For a quantum-mechanical system of $N$ identical fermions, the $N$ -representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric $N$ -particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, $N$ , is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be $N$ -representable is that its natural spin-orbital occupation numbers be equal in pairs.For a quantum-mechanical system of $N$ identical fermions, the $N$-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric $N$-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, $N$, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be $N$-representable is that its natural spin-orbital occupation numbers be equal in pairs.

Physical ReviewJul 29, 196667 citations

Representation of bosons by fermions

Fritz Bopp

Acta Physica Academiae Scientiarum HungaricaeDec 1, 19720 citations

Product-vector basis and occupation-number basis in Fock space for bosons and fermions

Boris Leaf

A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.

Journal of Mathematical PhysicsJul 1, 19731 citation

Unitary isomorphism of Fock spaces of bosons and fermions arising from a representation of the Cuntz algebra O2

Katsunori Kawamura

Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra O2 induces Fock representations HB and HF of bosons and fermions simultaneously. From this, a unitary operator U from HB to HF is obtained. We show the explicit formula of the action of U on the standard basis of HB. It is shown that U preserves the particle number of HB and HF.Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra O2 induces Fock representations HB and HF of bosons and fermions simultaneously. From this, a unitary operator U from HB to HF is obtained. We show the explicit formula of the action of U on the standard basis of HB. It is shown that U preserves the particle number of HB and HF.

arXiv:0807.2702Journal of Mathematical PhysicsOct 1, 20084 citations

Conservation of the number of fermions

Yoshio Ohnuki, S. Kamefuchi

Nuclear Physics BMar 1, 19695 citations

For a quantum-mechanical system of

N

identical fermions, the

N

-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric

N

-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles,

N

, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be

N

-representable is that its natural spin-orbital occupation numbers be equal in pairs.