eXact One-Body Approach in Quantum Systems

• eXact One-Body Approach is a theoretical framework that reduces many-body problems to one-body functionals, enabling tractable calculations.
• It leverages explicit mappings, inversion theorems, and effective Hamiltonian formalisms to capture key correlations and dynamics.
• Practical implementations like 1pEx-DFT and RBM mappings enhance energy accuracy across quantum, nuclear, and gravitational applications.

The eXact One-Body Approach (variously abbreviated XOB or 1pEx, depending on context) refers to a broad class of theoretical and computational techniques in many-body quantum physics, quantum chemistry, nuclear structure, and general relativity. The unifying feature of these methods is the exact or nearly exact reduction of many-body or two-body problems to functionals, mappings, or representations in terms of one-body quantities—most notably, the one-body reduced density matrix (1-RDM), one-body density, Green’s function, or a single effective Hamiltonian. In different domains, this approach enables explicit inversion theorems, tractable energy functionals, or effective descriptions that maintain the essential correlations or dynamical information present in the original problem.

1. Foundational Principles and Motivations

The eXact One-Body (XOB) approach leverages the representation of many-body correlation and dynamics at the level of one-body observables or mappings, aiming for mathematical exactness wherever possible. Motivating rationales include:

• Reduction of exponential complexity: Replacing the full $N$-body wavefunction ($\sim$ exponential in system size) by the 1-RDM or Green’s function ($\sim$ polynomial scaling), as in modern 1-RDM functional theory (Benavides-Riveros et al., 2023).
• Constructive mapping: Finding explicit functionals that reconstruct higher-order objects (e.g., two-body densities, total energies) from one-body objects (Gracia-Bondía et al., 2010, El-Sahili et al., 2023).
• Nonperturbative dynamics: Enabling analytic or nearly analytic time evolution and observables that encode full interaction physics, as in XOB methods for gravitational waveforms (Zeng, 20 May 2025, Zeng, 2023).
• Unified framework: Encapsulating both single-particle and correlation effects into universal functionals (e.g., constrained search over 1-RDM) or energy-mapping dictionaries (Damour, 2016).

2. Exact One-Body Functionals in Quantum Many-Body Systems

For systems with a known exact ground state or model interaction, explicit functionals connecting one-body and two-body densities or Wigner functions can be constructed:

• Harmonium (Moshinsky atom): For two interacting fermions in a harmonic trap, the exact two-body Wigner function $W_2$ is expressible as a closed functional $W_2[W_1]$ of the one-body Wigner function (Gracia-Bondía et al., 2010). The key form is a symplectic diagonal decomposition of $W_1$ into Laguerre–Gaussian modes with exact occupation numbers, and a corresponding expansion for $W_2$ via the SLK (Shull–Löwdin–Kutzelnigg) ansatz. The "phase dilemma"—i.e., ambiguity in the coefficients’ phases—is uniquely resolved by energy minimization, yielding alternating sign rules and exact energy recovery.

System	One-body quantity	Exact functional for two-body quantity	Key technical ingredient
Harmonium (2-electron)	$W_1$ (Wigner)	$W_2^{\rm SLK}[W_1]$	Laguerre expansion, sign rule, SLK ansatz

Such closed-form mappings are rare and generally restricted to integrable models (harmonium, helium atom, etc.) with quadratic or special two-body interactions.

3. One-Body Schemes for Spectral and Correlation Energies

Within many-body perturbation theory, the total ground-state energy and correlation spectra are exactly determined by the one-body Green’s function $G$ (if known). Formally,

$$E_0 = -i \lim_{2\to 1^+} \int dx_1[-\,\nabla^2/2 + v_{\rm ext}(x_1)]G(1,2) - \frac{i}{2} \int \Sigma_{xc}(1,3)G(3,1^+) \,dx_1 dx_3,$$

where $\Sigma_{xc}$ is the exchange-correlation self-energy (El-Sahili et al., 2023). In practice, the exact $G$ is unavailable, but if approximate $G$ and self-energy are constructed consistently—e.g., using a test-charge–test-electron (TCTE) form, with $\bar G$ and kernel $f$ ensuring correct densities—then the strictly one-body formalism yields exact total energies, provided input and output objects remain consistent.

This formalism generalizes to energy spectra: the exact removal/addition energies are pole positions of the one-body Green’s function. While two-point vertex corrections (e.g., $f$ kernels from TDDFT) improve quasiparticle energies, complete satellite structure demands higher-order functionals.

4. Exact One-Body Approaches in Effective Hamiltonian and Action Formalisms

A variant of the XOB approach involves mapping two-body or $N$-body Hamiltonians or actions to effective one-body problems whose dynamics or energetics are equivalent, or nearly equivalent, to those of the original systems.

• Gravitational two-body dynamics (post-Minkowskian): At first order in $G$ but all orders in $v/c$, the exact relativistic two-body gravitational interaction is precisely mapped to a geodesic motion in Schwarzschild spacetime via a quadratic energy map ($E_{\rm eff}(E_{\rm real})$), forming the core of the Effective One-Body (EOB) paradigm (Damour, 2016). Scalar–tensor gravitational theories extend this with explicit deformations in the effective potentials (Julié et al., 2017).
• Inspiral, merger, and ring-down in binary black holes: Refined eXact One-Body (XOB) treatments use nonperturbative, synchronous, Schwarzschild-patch–based formulations with exact synchronization and center-of-mass conditions, providing analytical waveforms throughout the full evolution (Zeng, 20 May 2025, Zeng, 2023). Error analysis shows deviations from PN/EOB are negligible on dissipative time scales, with inner-structure factors influencing late-time damping.

Domain	Mapping/Functional	Exactness criteria	Physical prediction
GR two-body binaries	$H_{\rm real} \to H_{\rm eff}$	1PM, all $v/c$ (Damour)	Scattering angle, energy map, QNM bands
Binary merger XOB	Action/Lagrangian	Full GR, almost-exact	Ringdown, symmetry enhancement, late-time QNM

5. Practical Frameworks and Algorithms

Recent methodologies operationalize these theoretical constructs for practical calculations:

• Single-Particle-Exact DFT (1pEx-DFT): An orbital-free density functional theory treating all single-particle energy contributions exactly, with variational degrees of freedom given by participation numbers of the one-body statistical operator (Trappe et al., 2023). The constrained minimization yields Hartree-Fock–level accuracy with transferability across atomic, ionic, and cold-gas systems.
• Restricted Boltzmann Machine (RBM) mappings: Many-body contact interactions in lattice models and EFTs can be written exactly as sums of one-body operators coupled to discrete auxiliary fields, with RBM architectures encoding the exact transformation (e.g., Hirsch transform for Hubbard models) (Rrapaj et al., 2020).
• Shell-model and UMOA treatments: In nuclear structure, introduction of an explicit one-body correlation operator enables exact optimization of the single-particle basis, eliminating spurious parameter dependencies (e.g., HO frequency $\hbar\omega$ in nuclear shell models) (Miyagi et al., 2017).

Approach	Core idea	Technical tool	Sample domain
1pEx-DFT	Exact single-particle energy	Participation number minimization	Atoms, ions, fermionic mesoscopics
RBM methods	One-body decomposition of k-body	Discrete auxiliary field mapping	Quantum lattice, Monte Carlo/Annealing
UMOA+S^{(1)}	Exact one-body correlation	Unitary cluster expansion	Nuclear ab initio structure

6. Limitations, Generalizations, and Outlook

The eXact One-Body approach achieves true exactness only in special settings:

• Model limitations: Analytic inversion or functional construction is generally possible only for two-body, quadratic, or integrable systems; for $N>2$, or non-quadratic interactions, the inversion problem (e.g., the phase dilemma in Wigner functionals) is generically intractable (Gracia-Bondía et al., 2010).
• Approximation regimes: For general many-body problems, approximate one-body functionals, machine-learning–assisted mappings (RBM), or variational constraints can deliver high accuracy but not mathematical exactness.
• Physical regime extensions: In gravitational physics, nearly exact mappings hold in the post-Minkowskian regime (all $v/c$, leading $G$) and in full weak-field contexts. Functional deformations (e.g., scalar–tensor gravity, inner-structure effects) are systematically encoded by exact one-body potential corrections (Julié et al., 2017, Zeng, 2023).

Nevertheless, these methods provide unparalleled insight into the structure of correlation, entanglement, and dynamical response, guiding the development of approximate two-body/one-body functionals and offering direct handles for high-precision many-body computation, waveform prediction, and fundamental tests of quantum and gravitational systems.