Geometric algebra as the input language of collider foundation models
Abstract
A hard hadron-collider event is represented here as a single geometric object: the kinematics and object-type labels of all reconstructed final-state particles in one multivector evMV, one Cl(1,3)⊗Vflav slot per object, whose higher grades the algebra generates from the measured four-momenta --- rather than as a list of four-momenta with label fields attached. The setting is geometric algebra, whose grade decomposition organises essentially every observable in current use: invariants at grade zero, four-momenta at grade one, decay-plane bivectors at grade two, oriented three-volumes at grade three, the CP-odd pseudoscalar at grade four. The high-level invariants, the low-level recipe and the equivariant-network inputs are recovered as projections onto specific grades. A per-grade dictionary of $32$ classical observables and an inventory of the symmetries acting on evMV are provided. Theorem settles which Lorentz invariants the higher grades unlock: none beyond {pi · pj, mi2}, the CP-odd sign of the pseudoscalar being the one genuine channel. The representation is intended as a uniform input layer for foundation models of collider physics, and the realisations demonstrated here occupy two rungs of a ladder: the higher grades are formed inside the network from the per-object slots, or selected grade-two and grade-three objects are supplied at the input layer. It is demonstrated on the resonance-topology separation of pp → tWb with a Lorentz-equivariant multivector transformer of L-GATr type. An ablation over eight input representations separates what supplying algebraic content explicitly can and cannot buy: content reconstructible from the measured momenta unlocks no new invariant, by the theorem, whereas a fixed reference blade encoding the beam plane carries structure the momenta do not contain.
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BibTeX
@misc{abasov2026geometric,
title = {{Geometric algebra as the input language of collider foundation models}},
author = {E. Abasov and L. Dudko and F. Grigoryev and P. Volkov and A. Zaborenko},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.15910},
eprint = {2605.15910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15910},
}
RIS
TY - UNPB
TI - Geometric algebra as the input language of collider foundation models
AU - E. Abasov
AU - L. Dudko
AU - F. Grigoryev
AU - P. Volkov
AU - A. Zaborenko
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2605.15910
UR - https://arxiv.org/abs/2605.15910
AB - A hard hadron-collider event is represented here as a single geometric object: the kinematics and object-type labels of all reconstructed final-state particles in one multivector evMV, one Cl(1,3)⊗Vflav slot per object, whose higher grades the algebra generates from the measured four-momenta --- rather than as a list of four-momenta with label fields attached. The setting is geometric algebra, whose grade decomposition organises essentially every observable in current use: invariants at grade zero, four-momenta at grade one, decay-plane bivectors at grade two, oriented three-volumes at grade three, the CP-odd pseudoscalar at grade four. The high-level invariants, the low-level recipe and the equivariant-network inputs are recovered as projections onto specific grades. A per-grade dictionary of $32$ classical observables and an inventory of the symmetries acting on evMV are provided. Theorem settles which Lorentz invariants the higher grades unlock: none beyond {p_i · p_j, m_i^2}, the CP-odd sign of the pseudoscalar being the one genuine channel. The representation is intended as a uniform input layer for foundation models of collider physics, and the realisations demonstrated here occupy two rungs of a ladder: the higher grades are formed inside the network from the per-object slots, or selected grade-two and grade-three objects are supplied at the input layer. It is demonstrated on the resonance-topology separation of pp → tWb with a Lorentz-equivariant multivector transformer of L-GATr type. An ablation over eight input representations separates what supplying algebraic content explicitly can and cannot buy: content reconstructible from the measured momenta unlocks no new invariant, by the theorem, whereas a fixed reference blade encoding the beam plane carries structure the momenta do not contain.
ER -