fermion swapping, it must be a function of even parity. Similarly for anti-symmetry under fermion swapping, it must be a function of odd parity.

It is not immediately obvious to me why parity symmetry $(-1)^l$ has anything to do with the bosonic or fermionic properties of the spatial wave function. So I suppose my questions are:

Is it true that spatial inversion is the same symmetry as swapping?

And if so, why?

Answer 1

When we talk about exchanging electron $i$ with electron $j$, we are actually changing the wavefunction according to

$$\Psi(..., x_i, ..., x_j, ...) \to \Psi(..., x_j, ..., x_i, ...).$$

The operation is taken by the parity operator $P$. Applying it twice would return the wavefunction to its original form. So the following eigenvalue equation is satisfied

$$ P^2 \Psi = 1 \Psi. $$

Then the $\lambda$ eigenvalue of the $P$ operator ($P\Psi = \lambda \Psi$) also satisfies an analogous equation

$$ \lambda^2 = 1. $$

$\lambda$ can only be either $\pm 1$. If it is positive the wavefunction is symmetric under swapping and if the eigenvalue is negative the wavefunction is antisymmetric under swapping. It can also be said that the symmetric (antisymmetric) wavefunction has even (odd) parity but do not confuse it with spatial symmetry.

It is assumed that fermions must be described by antisymmetric wavefunctions and bosons by symmetric wavefunctions. Many have argumented this fact alluding to the spin-statistics theorem but others have, as well, stated that such justification is not satisfactory. Besides, it is not completely clear why.