Many-fermion wave functions: structure and examples

TL;DR

This paper explores the algebraic and geometric structure of many-fermion wave functions, introducing shapes as fundamental building blocks and using algebraic geometry to understand correlation effects and generate wave functions.

Contribution

It presents a novel algebraic geometric framework for understanding many-fermion wave functions through shapes and their excitations, with algorithms for generating these shapes.

Findings

Shapes form a finite generating set for many-fermion Hilbert space.

Wave functions can be constructed from shapes and symmetric functions.

The framework allows algorithmic generation of complex many-body states.

Abstract

Many-fermion Hilbert space has the algebraic structure of a free module generated by a finite number of antisymmetric functions called shapes. Physically, each shape is a many-body vacuum, whose excitations are described by symmetric functions (bosons). The infinity of bosonic excitations accounts for the infinity of Hilbert space, while all shapes can be generated algorithmically in closed form. The shapes are geometric objects in wave-function space, such that any given many-body vacuum is their intersection. Correlation effects in laboratory space are geometric constraints in wave-function space. Algebraic geometry is the natural mathematical framework for the particle picture of quantum mechanics. Simple examples of this scheme are given, and the current state of the art in generating shapes is described from the viewpoint of treating very large function spaces.