(Anti-)Symmetrizing Wave Functions

Abstract: The construction of fully (anti-)symmetric states with many particles, when the single particle state carries multiple quantum numbers, is a problem that seems to have not been systematically addressed in the literature. A quintessential example is the construction of ground state baryon wave functions where the color singlet condition reduces the problem to just two (flavor and spin) quantum numbers. In this paper, we address the general problem by noting that it can be re-interpreted as an eigenvalue equation, and provide a formalism that applies to generic number of particles and generic number of quantum numbers. As an immediate result, we find a complete solution to the two quantum number case, from which the baryon wave function problem with arbitrary number of flavors follows. As a more elaborate illustration that reveals complications not visible in the two quantum number case, we present the complete class of states possible for a system of five fermionic particles with three quantum numbers each. Our formalism makes systematic use of properties of the symmetric group and Young tableaux. Even though our motivations to consider this question have their roots in SYK-like tensor models and holography, the problem and its solution should have broader applications.