The boson-fermion correspondence: Formulation, proof and application to Jack polynomials

Abstract

We have presented a proof of an algebraic version of the boson-fermion correspondence and used this correspondence to find an orthonormal basis in bosonic Fock space, namely, the Schur polynomials. We have expressed some Jack polynomials, which depend on a positive parameter \alpha, in this basis in order to find the elements in fermionic Fock space corresponding to these polynomials. We have concluded that the first few Jack polynomials are decomposable in F if and only if \alpha = 1, for which they correspond to the Schur polynomials. Lastly, we have transferred the inner product of the Jack polynomials to bosonic Fock space. In the future, investigation of a general proof of decomposability of the Jack polynomials and a general definition of these polynomials in terms of Schur polynomials might be interesting. Suggestions have been given for both of these potential research interests.