Is there a way to check that a wavefunction is antisymmetric?

Question

Is there a general method to check that a given wavefunction is antisymmetric or not? The solution would be simple for cases such as the single Slater determinant. Exchange the wavefunctions of two single particle wavefunction. If they change sign, they are antisymmetric. But what about cases where the wavefunction cannot be expressed as a combination of single particle wavefunctions.

Answer 1

If the wavefunction is provided by Nature then it is guaranteed to be antisymmetric (because that's how we believe Nature to behave).

If you have a (vector) state $\psi$ that, for some reasons, lives in a Hilbert space $H$ that contains the Fermionic Hilbert space $H_F$ as a proper subspace, then it is enough to check that $\psi$ is in the range of the projection onto $H_F$, that is, $P_H\psi = \psi$ (here $H$ and $H_F$ are assumed to be Fock spaces, i.e. $H=\Gamma(K)$ and $H_F=\Gamma_F(K)$, where $K$ is some single-particle Hilbert space, $\Gamma$ is the general Fock space construction and $\Gamma_F$ is the fermionic second-quantisation functor).

A possible reduction comes from the knowledge that $\psi$ describes an $N$-particle system, i.e. $\psi$ is an eigenvector of the Number operator on $H$ with eigenvalue $N$. Then it is enough to show that $\psi$ is in the range of the smaller projection $P_{H_F^N}\subset P_{H^N}$.