# Fundamental invariants of many-body Hilbert space

## TL;DR

This paper explores the algebraic structure of many-body Hilbert space, introducing 'shapes' as fundamental invariants that generalize the Slater determinant to multiple dimensions, offering new insights into complex many-body phenomena.

## Contribution

It introduces the concept of shapes as fundamental invariants of many-body Hilbert space, providing a new algebraic framework for understanding many-body physics.

## Key findings

- Shapes are finite generators of many-body Hilbert space.
- Physical states are superpositions of shapes with symmetric-function coefficients.
- The structure offers insights into the fermion sign problem and spectral bands.

## Abstract

Many-body Hilbert space is a functional vector space with the natural structure of an algebra, in which vector multiplication is ordinary multiplication of wave functions. This algebra is finite-dimensional, with exactly $N!^{d-1}$ generators for $N$ identical particles, bosons or fermions, in $d$ dimensions. The generators are called shapes. Each shape is a possible many-body vacuum. Shapes are natural generalizations of the ground-state Slater determinant to more than one dimension. Physical states, including the ground state, are superpositions of shapes with symmetric-function coefficients, for both bosons and fermions. These symmetric functions may be interpreted as bosonic excitations of the shapes. The algebraic structure of Hilbert space described here provides qualitative insights into long-standing issues of many-body physics, including the fermion sign problem and the microscopic origin of bands in the spectra of finite systems.