Fundamental invariants of many-body Hilbert space

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.Comment: Invited review, author's version as accepted by Modern Physics Letters

Fundamental Building Blocks of Strongly Correlated Wave Functions

“…Once the ground state is selected, perhaps as a superposition of the vacua, the remaining shapes may still make their presence felt as bandheads of higher-energy excitation bands, such as are ubiquitous in the spectra of finite systems. In this way their "exceptionalism" persists, giving them a special role in the excitation spectrum, even if some other state is the ground state [17].…”

“…Most pragmatically, one can regard the algorithm as just another way to obtain shapes, more practical than the other known [16] one, but in any case a means to an end. With the shapes in hand, the really interesting insight is to represent physical states as a free module (17), rather than a vector space. This is in some sense the furthest one can take Heisenberg's matrix mechanics.…”

“…Here it is described in detail for the particular case of three particles in three dimensions. An introductory review of the shape formalism can be found elsewhere [17].…”

Generic example of algebraic bosonisation

“…where P , Q, R, S are functions of squares (t 2 , u 2 , v 2 ) only (noting that P = tu is the same as Q = t 2 ). One recognizes the constraint (1) in the basis (10), obtained by inspection here. Therefore, excitations of RM must be expressed in discriminants,…”

“…It has recently become apparent [9,10] that many-body Hilbert space is a finitely generated free module, such that any antisymmetric wave function of N particles may be a Email: dks@phy.hr written…”