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arXiv:1702.06399 · Modern Physics Letters B

Fundamental invariants of many-body Hilbert space

D. K. Sunko

Published Dec 15, 2016• 2 citations
arXivHTMLPDF

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 [Formula: see text] 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.

DOI
10.1142/s021798491630009x

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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 [Formula: see text] 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.
The basic question is addressed of how the space dimension ddd is encoded in the Hilbert space of NNN identical fermions. There appears a finite number $N{!}^{d\ensuremath{-}1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonized into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the NNN-fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.
Corresponding to a finite dimensional Hilbert space HHH with dim⁡H=n\dim H=ndimH=n, we define a geometric algebra G(H)\mathcal{G}(H)G(H) with dim⁡[G(H)]=2n\dim\left[\mathcal{G}(H)\right]=2^ndim[G(H)]=2n. The algebra G(H)\mathcal{G}(H)G(H) is a Hilbert space that contains HHH as a subspace. We interpret the unit vectors of HHH as states of individual fermions of the same type and G(H)\mathcal{G}(H)G(H) as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on G(H)\mathcal{G}(H)G(H) and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on HHH and G(H)\mathcal{G}(H)G(H) are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from HHH to G(H)\mathcal{G}(H)G(H) are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces. Quanta 2025; 14: 48–65.
Many-body Hilbert space has the algebraic structure of a finitely generated free module. All N-body wave functions in d dimensions can be generated by a finite number of N!d − 1 of generators called shapes, with symmetric-function coefficients. Physically the shapes are vacuum states, while the symmetric coefficients are bosonic excitations of these vacua. It is shown here that logical entropy can be used to distinguish fermion shapes by information content, although they are pure states whose usual quantum entropies are zero. The construction is based on the known algebraic structure of fermion shapes. It is presented for the case of N fermions in three dimensions. The background of this result is presented as an introductory review.
From the point of view of functional analysis, wave functions are understood as vector in Hilbert’s space of quantum states. When the number of identical particles is greater than one, there appears a specific algebraic structure, where the same Hilbert space is a graded algebra over the ring of symmetric polynomials (bosonic excitations). Generators of the algebra (vacuums) are known in the Cartesian basis. In this work we translate them to the basis of good angular momentum for the specific case of two particles. Then three generators (Ψ1; Ψ2; Ψ3) make up the ground-state vector and the fourth Ψ4 = Ψ1Ψ2Ψ3 is a pseudoscalar found among doubly excited states, of which there are 28. When all those states are written in the form of bosonic excitations of the vacuums and in the basis of good angular momentum, it is revealed that Ψ4 is a component of a specific excited state of angular momentum l=3 and projection m = ±2. Therefore this state is identified as a band head of excited states with different symmetry then the band which is generated by excitations of the ground state.