Fermionic Wave Functions on Unordered Configurations

Abstract

Quantum mechanical wave functions of N identical fermions are usually represented as anti-symmetric functions of ordered configurations. Leinaas and Myrheim proposed how a fermionic wave function can be represented as a function of unordered configurations, which is desirable as the ordering is artificial and unphysical. In this approach, the wave function is a cross-section of a particular Hermitian vector bundle over the configuration space, which we call the fermionic line bundle. Here, we provide a justification for Leinaas and Myrheim's proposal, that is, a justification for regarding cross-sections of the fermionic line bundle as equivalent to anti-symmetric functions of ordered configurations. In fact, we propose a general notion of equivalence of two quantum theories on the same configuration space; it is based on specifying a quantum theory as a triple $(\mathscr{H},H,Q)$ (``quantum triple'') consisting of a Hilbert space $\mathscr{H}$, a Hamiltonian $H$, and a family of position operators (technically, a projection-valued measure on configuration space acting on $\mathscr{H}$).