Fourier analysis of many-body transition amplitudes and states

We decompose the counting statistics of many-body interference experiments into contributions associated with distinct irreducible exchange symmetries. To do so, we perform a Fourier transform over the symmetric group $S_N$ on the collection of $N!$ many-body transition amplitudes connecting two states of a system of $N$ particles. We apply our formalism to the interference of partially distinguishable bosons and fermions and describe mechanisms responsible for completely destructive interference in many-body systems obeying specific exchange symmetries, including, but not limited to, bosons and fermions.

💡 Research Summary

The paper presents a comprehensive framework for analyzing many‑body interference by applying a Fourier transform over the symmetric group Sₙ to the full set of N! transition amplitudes that connect two N‑particle states. The authors begin by recalling the familiar role of the Fourier transform for Abelian groups—expansion in eigenfunctions of commuting shift operators—and then extend the concept to non‑Abelian finite groups. In this non‑Abelian setting the transform is defined with respect to the irreducible representations (irreps) of the group; it maps convolution of functions on the group to ordinary matrix multiplication and preserves inner products via the Parseval–Plancherel identity.

The physical setting considered is a system of N identical particles (bosons, fermions, or partially distinguishable particles) each occupying an M‑dimensional single‑particle Hilbert space. The many‑body evolution is a non‑interacting unitary U acting identically on each particle, i.e. U^{⊗N}. For two product basis states |i⟩ and |o⟩ the naïve transition amplitude factorizes as a product of single‑particle matrix elements. However, because identical particles are indistinguishable, one must symmetrize (or antisymmetrize) the states. The authors introduce general symmetrization operators \