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arXiv:2609.22502 · Теоретическая и математическая физика

Spin-Boson Mappings in the Formalism of $f$-Deformations

Vladimir A. Orlov, Liubov A. Markovich, Alexey V. Mikheenkov, Vladimir I. Man'ko

quant-phcond-mat.othermath-phSubmitted Sep 18, 2026• Announced Sep 22, 2026• 0 citations
v1 · Sep 18, 2026v2 · Sep 22, 2026
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Abstract

We develop a unified algebraic approach to spin-boson transformations based on the formalism of $f$-deformed oscillators. In the single-mode case, we show that the Holstein-Primakoff and Dyson-Maleev transformations, together with the interpolating $α$-family, arise as different factorizations of the same algebraically determined object. The standard spin-boson mappings can thus be interpreted as realizations of a common structure, which makes it possible to clearly separate the exact algebraic content on the physical subspace from effects associated with non-Hermiticity, the choice of metric, and extensions beyond the physical subspace. In the two-mode case, the same approach yields both deformed versions of the Jordan-Schwinger transformation and new exact two-mode realizations. Our results provide a unified description of known spin-boson transformations and naturally lead to new bosonic representations.

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21 pages, 1 figure

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Spin-Boson Mappings in the Formalism of fff-Deformations
We develop a unified algebraic approach to spin-boson transformations based on the formalism of fff-deformed oscillators. In the single-mode case, we show that the Holstein-Primakoff and Dyson-Maleev transformations, together with the interpolating ααα-family, arise as different factorizations of the same algebraically determined object. The standard spin-boson mappings can thus be interpreted as realizations of a common structure, which makes it possible to clearly separate the exact algebraic content on the physical subspace from effects associated with non-Hermiticity, the choice of metric, and extensions beyond the physical subspace. In the two-mode case, the same approach yields both deformed versions of the Jordan-Schwinger transformation and new exact two-mode realizations. Our results provide a unified description of known spin-boson transformations and naturally lead to new bosonic representations.
The Jordan-Schwinger map is widely employed to switch between bosonic or fermionic mode operators and spin observables, with numerous applications ranging from quantum field theories of magnetism and ultracold quantum gases to quantum optics. While the construction of observables obeying the algebra of spin operators across multiple modes is straightforward, a mapping between bosonic or fermionic Fock states and spin states has remained elusive beyond the two-mode case. Here, we generalize the Jordan-Schwinger map by algorithmically constructing complete sets of spin states over several bosonic or fermionic modes, allowing one to describe arbitrary multi-mode systems faithfully in terms of spins. As a byproduct, we uncover a deep link between the degeneracy of multi-mode spin states in the bosonic case and Gaussian polynomials. We demonstrate the feasibility of our approach by deriving explicit relations between arbitrary three-mode Fock and spin states, which provide novel interpretations of the genuinely tripartite entangled GHZ and W state classes.
Considering a simple generalization of the (p,q)-deformed boson oscillator algebra, which leads to a two-parameter deformed bosonic algebra in an infinite dimensional subspace of the harmonic oscillator Hilbert space without first finite Fock states, we establish a new harmonic oscillator realization of the deformed boson operators based on the Bogoliubov (p,q)-transformations. We obtain exact expressions for the transformation coefficients and show that they depend on arbitrary functions of p and q which can be interpreted as the parameters of the (p,q)-deformed GL(2,C) group. We also examine the existence and structure of the corresponding deformed Fock-space representation for our problem.
We present a unification of mixed-space quantum representations in Condensed Matter Physics (CMP) and Quantum Field Theory (QFT). The unifying formalism is based on being able to expand any quantum operator, for bosons, fermions, and spin systems, using a universal basis operator Y(u,v) involving mixed Hilbert spaces of P and Q, respectively, where P and Q are momentum and position operators in CMP (which can be considered as a bozonization of free Bloch electrons which incorporates the Pauli exclusion principle and Fermi-Dirac distribution), whereas these are related to the creation and annihilation operators in QFT, where ψ^{†}=-iP and ψ=Q. The expansion coefficient is the Fourier transform of the Wigner quantum distribution function (lattice Weyl transform) otherwise known as the characteristic distribution function. Thus, in principle, fermionization via Jordan-Wigner for spin systems, as well as the Holstein--Primakoff transformation from boson to the spin operators can be performed depending on the ease of the calculations. Unitary transformation on the creation and annihilation operators themselves is also employed, as exemplified by the Bogoliubov transformation. Moreover, whenever Y(u,v) is already expressed in matrix form, M_{ij}, e.g. the Pauli spin matrices, the Jordan--Schwinger transformation is a map to bilinear expressions of creation and annihilation operators which expedites computation of representations. We show that the well-known coherent states formulation of quantum physics is a special case of the present unification. A new formulation of QFT based on Q-distribution of functional-field variables is suggested. The case of nonequilibrium quantum transport physics, which not only involves non-Hermitian operators but also time-reversal symmetry breaking, is discussed in the Appendix.