Four-body problem in <i>d</i>-dimensional space: Ground state, (quasi)-exact-solvability. IV
Bibliographic record
Abstract
Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a 4-body system with equal masses in d-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For d ≥ 3, this describes a 6-dimensional quantum particle moving in a curved space with special d-independent metric in a certain d-dependent singular potential, while for d = 1 it corresponds to a 3-dimensional particle and coincides with the A3 (4-body) rational Calogero model; the case d = 2 is exceptional and is discussed separately. The kinetic energy of the system has a hidden sl(7, R) Lie (Poisson) algebra structure, but for the special case d = 1, it becomes degenerate with hidden algebra sl(4, R). We find an exactly solvable 4-body S4-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasiexactly solvable 4-body sextic polynomial type potential with singular terms. The tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, which lead to exactly solvable models. Their generalization to the n-body system as well as the case of nonequal masses is briefly discussed.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".