Playing with Numbers, with Fermions and Bosons

Abstract

We construct nonlinear maps which realize the fermionization of bosons and the bosonization of fermions with the view of obtaining states coding naturally integers in standard or in binary basis. Specifically, with reference to spin 1 2 systems, we derive raising and lowering bosonic operators in terms of standard fermionic operators and vice versa. The crucial role of multiboson operators in the whole construction is emphasized.

Natural Numbers and Quantum States in Fock Space

“…[5] for spin Omitting computational details, one finds that S (ℓ) = 1 √γ ℓ + 1 a ℓ satisfies both conditions (14); a ℓ andγ ℓ denote the annihilation and the number operators of the (ℓ+1)-th slot, a ℓ |γ ℓ = √ γ ℓ |γ ℓ −1 ,γ ℓ |γ ℓ = γ ℓ |γ ℓ . However, the raising slot operator S † (ℓ) verifies only the first of conditions (15) while the second one needs somehow to be forced .…”

“…One derives the expression of S (ℓ) extending the formulation reported in Ref. [5] for spin 1 2 systems to general spin j systems. Specifically, S (ℓ) and S † (ℓ) are expressed in terms of the spin j representations of the generators J − and J + = J † − of the su(2) algebra {J + , J − , J z }, [J + , J − ] = 2J z , [J z , J ± ] = ±J ± , the standard basis |j, m , −j ≤ m ≤ j, being replaced with the number basis through the Holstein-Primakoff representation of su(2) [10].…”

“…An example of explicit realization is reported in Ref. [5] where, with reference to spin 1 2 systems corresponding to the binary base, fermionized translation operators are constructed, which satisfy the anticommutation relations through appropriate phase factor operators.…”

“…Both are based on the one-to-one correspondence between natural numbers and number states in the Fock space, though their realizations are distinct: indeed, one of them entails the definition of specific multiboson operators, while the other one utilizes translation operators which, unlike those reported in Ref. [5], are not endowed with a definite algebraic structure.…”